Noncommutative Riesz Transforms— Dimension Free Bounds and Fourier Multipliers

J. Eur. Math. Soc. 20, 529–595 c European Mathematical Society 2018 DOI 10.4171/JEMS/773

Marius Junge Tao Mei Javier Parcet · · Noncommutative Riesz transforms— dimension free bounds and Fourier multipliers

Received April 12, 2015

Abstract. We obtain dimension free estimates for noncommutative Riesz transforms associated to conditionally negative length functions on group von Neumann algebras. This includes Poisson semigroups, beyond Bakry’s results in the commutative setting. Our proof is inspired by Pisier’s method and a new Khintchine inequality for crossed products. New estimates include Riesz trans- n forms associated to fractional laplacians in R (where Meyer’s conjecture fails) or to the word length of free groups. Lust-Piquard’s work for discrete laplacians on LCA groups is also generalized in several ways. In the context of Fourier multipliers, we will prove that Hormander–Mikhlin¨ multipliers are Littlewood-Paley averages of our Riesz transforms. This is highly surprising in the Eu- clidean and (most notably) noncommutative settings. As application we provide new Sobolev/Besov type smoothness conditions. The Sobolev-type condition we give reﬁnes the classical one and yields dimension free constants. Our results hold for arbitrary unimodular groups.

Keywords. Riesz transform, group von Neumann algebra, dimension free estimates, Hormander–¨ Mikhlin multiplier

Contents

Introduction ...... 530 1. Riesz transforms ...... 540 1.1. Khintchine inequalities ...... 540 1.2. Riesz transforms in Meyer form ...... 543 1.3. Riesz transforms in cocycle form ...... 547 1.4. Examples, commutative or not ...... 549 2. Hormander–Mikhlin¨ multipliers ...... 554 2.1. Littlewood–Paley estimates ...... 554 2.2. A reﬁned Sobolev condition ...... 558 2.3. A dimension free formulation ...... 564 2.4. Limiting Besov-type conditions ...... 566

M. Junge: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green St. Urbana, IL 61891, USA; e-mail: [email protected] T. Mei: Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, MI 48202, USA; e-mail: [email protected] J. Parcet: Instituto de Ciencias Matematicas,´ CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Cient´ıﬁcas, C/Nicolas´ Cabrera 13-15, 28049, Madrid, Spain; e-mail: [email protected] Mathematics Subject Classiﬁcation (2010): Primary 43A50, 42B15; Secondary 46L52 530 Marius Junge et al.

3. Analysis in free group branches ...... 569 Appendix A. Unimodular groups ...... 572 Appendix B. Operator algebraic tools ...... 579 Appendix C. A geometric perspective ...... 585 Appendix D. Meyer’s problem for Poisson ...... 590 References ...... 592

Introduction

1/2 The classical Riesz transforms Rj f @j ( 1) f are higher-dimensional forms of = the Hilbert transform in R. Dimension free estimates for the associated square functions f ( 1) 1/2f were first proved by Gundy/Varopoulos [28] and shortly after by R = |r | Stein [71], who pointed out the significance of a dimension free formulation of Euclidean harmonic analysis. The aim of this paper is to provide dimension free estimates for a much broader class of Riesz transforms and apply them for further insight in Fourier multiplier Lp-theory. Our approach is surprisingly simple and it is valid in the general context of group von Neumann algebras. A relevant generalization appeared in the work of P. A. Meyer [50], continued by Bakry, Gundy and Pisier [3, 4, 27, 55] among others. The probabilistic approach consists in replacing 1 by the infinitesimal generator A of a nice semigroup acting on a probability space (, µ). The gradient form f1, f2 is also replaced by the so-called “carre´ 1 hr r i du champs” 0A(f ,f ) (A(f )f f A(f ) A(f f )), and Meyer’s problem for 1 2 = 2 1 2 + 1 2 1 2 (, µ, A) consists in determining whether 1/2 1/2 0A(f, f ) p c(p) A f p (1 0. We write A c B when max M,1/ c. Meyer proved ⇠ { } this for the Ornstein–Uhlenbeck semigroup, while Bakry considered other diffusion semigroups assuming the 02 0 condition, which yields in turn a lower bound for the Ricci curvature in Riemannian manifolds [4, 43]. Clifford algebras were considered by Lust- Piquard [46, 47], and other topics concerning optimal linear estimates can be found in [12, 22] and the references therein. Further dimension free estimates for maximal functions appear in [6, 10, 51, 70]. n Contrary to what might be expected, (MP) fails for the Poisson semigroup in R when 1

convolution processes have not been studied sys- tematically. Lust-Piquard’s theorem on discrete laplacians for LCA groups [48] seems to be the only exception. Our first goal is to fill this gap and study Meyer’s problem for Markov convolution semigroups in the Euclidean case and other group algebras. In this paper, we introduce a new form of (MP) which holds in much larger generality. As we shall see, this requires following the tradition of noncommutative Khintchine inequalities [49, 60] which involves considering an infimum over decompositions into two terms when Riesz transforms and Fourier multipliers 531 p<2. In the terminology of noncommutative geometry, our decomposition takes place in the space of differential forms of order 1 (see Appendix, Lemma C1). Indeed, the deeper understanding of derivations in noncommutative analysis provides a better understanding of Riesz transforms, even for classical semigroups of convolution type. Let us first consider a simple model. Given a discrete abelian group G, let (, µ) = (G,⌫) be the compact dual group with its normalized Haar measure and construct the group characters g T. By Schoenberg’s theorem [64] a given convolution semi- : !t (g) groupb ,t g e g is Markovian in iff (e) 0 for the identity e, S : 1 7! 1 = (g) (g ), and ag 0 agah (g h) 0. Any such function = g = ) g,h  is called a conditionally negative length. A (g) (g)g is the generator, which de- P P = termines the gradient form 0 . Does (MP) or a generalization of it hold for arbitrary pairs (G, )? To answer this question we first widen the scope of the problem and consider its formulation for nonabelian discrete groups G. The roleˆ of L (, µ) is now played by the group von Neumann algebra (G), widely studied in noncommutative1 geometry and L operator algebras [9, 18, 20]. Let G be a discrete group with left regular representation G (` (G)) given : ! B 2 by (g)h gh, where the g’s form the unit vector basis of ` (G). Write (G) for = 2 L its group von Neumann algebra, the weak operator closure of the linear span of (G) in (`2(G)). Consider the standard trace ⌧((g)) g e where e denotes the identity of G. B = = Any f (G) has a Fourier series expansion 2 L f (g)(g) with ⌧(f ) f (e). = g G X2 b b Define the Lp space over the noncommutative measure space ( (G), ⌧) as L p 1/p Lp(G) Lp( (G), ⌧) closure of (G) in the norm f (⌧ f ) . = L ⌘ L k kLp(G) = | | In general,b the (unbounded) operator f p is obtained from functionalb calculus on the | | Hilbert space `2(G) (see [60] or Appendix B for further details). It turns out that

Lp(G) Lp(G) Lp(, µ) ' = for abelian G. Indeed, the map (g)b ( (Gb), ⌧) g L (G,⌫)extends to a trace 2 L ! 2 1 preserving -homomorphism, hence to an Lp isometry for p 1. This means that we can ⇤ identify Fourier series in both spaces sending (g) to the group characterb g and

f (g)(g) f (g)g . Lp(G) = Lp(G) g G g G X2 X2 b b b b Harmonic analysis on (G) places the group on the frequency side. This approach is L partly inspired by the remarkable results of Cowling/Haagerup [19, 29] on approximation properties and Fourier multipliers on group algebras. This paper is part of an effort [34, 35, 53] to extend modern harmonic analysis to the unexplored context of group von 532 Marius Junge et al.

Neumann algebras. Markovian semigroups acting on (G) are composed of self-adjoint, L completely positive and unital maps. Schoenberg’s theorem is still valid and

t (g) ,t f e f (g)(g) S = g G X2 b will be Markovian if and only if G R is a conditionally negative length. : ! + 1/2 Riesz transforms should look like R ,j f @ ,j A f where the laplacian is re- = placed by A ((g)) (g)(g) and @ ,j is a certain differential operator playing the = n role of a directional derivative. Unlike for R , there is no standard differential structure for an arbitrary discrete G. The additional structure comes from the length , which allows a broader interpretation of tangent space in terms of the associated cocycle. Namely, conditionally negative lengths are in one-to-one correspondence with affine representations ( ,↵ ,b ), where ↵ G O( ) is an orthogonal representation over a H : ! H real Hilbert space and b G is a mapping satisfying the cocycle law (see H : ! H Appendix B for further details) b (gh) ↵ (b (h)) b (g) b (g) 2 (g). ,g and = + k kH = Since @j (exp(2⇡i x, )) 2⇡ixj exp(2⇡i x, ), it is natural to define h ·i = h ·i b (g), ej 1/2 R ,j f @ ,j A f 2⇡i h iH f (g)(g) = = p (g) g G X2 b for some orthonormal basis (ej )j 1 of . Recalling that b (g)/p (g) is always a H normalized vector, we recover the usual symbol of Rj as a Fourier multiplier. Note also that classical Riesz transforms can be seen from this viewpoint. Namely, de Leeuw’s n theorem [21] allows us to replace R by its Bohr compactification, whose Lp spaces come n n from the group von Neumann algebra (R ) of R equipped with the discrete topology. L disc Then the classical Riesz transforms arise from the standard cocycle where (⇠) ⇠ 2 n =| | (generating the heat semigroup) and R with the trivial action ↵ and the identity n H = map b on R . Moreover, the classical Riesz transforms vanish on the Lp-functions fixed by the heat semigroup: the constant functions on the n-torus and the zero function on the Euclidean space. This is also the case here and R ,j will be properly defined on

L (G) f Lp(G) f (g) 0 whenever b (g) 0 . p ={ 2 | = = } An elementary calculationb shows thatb b 1/2 1/2 2 0 (A f, A f) R ,j f . = | | j 1 X By Khintchine’s inequality, (MP) in the commutative setting is equivalent to

R f f j ,j c(p) Lp(G) Lp( G) ⇠ k k j 1 ⇥ X b b Riesz transforms and Fourier multipliers 533 for any family (j )j 1 of centered independent gaussians in . We have pointed out that this fails when A ( 1)1/2 is the generator of the Poisson semigroup. According to the = standard gaussian measure space construction (see below), we may construct a canonical action G y L () determined by . As we shall justify in this paper, a natural : 1 version of Meyer’s problem (MP) is to ask whether R f f . j o ,j c(p) Lp(G) Lp(L ()oG) ⇠ k k j 1 1 X b Our ﬁrst result claims that this form of (MP) holdsb for all Markov convolution semigroups on group von Neumann algebras, including the Poisson semigroup in the Euclidean space. The Riesz transforms above were introduced in [34] under the additional assumption that dim < . Our dimension free estimates below—in cocycle form, see Theorem H 1 A2 for a Meyer-type formulation—allow one to consider Riesz transforms associated to inﬁnite-dimensional cocycles. Theorem A1. Let G be a discrete group, f L (G) and 1

n (a) Fractional laplacians in R . Recall that Theorem A1 also holds for group algebras n over arbitrary unimodular groups. In the particular case G R we may consider = conditionally negative lengths of the form

(⇠) 2 (1 cos(2⇡ x,⇠ )) dµ (x) = n h i ZR n for a positive Borel measure µ satisfying (⇠) < for all ⇠ R . If dµ (x) n 2 1 22 = dx/ x for any 0 <<1, we get (⇠) kn() ⇠ . This will provide | | + = | | us dimension free estimates for Riesz transforms associated to fractional laplacians, which are new. The estimates predicted by Meyer fail for 1/2 (see Appendix D). = In contrast to the case 1, we ﬁnd highly nontrivial cocycles. The vast family of = measures µ are explored in further generality in the second part of this paper. (b) Discrete laplacians in LCA groups. Let 00 be a locally compact abelian group and s 0 be torsion free. If @j f ( ) f ( ) f ( ,...,s j ,...,n) stand for dis- 0 2 0 = 1 0 crete directional derivatives in 0 00 00, we may consider the laplacian 1/2= ⇥···⇥ L @ @j and Rj @j L . Lust-Piquard [48] provided dimension free esti- = j j⇤ = mates for these Riesz transforms. If we set j (0,...,0,s0, 0,...,0) with s0 in P = the j-th entry, consider the sum of point-masses µ . Then we shall recover = j j Lust-Piquard’s theorem via Theorem A1 taking P

(g) L(g) (2 g 1 )( ) dµ ( ) for g G 0. = = g 2 = Z0 The advantage isb that we do not need to require s0 to be torsion free. Moreover,b our formulation holds for any ﬁnite sum of point-masses, so that we may allow the shift s0 to depend on the entry j or even the group 0 not to be given in a direct product form. . . This solves the problem of discrete laplacians of a very general form; continuous analogues can also be given. (c) Word-length laplacians. Consider a ﬁnitely generated group G and write g to | | denote the word length of g, its distance to e in the Cayley graph. If it is conditionally negative—like for free, cyclic, Coxeter groups—a natural laplacian is A ((g)) g (g), and the Riesz transforms || =| | 1/2 b (g), ej R ,j f @ ,j A f 2⇡i h || iH||f (g)(g) || = || = p g || g G X2 | | satisfy Theorem A1. Many other transforms arise from other conditionallyb negative lengths. The natural example given above is out of the scope of [34]. It yields new interesting inequalities; here are two examples in the (simpler) case p 2. When G Z2m, = 2 1/2 2⇡i j f (j) 2⇡i j f (j)e 2m · c(p) e 2m · p ⇠ pj (2m j) p j Z2m ✓k Z2mj 3k ^ ◆ 2X 2X X2 b for 3k jb Z2m j k s mod 2m with 0 s m 1 . When G Fn, ={ 2 | ⌘   } = 2 2 1/2 1 1 1 f p c(p) f (g)(g) f (g )(g) . k k ⇠ p g + p g ✓h e g h g h ◆ p X6= X | | X | | b b Riesz transforms and Fourier multipliers 535

Let us now go back to Meyer’s problem (MP) for convolution Markov semigroups. In the Euclidean case, integrating by parts we get 1 . According to Sauvageot’s =r⇤ r theorem [63], a similar factorization takes place for Markovian semigroups. Namely, there exists a Hilbert (G)-bimodule M and a densely deﬁned closable symmetric derivation L L (G) M such that A ⇤ . : 2 ! = If B ej j L (, 6, µ) denotes the standard gaussian measure space : H 3 7! 2 b2 construction, we will ﬁnd in our case that M L (, 6, µ) o G where G acts via = 1 the cocycle action as ,g(B(h)) B(↵ ,g(h)). The derivation is = (g) B(b (g)) o (g), ⇤ ⇢ o (g) ⇢, B(b (g)) (g). : 7! : 7! h i If we consider the conditional expectation onto (G), L

E (G) M ⇢g o (g) ⇢g dµ (g) (G), L : 3 7! 2 L g G g G✓ ◆ X2 X2 Z and recall the identity

0 (f1,f2) E (G)(( f1)⇤ f2) = L from Remark 1.3, we obtain the following solution to (MP) for (G, ). Theorem A2. The following norm equivalences hold for G discrete: (i) If 1

uj Gp(C) o G g, B(ej ) L ()(g) Lp(G). : 3 7! h i 2 2 g G X2 b Then R ,j f aj bj runs over (aj ,bj ) (uj (a), uj (b)) for a,b Gp(C) o G. = + = 2 536 Marius Junge et al.

As in Theorem A1, we recover Meyer’s inequalities (MP) when G is abelian and the cocycle action is trivial; the general case is more involved. The inﬁmum cannot be reduced to decompositions f f f (see Remark 1.4). The main result in [33] provides lower = 1 + 2 estimates for p 2 and regular Markov semigroups satisfying 02 0. In the context of group algebras, Theorem A2 goes much further. We refer to Remark 1.5 for a brief analysis of optimal constants. Theorem A1 follows from Theorem A2 by standard manipulations. The proof of the latter is inspired by a crossed product extension of Pisier’s method [55], which ultimately relies on a Khintchine-type inequality of independent interest. The key point in Pisier’s argument is to identify the Riesz transform as a combination of the transferred Hilbert transform 1 dt Hf (x, y) p.v. t f (x, y) where t f (x, y) f (x ty) = ⇡ t = + ZR and the gaussian projection

n n Q Lp(R ,) Lp-span B(⇠) ⇠ R . : ! { | 2 } Here the gaussian variables are given by B(⇠)(y) ⇠,y , homogeneous polynomials of =h i n degree 1. The following identity can be found in [55] for any smooth f R C: : !

/ 1 dt 1 2 n 2/⇡ ( 1) f (idL (R ) Q) p.v. t f , (RI) = 1 ⌦ ⇡ t ✓ R ◆ p Z n n n where C1(R ) C1(R R ) is the derivation : ! ⇥ n @f (f )(x, y) yk f (x), y . = @x = hr i k 1 k X= Our Khintchine inequality allows us to generalize this formula to pairs (G, ). It seems fair to say that for the Euclidean case, this kind of formula has its roots in the work of Duoandikoetxea and Rubio de Francia [24] through the use of Calderon’s´ method of ro- tations. Pisier’s main motivation was to establish similar identities involving Riesz transforms for the Ornstein–Ulhenbeck semigroup. Our class of -Riesz transforms becomes very large when we vary . This yields a fresh perspective in Fourier multiplier theory, mainly around Hormander–Mikhlin¨ smoothness conditions in terms of Sobolev and (limiting) Besov norms. We refer to [34] for a more in-depth discussion of smoothness conditions for Fourier multipliers deﬁned on discrete groups. The main idea is that this smoothness may be measured through the use of cocycles, via lifting multipliers m living in the cocycle Hilbert space (identiﬁed n with R for some n 1), so that m m b . = Let Mp(G) be the space of multiplierse m G C equipped with the p p norm of : ! !n the map (g) m(g)(g). Consider the classical differential operators in R given by 7! e ↵/2 ↵ n D↵ ( 1) so that D↵f (⇠) ⇠ f (⇠) for ⇠ R . = =| | 2 = H d b Riesz transforms and Fourier multipliers 537

We shall also need the fractional laplacian lengths

dx 2" "(⇠) 2 (1 cos(2⇡ ⇠,x )) kn(") ⇠ . = n h i x n 2" = | | ZR | | + Our next result provides new Sobolev conditions for the lifting multiplier.

Theorem B1. Let (G, ) be a discrete group equipped with a conditionally negative length giving rise to an n-dimensional cocycle ( ,↵ ,b ). Let ('j )j Z denote a stan- H n 2 dard radial Littlewood–Paley partition of unity in R . If 1 0, then 1

m Mp(G) .c(p,n) m(e) inf sup Dn/2 " " 'j m L ( n) . k k | |+m m b + 2 R = j Z n 2 p o The infimum runs over all lifting multiplierse m C such thatem m b . : H ! = Our Sobolev-type condition in Theorem B1 is formally less demanding than the stan- e e dard one (see below), and our argument is also completely different from the classical approach used in [34]. As a crucial novelty, we will show that every Hormander–Mikhlin¨ type multiplier (those for which the term on the right hand side is finite, in particular the classical ones) is in fact a Littlewood–Paley average of Riesz transforms associated to a single infinite-dimensional cocycle! The magic formula comes from an isometric isomorphism between the Sobolev-type norm in Theorem B1 and mean-zero elements of n (n 2") n n L2(R ,µ") with dµ"(x) x + dx. In other words, if b" R L2(R ,µ") =|| n : ! denotes the cocycle map associated to " then m R C satisfies : !

Dn/2 " " m n < + eL2(R ) 1 np iff there exists a mean-zero h L2(R ,µ") such that 2 e

h, b"(⇠) µ" n m(⇠) h i and h L2(R ,µ") Dn/2 " " m L ( n) = p "(⇠) k k = + 2 R p (see Lemmae 2.5 for further details). A few remarks are in order: e Theorem B1 holds for unimodular ADS groups (see Appendix A). • Our bound is majorized by the classical one • n/ "/ j 2 4 2 n sup (1 ) + ('0m(2 ))^ L2(R ). j k +|| · k 2Z A crucial fact is that our Sobolev norm is dilatione invariant; more details in Corol- lary 2.7. Moreover, our bound is more appropriate in terms of dimensional behavior of the constants (see Remark 2.8). Our result is stronger than the main result in [34] in two respects. First we obtain • Sobolev-type conditions, which are way more ﬂexible than the Mikhlin assumptions

sup sup ⇠ | | @ m(⇠) < . ⇠ n 0 n/2 1 | | | | 1 2R \{ } | |[ ]+ e 538 Marius Junge et al.

Second, we avoid the modularity restriction in [34]. Namely, there we needed a simul- taneous control of left/right cocycles for nonabelian discrete groups, when there is no spectral gap. In the lack of that, we could also work only with the left cocycle at the price of extra decay in the smoothness condition. In Theorem B1, it sufﬁces to satisfy our Sobolev-type conditions for the left cocycle. On the other hand, the approach in [34] is still necessary. First, it explains the connection between Hormander–Mikhlin¨ multipliers and Calderon–Zygmund´ theory for group von Neumann algebras. Second, the Littlewood–Paley estimates in [34, Theorem C] are crucial for this paper and [53]. Third, our approach here does not give L BMO bounds. 1 ! The dimension dependence in the constants of Theorem B1 has its roots in the use of certain Littlewood–Paley inequalities on (G), but not on the Sobolev-type norm itself. L This yields a form of the Hormander–Mikhlin¨ condition with dimension free constants, replacing the compactly supported smooth functions 'j by a certain class of analytic J functions which arises from Cowling/McIntosh holomorphic functional calculus, the sim- plest of which is x xe x that already appears in the work of Stein [68]. Our result is 7! the following.

Theorem B2. Let G be a discrete group and 3G the set of conditionally negative lengths G R giving rise to a finite-dimensional cocycle ( ,↵ ,b ). Let ' R C : ! + H : + ! be an analytic function in the class . If 1 0, then J 1 2 m c(p) m(e) inf ess sup D " '(s )m . Mp(G) . (dim )/2 " L ( ) k k | |+ 3G s>0 H + |·| 2 H m 2m b = n p o e The infimum runs over all 3G and all m C such that m m b . 2 e : H ! = Theorem B2 also holds for unimodular groups (see Appendix A). Taking the trivial co- n cycle in R whose associated length functione is ⇠ 2, we find a Sobolev conditione which | | works up to dimension free constants; we do not know whether this statement is known in the Euclidean setting. The versatility of Theorems B1 and B2 for general groups is an illustration of what can be done using other conditionally negative lengths to start with. Replacing for instance the fractional laplacian lengths by some others associated to limiting measures when " 0, we may improve the Besov-type conditions a` la Baern- ! stein/Sawyer [2] (see also the related work of Seeger [66, 67] and [11, 44, 65]). The main (n 2") idea is to replace the measures dµ"(x) x dx, used to prove Theorem B1, by =| | + the limiting measure d⌫(x) u(x)dx with = 1 1 u(x) 1B (0)(x) 1 n B (0)(x) . = x n 1 + 1 log2 x R \ 1 | | ✓ + | | ◆ Let us also consider the associated length

`(⇠) 2 (1 cos(2⇡ ⇠,x ))u(x) dx. = n h i ZR Then, if 1

n where ('j )j Z is a standard radial Littlewood–Paley partition of unity in R and the 2 2 weights wk are of the form k 0 k k>0 for k Z. A more detailed analysis of this  + 2 result will be given in Section 2.4. In fact, an even more general construction is possible which relates Riesz transforms to ‘Sobolev-type norms’ directly constructed in group von Neumann algebras. We will not explore this direction here; further details in Remarks 2.11 and 2.12. Let us now consider a given branch in the Cayley graph of F , the free group with inﬁnitely many generators. Of particular interest are two applications1 we have found for operators (frequency) supported by such a branch. If we ﬁx a branch B of F let us set 1 Lp(B) f Lp( (F )) f (g) 0 for all g/B . ={ 2 L 1 | = 2 } As usual, we shall write for the word length of the free group F . b || b 1 Theorem C. Given any branch B of F , we have 1 (i) (Hormander–Mikhlin¨ multipliers) If m Z C, then : + ! m c(p) sup m(j) j m(j) m(j 1) , Mp(B) . || j 1 | |+ | | e e e e where Mp(B) denotes the space of Lp(B)-bounded Fourier multipliers. (ii) (Twisted Littlewood–Paley estimates) Consider a standard Littlewood–Paley partition of unity ('j )j 1 in R , generatedb by dilations of a function with p Lip- + schitz. Let 3j (g) 'j ( g ) (g) denote the corresponding radial multipliers : 7! | | in (F ). Then, for any f Lp(B) and 1 2 (see Corol- lary 3.2). At the time of this writing, we do not know of an appropriate upper bound for f p (p>2) since standard duality fails due to the twisted nature of square func- k k tions. Bozejko–Fendler’s˙ theorem [8] indicates that sharp truncations might not work for all values of 1

1. Riesz transforms

In this section we shall focus on our dimension free estimates for noncommutative Riesz transforms. More speciﬁcally, we will prove Theorems A1 and A2. We shall also illustrate our results with a few examples which provide new estimates both in the commutative and in the noncommutative settings.

1.1. Khintchine inequalities Our results rely on Pisier’s method [55] and a modified version of Lust-Piquard/Pisier’s noncommutative Khintchine inequalities [45, 49]. Given a noncommutative measure space ( ,'), we let RCp( ) be the closure of the finite sequences in Lp( ) equipped M M M with the norm 1/2 1/2 inf gk⇤gk hkhk⇤ if 1 p 2, fk gk hk p + p   = + k k (fk) RCp( ) 8 ⇣X 1/⌘2 ⇣X 1/2⌘ k k M = >max fk⇤fk , fkfk⇤ if 2 p< . < p p k k  1 n⇣X ⌘ ⇣X ⌘ o > The noncommutative: Khintchine inequality reads as Gp( ) RCp( ), where M = M Gp( ) denotes the closed span in Lp(, µ Lp( )) of a family (k) of centered in- M ; M dependent gaussian variables on (, µ). The specific statement for 1 p< is  1 p 1/p k(w)fk dµ(w) c(p) (fk) RCp( ). Lp( ) ⇠ k k M ✓Z k M ◆ X Our goal is to prove a similar result with a group action added to the picture. Let be H a separable real Hilbert space. Choosing an orthonormal basis (ej )j 1, we consider the linear map B L (, µ) given by B(ej ) j . Let 6 stand for the smallest : H ! 2 = Riesz transforms and Fourier multipliers 541

-algebra making all the B(h)’s measurable. Then the well known gaussian measure space construction [14] tells us that, for every real unitary ↵ in O( ), we can construct a H measure preserving automorphism on L (, 6, µ) such that (B(h)) B(↵(h)). 2 = Now, assume that a discrete group G acts by real unitaries on and isometri- H cally on some ﬁnite von Neumann algebra . In particular, G acts isometrically on M L (, 6, µ) and we may consider the space Gp( ) o G of operators of the 1 ⌦¯ M M form (B(ej ) fg,j ) o(g) Lp( ) ⌦ 2 A g G j 1 X2 X fg with (L (, 6, µ) | )oG and{z fg }Gp( ). We will also need the conditional A = 1 ⌦¯ M 2 M expectation E G(fg o (g)) ( fg dµ) o (g), which takes Lp( ) contractively Mo = A to Lp( o G). The conditional Lp norms M R r c rc Lp(E oG) Lp(E oG) if 1 p 2, Lp (E oG) r M + c M   M = (Lp(E G) Lp(E G) if 2 p< , Mo \ Mo  1 are determined by

1/2 1/2 f r (ff ) f c (f f) Lp(E G) E oG ⇤ p and Lp(E G) E oG ⇤ p k k Mo =k M k k k Mo =k M k (see [30, Section 2] for a precise analysis of these spaces). In particular, if 1

Theorem 1.1. Let G be a discrete group. If 1

Gp( ) o G is complemented in M Lp(L (, 6, µ) o G) 1 ⌦¯ M and the norm of the corresponding projection Q is p2/(p 1). ⇠ Proof. Let us first assume p>2; the case p 2p is trivial. Then the lower estimate b= holds with constant 1 from the continuity of the conditional expectation on Lp/2. The upper estimate relies on a suitable application of the central limit theorem. Indeed, assume first that f is a finite sum g,h(B(h) fg,h) o (g). Fix m 1, use the diagonal ⌦ m action (copying the original action on entrywise) on `2 ( ) and repeat the gaussian P H H m measure space construction on the larger Hilbert space resulting in a map Bm ` ( ) : 2 H ! 542 Marius Junge et al.

m L2(m,6m,µm). Let m `2 ( ) denote the isometric diagonal embedding h 1/2 : H ! H 7! m j m h ej and let F1,...,Fk be bounded functions on R. Then  ⌦ P⇡ F (B(h )) Fk(B(hk)) F (Bm(m(h ))) Fk(Bm(m(hk))) 1 1 ··· = 1 1 ··· extends to a measure preserving -homomorphism L (, 6, µ) L (m,6m,µm) ⇤ 1m ! 1 which is in addition G-equivariant, i.e. ⇡(g(f )) (⇡(f )). The action here is given = g by g(B(h)) B(↵g(h)) with ↵ the cross-product action. Thus, we obtain a trace pre- = serving isomorphism ⇡G (⇡ id )oidG from (L (, 6, µ) )oG to the larger = ⌦ M 1 ⌦¯ M space (L (m,6m,µm) ) o G. This implies 1 ⌦¯ M 1 m f Gp( ) G (Bm(h ej ) fg,h) o (g) . k k M o = pm ⌦ ⌦ j 1 g,h p X= X The random variables

fj (Bm(h ej ) fg,h) o (g) = g,h ⌦ ⌦ X are mean-zero and independent over E oG (see Appendix B for precise deﬁnitions). Hence, the noncommutative Rosenthal inequality—(M B.1) in the Appendix—yields

f Gp( ) G k k M o m m m Cp p 1/p 1/2 1/2 fj p E G(fj⇤fj ) E G(fj fj⇤) .  pm k k + Mo p + Mo p h⇣j 1 ⌘ ⇣j 1 ⌘ ⇣j 1 ⌘ i X= X= X= Note that E G(fj fj⇤) E G(ff ⇤) and E G(fj⇤fj ) E G(f ⇤f) for all j. Mo = Mo Mo = Mo Moreover, fj p f p. Therefore, the second inequality with constant O(p) follows k k =k k by letting m . An improved Rosenthal inequality [40] actually yields !1

f Gp( ) G Cpp f RCp( ) G, k k M o  k k M o which provides the correct order of the constant in our Khintchine inequality. Let us now consider the case 1

Q(f ) fk dµ k, = k X✓Z ◆ which is independent of the choice of the basis. Let Q (Q id )oidG be the ampliﬁed = ⌦ M gaussian projection on Lp(L (, 6, µ) o G). It is clear that Gp( ) o G is the 1 ⌦¯ M M image of this Lp-space under the gaussian projection.b Similarly RCp( ) o G is the rc M image of Lp (E G). Note that Mo rc Q Lp (E G) RCp( ) o G : Mo ! M is a contraction. Indeed, we have b E G(ff ⇤) E G(Qf Qf ⇤) E G(Q?f Q?f ⇤) E G(Qf Qf ⇤) Mo = Mo + Mo Mo b b b b b b Riesz transforms and Fourier multipliers 543 by orthogonality, and the same holds for the column case. By the duality between rc rc Lp (E G) and Lq (E G) and also between Lp( ) and Lq ( ), we obtain Mo Mo A A

f RCp( )oG sup tr(fg) sup tr(f Qg) k k M = g rc 1 | |= g rc 1 | | k kLq  k kLq  b sup tr(fg) sup g Gq ( )oG f Gp( )oG = g RC 1 | | g RC 1 k k M k k M k k q  ⇣k k q  ⌘ with 1/p 1/q 1. In conjunction with our estimates for q 2, this proves the lower + = estimate for p 2. The upper estimate is a consequence of the continuous inclusion rc  Lp (E G) Lp for p 2[38, Theorem 7.1]. Indeed, Mo !  2 1/2 1/(2p) 2 1/2 1/(2p) f G ( ) f 2 E ( f ) 2 f c . p oG Lp/ ( ) oG Lp/ ( ) Lp(E oG) k k M = | | 2 A  k M | | k 2 A = k k M It remains to prove the complementation result. Since the gaussian projection is self- adjoint, we may assume p 2. Moreover, the upper estimate in the ﬁrst assertion together  rc with the contractivity of the gaussian projection on Lp (E G) give rise to Mo

Qf Gp( )oG . Qf RCp( )oG sup tr((Qf )g) sup tr(f Qg) k k M k k M = g RC 1 | |= g RC 1 | | k k q  k k q  b b b b sup Qg Gq ( )oG f Lp( ) C2pq f Lp( ).  g RC 1 k k M k k A  k k A k k q  The last inequality follows fromb the ﬁrst assertion for q. ut

1.2. Riesz transforms in Meyer form Denote by the Lebesgue measure and write for the normalized gaussian measure n in R . With this choice, the maps t f (x, y) f (x ty) are measure preserving - n n = n + ⇤ homomorphisms from L (R , ) to L (R R , ). We may also replace by 1 1 ⇥ n ⇥ n the Haar measure ⌫ on the Bohr compactiﬁcation Rbohr of R , this latter case including n n . Moreover, if G acts on R then t commutes with the diagonal action. As we =1 already recalled in the Introduction, (RI) takes the form

/ 1 dt 1 2 n 2/⇡ ( 1) f (idL (R ,) Q) p.v. t f = 1 ⌦ ⇡ t ✓ ZR ◆ p n n n with Q the gaussian projection and C1(R ) C1(R R ) the derivation : ! ⇥ n @f (f )(x, y) yj f (x), y B( f (x))(y). = @x = hr i= r j 1 j X= n Lemma 1.2. Let G be a discrete group acting on L (R ,⌫). If 1

Proof. The cross-product extension of (RI) reads

1/2 1 dt 2/⇡ (( 1) o idG)f Q p.v. (t o idG)f . = ⇡ t ✓ R ◆ p Z 1/2 b This gives ( 1) o idG p⇡/2 Q(H o idG) where H is the transferred Hilbert = transform 1 dt Hf (x, y) p.v.b t f (x, y) . = ⇡ t ZR By de Leeuw’s theorem [21, Corollary 2.5], the Hilbert transform is bounded on Lp(Rbohr,⌫) with the same constants as in Lp(R, ). The operator above can be seen as a directional Hilbert transform at x in the direction of y, which also preserves the same constants for y ﬁxed. In particular, a Fubini argument combined with a gaussian average easily gives n n n H Lp(R ,⌫) Lp(R R ,⌫ ) : bohr ! bohr ⇥ ⇥ again with the classical constants, even for n . =1 To analyze the crossed product H o idG we note that t o idG is a trace preserving -automorphism ⇤ n n n L (R ,⌫)o G L (R R ,⌫ )o G. 1 bohr ! 1 bohr ⇥ ⇥ According to the Coifman–Weiss transference principle [17, Theorem 2.4] and the fact that H is G-equivariant, we see that H o idG extends to a bounded map on Lp with constant c(p) p2/(p 1). Indeed, it is straightforward that the proof of the transfer- ⇠ ence principle for one-parameter automorphisms translates verbatim to the present setting. Then the assertion follows from the complementation result in Theorem 1.1 for n L (R ,⌫). M = 1 bohr ut n Given a length G R , consider its cocycle map b G R and the crossed n : ! + : ! product L (R ,)o G deﬁned via the cocycle action ↵ . The derivation (G) n 1 : L ! L (R ,)o G is determined by ((g)) B(b (g)) o (g). We include the case 1 = n , so that any length function/cocycle is admissible. The cocycle law yields the =1 Leibniz rule ((gh)) ((g))(h) (g) ((h)). = + An alternative argument to the proof given below can be found in Appendix A.

Proof of Theorem A2. Given the infinitesimal generator A ((g)) (g)(g) and ac- = cording to the definition of the norm in RCp(C) o G, it suffices to prove that

1/2 A f c (p) f G ( ) c (p) f RC ( ) . k kLp(G) ⇠ 1 k k p C oG ⇠ 2 k k p C oG The second norm equivalenceb follows from the Khintchine inequality in Theorem 1.1 2 with constant c (p) p /(p 1); let us prove the ﬁrst one. Since b is a cocycle, the 2 ⇠ map p n ⇡ (G) (g) exp(2⇡i b (g), ) o (g) L (R ,⌫)o G : L 3 7! h ·i 2 1 bohr Riesz transforms and Fourier multipliers 545 is a trace preserving -homomorphism which satisﬁes ⇤ / 1/2 1 2 n (( 1) o idG) ⇡ i(idL (R , ) o ⇡) A . = 1 Indeed, if we let the left hand side act on (g) we obtain

1/2 1/2 (( 1) o idG) ⇡((g)) ( 1) exp(2⇡i b (g), ) o (g) = h ·i 1 exp(2⇡i b (g), ) o (g) = 2⇡ b (g) h ·i k kH i exp(2⇡i b (g), ) B(b (g)) o (g), = p (g) h ·i ⌦

1/2 n which coincides with i(idL (R , ) o ⇡) A ((g)). By Lemma 1.2, both sides in 1 n n this intertwining identity are bounded Lp( (G)) Lp(L (R R ,⌫ )o G) for L ! 1 bohr ⇥ ⇥ 1

1/2 1 dt n 2/⇡ i(idL (R , ) o ⇡) A f Q p.v. (t o idG)⇡f . 1 = ⇡ t ✓ R ◆ p Z n b Since idL (R , ) o ⇡ is also a trace preserving -homomorphism, this yields 1 ⇤ 1/2 1/2 n n n n f Lp(L (R , )oG) (idL (R , ) o ⇡) A (A f) Lp(L (R R ,⌫ )oG) k k 1 =k 1 k 1 bohr⇥ ⇥ 3 p 1/2 . A f . (p 1)3/2 k kLp(G) b The constant above also follows from Lemma 1.2. The reverse estimate follows with the same constant from a duality argument. Indeed, if we ﬁx f to be a trigonometric polynomial, there exists another trigonometric polynomial f with f p 1 such that 0 k 0k 0 = 1/2 1/2 (1 ") A f p ⌧(f 0⇤A f ). k k  1/2 Note that A is only well-deﬁned on f 00 (g) 0 f 0(g)(g). However, since = 6= G0 g G (g) 0 is a subgroup, we may consider the associated conditional ={ 2 | = } P expectation E on (G) and obtain f f E f so thatb f p 2. On the other G0 L 00 = 0 G0 0 k 00k 0  hand, we note the crucial identity

trL ()oG(( f1)⇤ f2) 1 1 f1(g)f2(h) ↵ 1 B(b (g))B(b (h)) dµ ⌧((g h)) = g g,h G ✓Z ◆ X2 b b 1/2 1/2 f (g)f (g) b (g), b (g) f (g)f (g) (g) ⌧((A f )⇤A f ). = 1 2 h i = 1 2 = 1 2 g G g G X2 X2 b b b b 546 Marius Junge et al.

Combining both results we get

1/2 1 1/2 1 1/2 A f p ⌧(f 0⇤A f) ⌧(f 00⇤A f) k k  1 " = 1 " 1 1/2 trL ()oG (( A f 00)⇤ f) = 1 " 1 3 1 1/2 p A f 00 p f p . f p.  1 " k k 0 k k (p 1)3/2 k k The last estimate was already obtained in the ﬁrst part of this proof, it also follows from Lemma 1.2 and yields c (p) p3/(p 1)3/2. This completes the proof. 1 . ut Remark 1.3. Let

1 0 (f ,f ) A (f ⇤)f f ⇤A (f ) A (f ⇤f ) . 1 2 = 2 1 2 + 1 2 1 2 In the Introduction we related Theorem A2 to Meyer’s formulation in terms of 0 via the identity E (G)(( f1)⇤ f2) 0 (f1,f2). The proof follows by arguing as above, and L = we ﬁnd

1 E (G)(( f1)⇤ f2) f1(g)f2(h) b (g), b (h) (g h) 0 (f1,f2) L = g,h h i = X b b 1 1 since b (g), b (h) ( (g) (h) (g h)), as explained in Appendix B. h iH = 2 + Remark 1.4. When 1

Remark 1.5. Our constants grow like p3/2 as p with a dual behavior as p 1. !1 ! According to the results in the literature, one might expect a linear growth of the constant p. It is however not clear to us whether this is true in our context since we admit ⇠ semigroups which are not diffusion semigroups in the sense of Bakry, like the Poisson semigroup. It is an interesting problem to determine the optimal behavior of dimension tp 1 free constants for (say) the Riesz transform associated to the Poisson semigroup (e ) n in R . On the other hand, we are not aware of any dimension free estimates with constants / n t1 better than p3 2 for matrix-valued functions in R , even for the heat semigroup (e ). Riesz transforms and Fourier multipliers 547

1.3. Riesz transforms in cocycle form We are now ready to prove Theorem A1. The main ingredient comes from a factorization of the conditional expectation E (G) L (, µ) o G (G) in terms of a certain L : 1 ! L right (G)-module map. As predicted by Hilbert module theory [30], this factorization is L always possible. In our case, when 1,2 Gp(C) o G, it takes the form 2

E (G)(1⇤2) u(1)⇤u(2), L = where u Gp(C) o G Cp(Lp(G)) is deﬁned as follows: : !

u() u g o (g)b g, B(ej ) L (,µ)(g) ej1. = = h i 2 ⌦ g G j 1 g G ⇣X2 ⌘ X ⇣X2 ⌘ uj ()

Before proving Theorem A1, it is convenient| to explain{z where the infimum} is taken and how we define the twisted form of (bj ). The infimum R ,j f aj bj runs over all = + possible families of the form aj uj (1) and bj uj (2) for some 1,2 Gp(C)oG. = = 2 This is equivalent to requiring that B(ej ) o aj Gp(C) o G, and the same for (bj ). j 2 Once this is settled, if we note that P

bj bk(g)ek,ej (g), = g G k 1 H X2 DX E b then the twisted form of the family (bj )j 1 is determined by the formula

bj bk(g)ek,↵ ,g(ej ) (g). = g G k 1 H X2 DX E e b Proof of Theorem A1. If f L (G), we have 2 p

1/2 b b(g), ej u( A f) h iH f (g)(g) ej R ,j f ej . = p (g) ⌦ 1 = ⌦ 1 j 1✓g G ◆ j 1 X X2 X b When p 2, the assertion follows directly from Theorem A2 since 1/2 1/2 1/2 1/2 0 (A f, A f) E (G)(( A f)⇤ A f) = L 1/2 2 2 u( A f) R ,j f , =| | = | | j 1 X and similarly with f replaced by f . When 1

Note that R ,j (f ) R (f ) so that (R ,j f ) (R ,j f ) can be written as a ⇤ = 0 ,j ⇤ j ⇤ ⇤ ⇤ row square function in terms of the row Riesz transforms R . Although these two for- P 0 ,j mulations become identical for G abelian (left and right cocycles coincide), the statement does not simplify for nontrivial cocycle actions and 1

1.4. Examples, commutative or not In order to illustrate Theorems A1 and A2, it will be instrumental to present conditionally negative lengths in a more analytic way. As will be justified in Appendix B (Theorem B.4), these lengths are all deformations of the standard inner cocycle which acts by left multiplication. Namely, G R is conditionally negative iff it can be written as : ! + 1 (g) ⌧ 2(e) (g) (g ) (CN) = for a positive linear functional ⌧ 5 0 C defined on the space 50 of trigonometric : ! polynomials in (G) whose Fourier coefficients have vanishing sum. Having this in mind, L let us consider some examples illustrating Theorems A1 and A2. n /2 A. Fractional laplacians in R . The Riesz potentials f ( 1) f are classical 7! operators in Euclidean harmonic analysis [69]. It seems however that dimension free es- n timates for associated Riesz transforms in R are unexplored. If we let our infinitesimal n generator be A ( 1) and p 2 (for simplicity), the problem in R consists in = showing that 1/2 1/2 2 @ A f f n ,j n c(p) Lp(R ) | | Lp(R ) ⇠ k k ⇣j 1 ⌘ X for some differential operators @,j and constants c(p) independent of the dimension n. A generates a Markov semigroup for 0 1. Indeed, the nonelementary cases   2 0 <<1 require to know that the length 0 (⇠) ⇠ is conditionally negative. Since n =| | (CN) holds for G R , the claim follows from the simple identity = 2 1 2⇡i x,⇠ 2⇡i x,⇠ 1 0 (⇠) ⇠ (2 e h i e h i)dµ (x) (⇠) =| | = k () n = k () n ZR n n 2 with dµ (x) dx/ x and = | | + ds ⇡ n/2 1 1 kn() 2 1 cos(2⇡ ⇠/ ⇠ ,s ) max , . = n h | | i s n 2 ⇠ 0(n/2) 1 ZR + ⇢ | | The constant kn() only makes sense for 0 <<1 and is independent of ⇠. The last equivalence follows from the fact that the integral on the left hand side is comparable to s,⇠/ ⇠ 2 1 |h | |i| ds ds, n 2 n 2 B (0) s + + n B (0) s + Z 1 | | ZR \ 1 | | 550 Marius Junge et al. and the equivalence follows by using polar coordinates. The associated cocycle ( ,↵ ,b ) is given by the action ↵,⇠ (f ) exp(2⇡i ,⇠ )f and the cocycle map H n = h· i ⇠ 1 exp(2⇡i ,⇠ ) L2(R ,µ /kn()). Of course, we may regard as a 7! h· i 2 H = H real Hilbert space by identifying exp(2⇡i x,⇠ ) C with (cos(2⇡ x,⇠ ), sin(2⇡ x,⇠ ) h i 2 h i h i in R2 and the product exp(2⇡i x,⇠ ) with a rotation by 2⇡ x,⇠ . In contrast to the h i ·[] n h i standard Riesz transforms for 1, for which R and the cocycle is trivial, n = H = we need to represent R in infinitely many dimensions to obtain the right differential operators b (⇠), e \1/2 j R\,j f (⇠) @,j A f (⇠) h iH f (⇠). = = ⇠ | | In particular, Theorem A1 (or its extension to unimodular groupsb in Appendix A) gives norm equivalences for all 1

d 1/2 b /2 2⇡i ,⇠ ( 1) f p Mf, ( , ⇠)e h· i d⇠ k k ⇠ n · ✓ZR ◆ p for f smooth enough and 1 2 2 Mf, (x, ⇠) f (x)f( ⇠) f (x)f (⇠) f (⇠) ⇠ . = 2 + | | | | More applications to EuclideanL multipliers will be analysed in the next section. p b b d B. Discrete laplacians in LCA groups. Let 00 be a locally compact abelian group and s 0 be torsion free. If @j '( ) '( ) '( ,...,s j ,...,n) stand for discrete 0 2 0 = 1 0 directional derivatives in 0 0 0 , we may construct the laplacian L @ @j . = 0 ⇥···⇥ 0 = j⇤ Lust-Piquard’s [48] main result establishes dimension free estimates in this context for the 1/2 1/2 P family of discrete Riesz transforms given by Rj @j L and R L @ . If p 2, = j⇤ = j⇤ she obtained n 1/2 2 2 ( Rj ' Rj⇤' ) c(p) ' Lp(0). | | +| | Lp(0) ⇠ k k ⇣j 1 ⌘ X= It is not difficult to recover and generalize Lust-Piquard’s theorem from Theorem A1 for 0 LCA, justified in Appendix A. Indeed, let 0 be any LCA group equipped with a positive measure µ . If G denotes the dual group of 0, let us write g 0 T for the associated : ! characters and ⌫ for the Haar measure on G. Define

A ' '(g)A (g) d⌫(g) '(g) (2e g 1 )( ) dµ ( ) g d⌫(g). = = g ZG ZG Z0 2 (g) 1 g b b =k kL2(0,µ ) | {z } If G R , then it is a conditionally negative length which may be represented by : ! + the cocycle ( ,↵ ,g,b (g)) L (0, µ ), g , 1 g . H = 2 ·[] Riesz transforms and Fourier multipliers 551

In other words, we have (g) b (g), b (g) . Again, we may regard as a =h iH 2H real Hilbert space by identifying C g( ) (Re(g( )), Im(g( ))) R and the 3 7! 2 product g( ) with a rotation by arg(g( )). Set ·[] b (g), e1 1 Re b (g, ) R ' h iH '(g)g d⌫(g) '(g)g d⌫(g), = p (g) = p (g) ZG ZG b (g), e2 2 b Im b (g, )b R ' h iH '(g)g d⌫(g) '(g)g d⌫(g). = p (g) = p (g) ZG ZG Then Theorem A1 takes the followingb form for LCA groups and pb 2: 1/2 1 2 2 2 ( R ' R ' )dµ ( ) c(p) ' L (0). | | +| | ⇠ k k p ✓Z0 ◆ Lp(0) Now, let us go back to Lust-Piquard’s setting 0 00 00 and write j = ⇥···⇥ = (0,...,0,s0, 0,...,0) with s0 in the j-th entry. If we pick our measure µ to be the sum j j of point-masses and recall the simple identities

P 1/2 1 2 1/2 1 2 Rj @j L R iR ,R⇤ L @⇤ R iR , = = j j j = j = j + j then we deduce Lust-Piquard’s theorem as a particular case of Theorem A1 for LCA groups. We may also recover her inequalities for 1

1 1 f1,f2 f1(g)f2(h) g h h i||= | | 2 g,h G X2 b b and (↵ ,g(f ), b (g)) ((g)f, (e) (g)), as usual. The identiﬁcation of this Hilbert || || = space as a real one requires taking real and imaginary parts as in the previous examples. The Riesz transforms

1/2 b (g), ej R ,j f @ ,j A f 2⇡i h || iH||f (g)(g) || = || = p g || g G X2 | | b 552 Marius Junge et al. satisfy Theorem A1 for any orthonormal basis (ej )j 1 of . Theorem A2 also applies H|| with A ((g)) g (g) and the associated gradient form 0 . Further Riesz transforms || =| | || arise from other positive linear functionals ⌧ on 50. Let us recall a few well-known groups for which is conditionally negative: || (i) Free groups. The conditional negativity of the length function for the free group Fn with n generators was established by Haagerup [29]. A concrete description of the associated cocycle yields an interesting application of Theorem A1. The Gromov form is g h g 1h K(g, h) b (g), b (h) | |+| || | min(g, h) =h || || iH|| = 2 =| | where min(g, h) is the longest word inside the common branch of g and h in the Cayley graph. If g and h do not share a branch, we let min(g, h) be the empty word. Let R Fn stand for the algebra of ﬁnitely supported real valued functions on Fn and [ ] consider the bracket g,h K(g, h). If N denotes its null space, let be the h i= H completion of R Fn /N. Then, writing g for the word which results after deleting [ ] the last generator on the right of g, the system ⇠g g g N for all g Fn e = + 2 \{ } is orthonormal and generates . Indeed, it is obvious that e belongs to N and we H may write

agg ah ⇠g = = g n g n h g X2F X2F ⇣X ⌘ for any R Fn . Here h g iff g belongs to the (unique) path from e to h in the 2 [ ] Cayley graph and we set ⇠e e. This shows that N Re and dim . It = = H =1 yields a cocycle with ↵ and = b Fn g g Re R Fn /Re. : 3 7! + 2 [ ]

Considering the ONB (⇠h)h e we see that b(g), ⇠h g h, and the operators 6= h i= 1 R ,hf f (g)(g) || = g h p g X | | b form a natural family of Riesz transforms in (Fn) with respect to word length. If L p 2, Theorem A1 gives rise to the inequalities below in the free group algebra with constants c(p) only depending on p:

2 1 2 1/2 f (g) f (g ) f c(p) (g) (g) . k kLp(Fn) ⇠ p g + p g ✓h e g h g h ◆ Lp(Fn) X6= X b| | X b | | b The case 1

(ii) Finite cyclic groups. The group Zn Z/nZ with the counting measure is a cen- = tral example. Despite its simplicity it may be difﬁcult to provide precise bounds for Fourier multipliers (see for instance [35] for a discussion on optimal hypercontractivity bounds). The conditional negativity for the word length k min(k, n k) in | |= Zn was justiﬁed in that paper. For simplicity we will assume that n 2m is an even = integer. Its Gromov form (j (2m j)) (k (2m k)) ((k j) (2m k j)) K(j, k) ^ + ^ ^ + = 2 can be written as follows for 0 j k 2m 1:    K(j, k) j ,k j (2m k) (m k j) =h i= ^ ^ + + with s 0 s; the details are left to the reader. Note that K(j, j) (j) as + = _ = it should be. Using the above formula, we may consider the associated bracket in 50 R Z2m R0 and deduce, after rearrangement, ' [ ] 2m 1 2m 1 m k m 1 k m 1 m + + aj j , bj j aj bj ↵kk. = = j 1 j 1 k 1 j k j k k 1 D X= X= E X= ⇣ X= ⌘⇣ X= ⌘ X= This shows that the null space N of the bracket above is the space of elements j aj j in R Z2m R0 with vanishing coordinates ↵k. If we quotient out this subspace we [ ] end up with a Hilbert space of dimension (2m 1) (m 1) m. Our discussionP H = establishes an isometric isomorphism 2m 1 m m 8 aj j ↵kek R , : H 3 7! 2 j 1 k 1 X= X= where ↵k j 2m 1 aj j , 13k with =h  i 3k P j Z2m j k s mod 2m for some 0 s m 1 , ={ 2 | ⌘   } 1 so that 13 8 (ek). Consider the ONB of given by 13 for 1 k m.We k = H k   are ready to construct explicit Riesz transforms associated to the word length in Z2m. Namely, let Z2m T2m be the group of 2m-roots of unity with the normalized count- = ing measure. Then using the ONB above we set b 1 j R ,kf f (j)j with j (z) z . || = pj (2m j) = j 3k X2 ^ b If p 2, Theorem A1 establishes the following equivalence in Lp(Z2m): m f (j) 2 1/2 f (j)j c(p) j b Lp(Z2m) ⇠ pj (2m j) L ( ) j Z2m ✓k 1 j 3k ^ ◆ p Z2m 2X X= X2 b with constantsb independent b of m. Similar computations can be performed b for n = 2m 1 odd, and Theorem A1 can be reformulated in this case. We leave the details + to the reader. 554 Marius Junge et al.

(iii) Inﬁnite Coxeter groups. Any group presented by

sjk G g ,...,gm (gj gk) e =h 1 | = i with sjj 1 and sjk 2 for j k is called a Coxeter group. Bozejko˙ [7] proved = 6= that word length is conditionally negative for any inﬁnite Coxeter group. The Cayley graph of these groups is more involved and we will not construct here a natural ONB for the associated cocycle; we invite the reader to do it.

Other interesting examples include discrete Heisenberg groups or symmetric groups.

2. Hormander–Mikhlin¨ multipliers

In this section we shall prove Theorems B1 and B2. Before that we obtain some pre- liminary Littlewood–Paley type inequalities. Afterwards we shall also study a few Besov limiting conditions in the spirit of Seeger for group von Neumann algebras that follow from Riesz transforms estimates.

2.1. Littlewood–Paley estimates Let (G, ) be a discrete group equipped with a conditionally negative length and associated cocycle ( ,↵ ,b ). If h we shall write R ,h for the h-directional Riesz H 2 H transform b (g), h R ,hf 2⇡i h iH f (g)(g). = p (g) g G X2 We begin with a consequence of Theorem A1 which alsob generalizes it.

Lemma 2.1. Given (hj )j 1 in and p 2, H 1/2 1/2 2 2 R ,hj fj .c(p) sup hj fj . | | Lp(G) k kH | | Lp(G) j 1 j 1 j 1 ⇣X ⌘ ⇣ ⌘⇣X ⌘ b b Proof. Given an ONB (ek)k 1 of , we have H 1/2 2 R ,hj fj | | Lp(G) ⇣j 1 ⌘ X hj ,eb k b (g), ek h iH h iH fj (g)(g) ej, = p (g) ⌦ 1 j 1✓g G k 1 ◆ Sp(Lp) X X2 X b R ,kfj hj ,ek ej,1 R ,kfj 3 (e(j,k),1) , = ⌦h iH Sp(Lp) = ⌦ Sp(Lp) j,k 1 j,k 1 X X c where 3 `2(N N) (j,k) hj ,ek j `2(N). Since ` (C,`2) is a : ⇥ 3 7! h iH 2 2 := B homogeneous Hilbertian operator space [58], the cb-norm of 3 coincides with its norm, Riesz transforms and Fourier multipliers 555

which in turn equals supj hj . Altogether, we deduce k kH

1/2 2 R ,hj fj sup hj R ,kfj e(j,k),1 | | Lp(G)  k kH ⌦ Sp(Lp) j 1 j 1 j,k 1 ⇣X ⌘ ⇣ ⌘ X 1/2 b 2 sup hj R ,kfj = k kH | | Lp(G) j 1 j,k 1 ⇣ ⌘⇣ X ⌘ 1/2 2 b sup hj R ,kf = j 1 k kH | | Sp(Lp) ⇣ ⌘⇣k 1 ⌘ X e for R ,k R ,k id (` ) and f j fj ej,1. Now, since Theorem A1 also holds = ⌦ B 2 = ⌦ in the category of operator spaces, the last term on the right hand side is dominated by P c(p)ef S (L ), which yields the inequality we are looking for. k k p p ut We need to ﬁx some standard terminology for our next result. Consider a sequence of k functions 'j R C in n (R 0 ) for kn n/2 1 such that : + ! C + \{ } =[ ]+ 2 1/2 d k 'j (⇠) ⇠ for all 0 k kn. d⇠k . | |   ✓ j ◆ X Let Mp(G) stand for the space of symbols m G C associated to Lp-bounded Fourier : ! multipliers in the group von Neumann algebra (G); equip any such symbol m with the L p p norm of the multipler (g) m(g)(g). Now, given any conditionally negative ! 7! length G R and h in the associated cocycle Hilbert space , let us consider the : ! + H symbols

m ,h(g) b (g), h / (g) so that R ,h((g)) 2⇡im ,h(g)(g). =h i = p Then, we may combine families of these symbols into a single Fourier multiplier patching them via the Littlewood–Paley decompositions provided by the families ('j ) and ﬁnite- dimensional cocycles on G. The result is the following.

Lemma 2.2. Let G be a discrete group equipped with two n-dimensional cocycles with associated length functions 1, 2 and an arbitrary cocycle with associated length function 3. Let ('1j ), ('2j ) be Littlewood–Paley decompositions satisfying the assumptions above and (hj )j 1 in . Then, for any 1

' j ( ( ))' j ( ( ))m ,h c(p,n) sup hj . 1 1 2 2 3 j . 3 · · Mp(G) j 1 k kH j 1 X Proof. Consider the multipliers

3 ,' (g) '( (g))(g). : 7! 556 Marius Junge et al.

When p 2, let f Lp(G) and f Lq (G) with 1/p 1/q 1 so that 2 2 + = b e b ⌧ f ⇤ 3 1,'1j 3 2,'2j R 3,hj (f ) j ⇣ X ⌘ e ⌧ (3 ,' f)⇤R ,h (3 ,' f) = 1 1j 3 j 2 2j j X e 3 1,'1j f j R 3,hj (3 2,'2j f) j  ⌦ RCq ( (G)) ⌦ RCp( (G)) j L j L X X e by anti-linear duality. On the other hand, Lemma 2.1 trivially extends to RCp-spaces. Indeed, the row inequality follows from the column one applied to fj⇤’s together with R Remark 1.7 and the fact that the maps 0 ,hj satisfy the same estimates. We apply it to the last term of the right hand side. Combining that with the Littlewood–Paley inequality in [34, Theorem C] we obtain

⌧ f ⇤ 3 1,'1j 3 2,'2j R 3,hj (f ) .c(p,n) sup hj f L ( ) f L ( ). k kH 3 k k p G k k q G j 1 j 1 ⇣ X ⌘ ⇣ ⌘ b b e e The result follows by taking supremums over f running over the unit ball of Lq (G). On the other hand, when 1 2 since the multipliers involved are R-valued ande therefore self-adjoint. b ut The only drawback of Lemma 2.2 is that we need ﬁnite-dimensional cocycles to apply our Littlewood–Paley estimates from [34]. We may ignore that requirement at the price of using other square function inequalities from [32], where the former 'j are now dilations of a ﬁxed function. Namely, given 1 in . Then for any 1 2 3 1 2 2 J 0 H 3 1 0 Z + Riesz transforms and Fourier multipliers 557

rc Proof. Let us write Lp(G L (R )) for the space RCp( (G)) in which we replace ; 2 + L discrete sums over Z by integrals on R with the harmonic measure. Arguing as in + + Lemma 2.1, it is not difﬁcultb to show that Theorem A1 implies

ds rc rc R 3,hs Lp(G L (R )) Lp(G L (R )) .c(p) ess sup hs , 2 + 2 + 3 R s : ; ! ; s>0 k kH Z + for p 2. According to [32b , Theorem 7.6], theb maps rc 81 Lp(G) f ('1( A )f )s>0 Lp(G L (R )), : 3 7! · 1 2 ; 2 + rc 82 Lp (G) f ('2( A )f )s>0 Lp (G L (R )), : 0 b 3 7! · 2 2 0 b ; 2 + will be bounded as longb as A are sectorial of type ! (0b,⇡) on Lp(G) and admit a j 2 bounded H (6✓ ) functional calculus for some ✓ (!p,⇡) with !p ⇡ 1/p 1/2 . 1 2 = | | Sectoriality is conﬁrmed by [32, Theorem 5.6], whereas the H 1(6✓ ) calculusb follows from the existence of a dilation as proved in [32, Proposition 3.12]. The fact that ,t S admits a dilation is a consequence of the main result in [61] (see also [37]). The combination of these arguments is explained for the Poisson semigroup on the free group in [32, Theorem 10.12]. Therefore, the result follows for p 2 by noticing that ds ds '1(sA 1 )'2(sA 2 )R 3,hj 82⇤ R 3,hs 81. R s = R s Z + ✓Z + ◆ The case 1

Remark 2.4. Slight modiﬁcations also give: (i) If p 2, then 1/2 R f 2 ( h ,h ) 1/2 f . ,hj .c(p) j k (` ) Lp(G) | | Lp(G) k h i k 2 k k j 1 B ⇣X ⌘ b b (ii) If 1 0 [ ] Z + wherep 3(g) ms(g) hs,b (g) for all g and ms X hs . =h 3 iH 3 k k 3 =k kH 3 558 Marius Junge et al.

Assertion (i) follows as in Lemma 2.1 with 30 `2(N) k j hj ,ek j `2(N) : 3 7! h iH 2 instead of the map 3 used there. Note that 3 (3 ) ( hj ,hk )j,k. Moreover, by duality 0 0 ⇤ = h iP we also ﬁnd the following inequality for 1

2.2. A reﬁned Sobolev condition In order to prove Theorem B1, we start with a basic inequality for Euclidean Fourier ↵/2 multipliers which is apparently new. Recall the notation D↵ ( 1) and the fractional 2" = n laplacian lengths "(⇠) kn(") ⇠ from the Introduction. Here " L2(R ,µ") with n 2" = | | H = dµ"(x) x dx and (b",↵") stand for the corresponding cocycle map and cocycle | | + = action. The result below shows how large is the family of Riesz transforms of convolution n type in R when we allow inﬁnite-dimensional cocycles.

Lemma 2.5. If 1 0, then 1

n m Mp( ) .c(p) Dn/2 " " m n . k k R + L2(R ) p In fact, when the right hand side is ﬁnite we have

m(⇠) m ",h(⇠) h, b"(⇠) " / "(⇠) = =h iH n p for some h L(R ,µ") with 2 2

n h L ( ,µ") Dn/2 " " m n . k k 2 R = + L2(R ) p Moreover, the constants in the above inequality are independent of " and n. Riesz transforms and Fourier multipliers 559

Proof. If we consider the Sobolev-type space 2 n n W(n/ ,")(R ) m R C m measurable, Dn/2 " " m n < , 2 = : ! + L2(R ) 1 L ( n,µ ) L ( n,µ ) p and 2 R " is the mean-zero subspace of 2 R " , then we claim that n 2 n 3 L(R ,µ") h mh W (R ), : 2 3 7! 2 (n/2,") 1 2⇡i ⇠,x mh(⇠) h(x)(e h i 1)dµ"(x), = p (⇠) n " ZR n extends to a surjective isometry. Indeed, let h be a Schwartz function in L2(R ,µ") with n 2" compact support away from 0 and write !"(x) 1/ x + for the density dµ"(x)/dx. = | | n In that case, the function h!" is a mean-zero Schwartz function in L2(R ) and

2⇡i ,x " mh h(x)e h· i dµ"(x) (h!")_. n = R = p Z In particular, this implies that

3(h) W2 ( n) Dn/2 " " mh n k k (n/2,") R = + L2(R ) 1 p (h! ) h n . " _^ L(R ,µ") = p!" n =k k 2 L2(R ) Therefore, since smooth mean-zero compactly supported functions away from 0 are n clearly dense in L2(R ,µ"), we may extend 3 to an isometry. Now, to show surjectivity we observe from elementary facts on Sobolev spaces [1] that the class of Schwartz 1 functions is dense in our W-space. Therefore, it sufﬁces to show that 3 (m) exists for any such Schwartz function m. By Plancherel theorem, we ﬁnd

1 \ 1 \ Dn/2 " " m L ( n) "m "m . + 2 R = p!" n = w" n L2(R ) L2(R ,µ") p p p 1 1 Denoting h 3 (m) p\ "m, h is also a Schwartz function. We may write = = !" 2⇡i ⇠,x "(⇠)m(⇠) h(x)!"(x)e h idx n = R p Z 2⇡i ⇠,x h(x)(e h i 1)dµ"(x) h(x)!"(x) dx A(⇠) B. = n + n = + ZR ZR 2⇡i ⇠,x The function A(⇠) is well-deﬁned since both h and e h i 1 are elements of the n Hilbert space L2(R ,µ"), so its product is absolutely integrable. Moreover, since h!" n 2 (R ) we see that A(0) 0 and conclude B p "(0) m(0) 0. This means that S 1 = n= = h 3 (m) is a mean-zero function in L2(R ,µ"), as desired. In other words, every = n m W2 (R ) satisﬁes 2 (n/2,") 2⇡i ⇠, h, e h ·i 1 µ" m(⇠) m h n . h i and Dn/2 " " 2 L2(R ,µ") = p "(⇠) + =k k p 560 Marius Junge et al.

This shows that every m in our Sobolev-type space is a "-Riesz transform and its norm coincides with that of its symbol h. The only drawback is that we use a complex Hilbert space for our cocycle, so the inner product is not the right one. Consider the cocycle ( ",↵",b") with H n 2 n " L2(R ,µ" R ) L2(R ,µ" C), H = ; ' ; 2⇡i ⇠, b"(⇠) cos(2⇡ ⇠, ) 1, sin(2⇡ ⇠, ) e h ·i 1, = h ·i h ·i ' cos(2⇡ ⇠, ) sin(2⇡ ⇠, ) f1 2⇡i ⇠, ↵",⇠ (f ) h ·i h ·i e h ·if. = sin(2⇡ ⇠, ) cos(2⇡ ⇠, ) f ' ✓ h ·i h ·i ◆✓ 2◆ Then it is easily checked that: 2⇡i ⇠, 0 If h is R-valued and odd, then h, e h ·i 1 µ" i ,b"(⇠) . • h i = h " 2⇡i ⇠, h H If h is -valued and even, then h, e 1 µ ,b"(⇠) . R h ·i " ⌦0 ↵ " • h i = H Therefore, decomposing ⌦ ↵

m Re(m ) Re(m ) i Im(m ) Im(m ) = odd + even + odd + even and noticing the elementary inequalities

Re/Im(modd/even) W2 ( n) m W2 ( n) k k (n/2,") R k k (n/2,") R we see that every element in our Sobolev-type space decomposes as a sum of four Riesz transforms whose Mp-norms are all dominated by c(p) m W2 ( n). k k (n/2,") R ut Lemma 2.6. Let (G, ) be a discrete group equipped with a conditionally negative length giving rise to an n-dimensional cocycle ( ,↵ ,b ). If 1 0, H 1 then n m Mp(G) .c(p) inf Dn/2 "( " m) L2(R ). k k m m b k + k = p Proof. According to Lemma 2.5, e e

4 4 hj ,b" b (g) " 1 m(g) h iH hj ,b" b (g) = (b (g)) = (b (g)) " j 1 " " j 1 H X= DX= E 2 np p n for any m W (R ) satisfying m m b and certain hj L2(R ,µ"). Next we 2 (n/2,") = 2 observe that each of the four summands above can still be regarded as the Riesz transform on (G) with respect to the following cocycle: L e e n 2 ," L2(R ,µ" R ), H = ; b ,"(g) cos(2⇡ b (g), ) 1, sin(2⇡ b (g), ) , = h ·i h ·i cos(2⇡ b (g), ) sin(2⇡ b (g), ) f1 ↵ ,g 1 ( ) ↵ ,",g(f ) h ·i h ·i · . = sin(2⇡ b (g), ) cos(2⇡ b (g), ) f2 ↵ 1 ( ) ✓ h ·i h ·i ◆✓ ,g · ◆ Riesz transforms and Fourier multipliers 561

We conclude by noticing that

m Mp(G) c(p) hj . c(p) m W2 ( n). k k  " k k (n/2,") R j 1 H X= e Namely, since " b is a conditionally negative length (associated to the cocycle above), the ﬁrst inequality follows from Lemma 2.1 for j 1 when p 2 and from Re- = mark 2.4(iv) when 1

Since m m (m m(e) ) b and m m m(e) almost everywhere, we may 0 = 0 = 0 just proceed by assuming that m(g) 0 for all g G , . Let ⌘ be a radially decreasing = 2 0 smooth function oneR with 1( 1,1) ⌘ e1( 2,e2) and set (⇠) ⌘(⇠) ⌘(2⇠), so that   = j we may construct a standard Littlewood–Paley partition of unity j (⇠) (2 ⇠) for = j Z. Note that 2 1 if ⇠ 0, j (⇠) 6= j = (0 if ⇠ 0. X2Z = 1 Since supp j supp k for j k 2, we see that ⇢j (j 1 j j 1) \ =; | | := 3 + + + forms a partition of unity satisfying ⇢j 1/3 on the support of j . We shall be working ⌘ n 2 with the radial Littlewood–Paley partition of unity in R given by 'j ⇢j . On the = || other hand, if we set ' j ' j j we may write m as follows (recall that m vanishes 1 = 2 = where does): p

m (j )m 3 (j )(⇢j )m = j = j X2Z X2Z 3 ('1j )('2j )(⇢j )m 3 ('1j )('2j )mj . = j = j X2Z X2Z Since

2 mj (g) ⇢j ( (g))m(g) ⇢j ( b (g) )m(b (g)) ('j m)(b (g)), = = | | = we deduce that mj mj b with mj 'j m. We know by assumption that = = e e

sup Dn/2 " " 'j m < . e e+ e 2 j Z 1 2 p e 562 Marius Junge et al.

According to Lemma 2.5 and the proof of Lemma 2.6, this implies that we may write mj as a Riesz transform m,h with respect to the length function " b , whose associated j inﬁnite-dimensional cocycle was also described in the proof of Lemma 2.6. Since the families '1j ,'2j satisfy the assumptions of Lemma 2.2, we may combine Lemma 2.2 with Lemma 2.5 to obtain

m Mp(G) '1j ( 1( ))'2j ( 2( ))m,hj k k = · · Mp(G) j Z X2 c(p, n) sup hj c(p, n) sup Dn/2 " "'j m . + 2  j Z k kH  j Z 2 2 p The dependence on n of c(p, n) comes from the Littlewood–Paley inequalities.e ut In the following result, we use the standard notation

2 2 ↵/2 H () f supported by (1 ) f L ( n) < , ↵ ={ |k +|| k 2 R 1} ˚ 2 ↵ H↵() f supported by D↵f L ( n) f n < . = k k 2 R b= || L2(R ) 1 Corollary 2.7. If 1

" Dn/2 "( " 'j m) 2 kn(") Dn/2 "( 'j m) 2 k + k = k + || k " j p pkn(") Dn/2 "( '0m(2 )) 2 e = k + || e · k ⇡n/2 1 p 2 n with kn(") . Indeed, it is easy to check that W (R ) has a dilation invariant ⇠ 0(n/2) " (n/2,") e norm. It therefore sufﬁces to show that, up to a constant c(n, "),

" 2 (n/2 ")/2 Dn/2 "( f) 2 . (1 ) + f 2 k + || k k +|| k for functions supported by (say) the corona B (0) B (0), which is the form of the = 2 \ 1 b support of '0. In other words, we need to show that 2 " ˚ 2 3" Hn/2 "() f f Hn/2 "() : + 3 7! | | 2 + deﬁnes a bounded operator. These two families of spaces satisfy the expected interpolation identities in the variable ↵ (1 ✓)↵ ✓↵ with respect to the complex method. = 0 + 1 To prove that 3" is bounded we use Stein’s interpolation. Assume by homogeneity that f is in the unit ball of H2 () and let ✓ 2" for n odd and ✓ " for n even. Thus, n/2 " = = there exists F analytic on the+ strip satisfying F(✓) f and =

max sup F (it) 2 , sup F(1 it) 2 1. Hn/2() H n/2 1() t k k t k + k [ ]+  n 2R 2R o Riesz transforms and Fourier multipliers 563

Given 0 < 1, deﬁne the analytic family of operators  2 "z/✓ z(F ) exp((z ✓) ) F (z), L = || so that ✓ (F ) 3"(f ). We claim that L = n/2 t2 ✓2 it(F ) H˚ 2 () c(n)(1 t ) e e , kL k n/2  +| | n/2 1 t2 (1 ✓)2 1 it(F ) ˚ 2 () c(n)(1 t )[ ]+ e e . + H n/2 1 kL k [ ]+  +| |

t2 t2 /2 Then the trivial estimate e (1 t ) . e (p t ) in conjunction with +| | k +| | k1 the three lines lemma imply the statement of the corollary with the constant

e((1 ✓)✓2 ✓(1 ✓)2) c(n) (") + . kn n/4 "/2 + " p↵ If ↵ (1 it) and u↵(⇠) ⇠ , our second claim follows from the simple inequality = ✓ + =| |

1 it(F ) ˚ 2 () + H n/2 1 kL k [ ]+ (1 it ✓)2 . c(n) e + sup @ u↵ L () F(1 it) H2 (). 1 n/2 1 | | 0 n/2 1 k k k + k [ ]+ | |[ ]+ Indeed, since n/2 1 Z, the Sobolev norm above can be computed using ordinary [ ]+ 2 derivatives and the given estimate arises from the Leibniz rule and the triangle inequality. it A similar argument shows that the map f f is contractive on L2() and bounded 2 7! |n/ |2 1 on H n/2 1() up to a constant c(n)(1 t )[ ]+ . By complex interpolation, [ ]+ +| | it it n/2 f H˚ 2 () f H2 () c(n)(1 t ) f H2 (). || n/2  || n/2  +| | k k n/2 Since "/✓ 1/2,1 , our ﬁrst claim follows by taking f F (it). 2{ } = ut Remark 2.8. On the other hand, a quick look at the constant we obtain in the proof of Corollary 2.7 shows that our Sobolev-type norm is more appropriate than the classical one in its dimensional behavior. Namely, it is easily checked that the constant c(n) above grows linearly with n since it arises from applying the Leibniz rule n/2 1 times. In [ ]+ particular, we obtain a constant

n⇡ n/4 1 c(n) kn(") ⇠ p0(n/2) p" p which decreases to 0 very fast with n. We pay a price for small " though. This could be rephrased by saying that our Sobolev-type norm is “dimension free” (see Section 2.3 below) and encodes the dependence on ">0. Indeed, the constant c(p, n) in Theorem B1 is independent of ". 564 Marius Junge et al.

Remark 2.9. The Coifman–Rubio de Francia–Semmes theorem [16] shows that functions in R of bounded 2-variation deﬁne Lp-bounded Fourier multipliers for 1

Remark 2.10. Corollary 2.7 implies, for m m b , = m Mp(G) .c(p,n) m(e) sup | |@ m . k k | |+0 e n/2 1 || 1 | |[ ]+ Indeed, this follows from the well-known relation between Mikhline smoothness and Sobolev–Hormander¨ smoothness (see e.g. [23, Chapter 8]). This already improves the main result in [34] for 1

2.3. A dimension free formulation A quick look at our argument for Theorem B1 shows that the only dependence of the constant we get on the dimension of the cocycle comes from the use of our Littlewood– Paley inequalities in Lemma 2.2. The proof of Theorem B2 just requires replacing that result by Lemma 2.3. Proof of Theorem B2. As in the statement, let be a conditionally negative length whose associated cocycle ( ,↵ ,b ) is ﬁnite-dimensional; let m be a lifting multiplier for that H cocycle, so that m m b ; and let ' R C be an analytic function in the class = : + ! J considered in Section 2.1. Arguing as in Theorem B1 wee may assume with no loss of generality that m vanishese where does. In other words, certain noninjective cocycles may be used as long as m is constant on a nontrivial subgroup of G. Since ds/s is dilation invariant,

3 ds 2 ds m(g) k' '(s (g)) m(g) k' '(s (g)) ms(g) = R s = R s Z + Z + for some constant k' 0. Note that 6= 2 ms(g) '(s (g))m(g) ms ms b with ms '(s )m. = ) = = |·| Then we proceed again as in Theorem B1. Indeed, we know by assumption that e e e m < . ess sup D(dim )/2 " " s 2 s>0 H + 1 p e Riesz transforms and Fourier multipliers 565

According to the proof of Lemma 2.6, this implies that we may write ms as a Riesz transform m,h with respect to the length function " b for almost every s>0. s = Since the families ' ,' ' satisfy the assumptions of Lemma 2.3, we may combine 1 2 = this result with Lemma 2.5 to obtain

2 ds m k' '(s ( )) m,h k kMp(G) = · s s R Mp(G) Z + c(p) h c(p) D '(s 2)m . . ess sup s ess sup (dim )/2 " " 2 s>0 k kH  s>0 H + |·| ut p Remark 2.11. Lemma 2.5 admits generalizations for any conditionally negativee length n ` R R whose associated measure ⌫ is absolutely continuous with respect to the : ! + n n Lebesgue measure and such that 1 cos(2⇡ ⇠, ) L1(R ,⌫)for all ⇠ R . Indeed, if h ·i 2 2 we set d⌫(x) u(x)dx we also have = 1 n p[ m Mp(R ) .c(p) `m k k pu n L2(R ) as long as the space determined by the right hand side admits a dense subspace of functions m for which p`msatisfies the Fourier inversion theorem. The Schwartz class was enough for our choice (`, ⌫) ( ",µ"). Lemma 2.6 can also be extended when ⌫ is in- = variant under the action ↵ of the chosen finite-dimensional cocycle ( ,↵ ,b ). This H invariance is necessary to make sure that the construction in the proof of Lemma 2.6 yields a well-defined cocycle out of ` and .

Remark 2.12. Although the above mentioned applications are of independent interest, it is perhaps more signiﬁcant to read our approach as a way to relate certain kernel repro- ducing formulas to some differential operators/Sobolev norms in von Neumann algebras. Namely, consider any length 1 (g) ⌧ (2(e) (g) (g )) with ⌧ (f ) ⌧(f ! ) = = for some positive invertible density ! , and construct the spaces

2 1/2 L (G,⌧ ) h L (G) h , ⌧ ( h ) < , 2 ={ 2 0 |k k2 = | | 1} 2 1 W b(G,⌧) m ` (bG) m W, m < . = 2 1 k k = p! 1 ⇢ L2(G) p b Here stands for the left regular representation, playing the role of the Fourierb transform. This is the analogue of what we do in the Euclidean case. Then, there exists a linear isometry ⌧ (b ( )h) 2 3 L(G,⌧ ) h · W (G, ⌧), : 2 3 7! p ( ) 2 · where L2(G,⌧ ) denotes theb subspace of mean-zero elements. Theb surjectivity of 3 de- pends as above on the existence of a dense subspace of our Sobolev-type space admitting Fourier inversionb in (G) (in fact, this can be used in the opposite direction to identify L 566 Marius Junge et al. nice Schwartz-type classes in group algebras). Whenever the map 3 is surjective, it relates the “Sobolev norm” of a Riesz transform to the L2-norm of its symbol, which in turn dominates its multiplier norm up to c(p). We have not explored applications in this general setting.

Remark 2.13. Using Laplace transforms in the spirit of Stein [68], we can prove Littlewood–Paley estimates in discrete time and also Lp bounds for smooth Fourier multipliers even for inﬁnite-dimensional cocycles.

2.4. Limiting Besov-type conditions n Let ⌘ be a radially decreasing smooth function on R with 1B (0) ⌘ 1B (0) and set 1   2 '(⇠) ⌘(⇠) ⌘(2⇠), so that me may construct a standard Littlewood–Paley partition of = k unity 'k(⇠) '(2 ⇠) for k Z. Consider the function 1 j 1 'j and let ↵ R = 2 p n = 2 and 1 p, q . Then the Besov space B↵q(R ) and its homogeneous analogue are  1 n P defined as subspaces of tempered distributions f 0(R ) in the following way: 2 S 1/q p n n p kq↵ q B˚↵q(R ) f 0(R ) f ↵q 2 'k f p < , = 2 S ||| ||| = k k ⇤ k 1 n ⇣X2Z ⌘ o 1/q p n n p b kq↵ q B↵q(R ) f 0(R ) f ↵q f p 2 'k f p < . = 2 S k k =k ⇤ k + k ⇤ k 1 n ⇣k 1 ⌘ o X p p b b Note that ↵q is a norm while ↵q is a seminorm. Besov spaces refine Sobolev spaces kk ||| ||| in an obvious way. For instance, it is straightforward to show that n n n n B2 (R ) H2 (R ) and B˚ 2 (R ) H˚ 2 (R ) ↵2 ' ↵ ↵2 ' ↵ with constants depending on the dimension n. It is a very natural question to study how we can modify the Sobolev (n/2 ")-condition in the Hormander–Mikhlin¨ theorem when + " approaches 0. This problem has been studied notably by Seeger [66, 67] (see also [11, 44, 65] and the references therein). In terms of Lp-bounded Fourier multipliers for 1 < p< the best known result is 1 j n 2 m Mp(R ) .c(p,n) sup '0m(2 ) n/2,1, k k j k · k 2Z where '0(⇠) '(⇠) ⌘(⇠) ⌘(2⇠) as defined above. Of course, we cannot expect = = 2 n to replace the Besov space on the right hand side by the bigger one Bn/2,2(R ) or its homogeneous analogue

1/2 2 nk 2 sup 'j m n/2,2 sup 2 'k ('j m) 2 j Z ||| ||| = j Z k k ⇤ k 2 2 ⇣X2Z ⌘ 1/2 nk b j 2 sup 2 'k ('0m(2 )) 2 = j Z k k ⇤ · k 2 ⇣X2Z ⌘ which holds by the dilation invariance of the normb we use. Riesz transforms and Fourier multipliers 567

In the following theorem we show that a log-weighted form of this space sits n in Mp(R ). Similar results for Euclidean multipliers were already found by Baern- stein/Sawyer [2], the main difference being that they impose a more demanding `1- Besov condition. Our argument is also very different. We formulate our result in the context of discrete group von Neumann algebras, although it also holds for arbitrary unimodular groups. The main idea is to replace the measures dµ"(x) !"(x)dx with (n 2") = !"(x) x , used to prove Theorem B1, by the limiting measure d⌫(x) u(x)dx =| | + = with 1 1 u(x) 1B (0)(x) 1 n B (0)(x) . = x n 1 + 1 log2 x R \ 1 | | ✓ + | | ◆ Let us also consider the associated length

`(⇠) 2 (1 cos(2⇡ ⇠,x ))u(x) dx. = n h i ZR After Theorem 2.15 we give more examples and a comparison with Seeger’s results.

Lemma 2.14. The length above satisﬁes 1 `(⇠) 1B (0)(⇠) 1 log ⇠ 1 n B (0)(⇠). ⇠ 1 log ⇠ 1 + + | | R \ 1 + | | Sketch of the proof. We have `(⇠) dx 1 cos(2⇡ x,⇠ ) dx ( ( ⇡ x,⇠ )) h i 1 cos 2 n 2 n 2 = B (0) h i x + n B (0) 1 log x x Z 1 | | ZR \ 1 + | | | | A(⇠) B(⇠). = + By dilation invariance of x ndx, we can write | | ⇠ dx A(⇠) 1 cos 2⇡ x, n . = B ⇠ (0) ⇠ x Z | | ✓ ✓ ⌧ | |◆◆ | | By symmetry the direction of ⇠ is irrelevant and using polar coordinates we ﬁnd If ⇠ < 1/2, then • | | 2 x,e1 2 A(⇠) |h ni| dx c(n) ⇠ . ⇠ B ⇠ (0) x ⇠ | | Z | | | | If ⇠ 1/2, then • | | 2 x,e1 dx A(⇠) |h ni| dx n c(n) 1 log ⇠ . ⇠ B1/2(0) x + B ⇠ (0) B1/2(0) x ⇠ + | | Z Z | | | | \ | | On the other hand, using spherical symmetry we may also write 1 cos(2⇡ x ⇠ ,e ) dx B(⇠) h | | 1i . 2 n = n B (0) 1 log x x ZR \ 1 + | | | | 568 Marius Junge et al.

When ⇠ < 1/2, we ﬁnd • | | 1 dx ⇠ 2 x,e 2 dx B(⇠) | | |h 1i| . n 2 x n 2 x n ⇠ R B1/(2 ⇠ )(0) 1 log x + B1/(2 ⇠ )(0) B1(0) 1 log x Z \ | | + | | | | Z | | \ + | | | | By using polar coordinates it is easy to see that B(⇠) c(n)/ 1 log ⇠ . ⇠ + | | When ⇠ 1/2 we just get • | | dx B(⇠) c(n). 2 n ⇠ n B (0) (1 log x ) x ⇠ ZR \ 1 + | | | | Combining the estimates above we get the equivalence in the statement with B1(0) replaced by B / (0). However, since 1 log ⇠ 1 when ⇠ 1/2, 1 , this is equivalent 1 2 + | | ⇠ | |2[ ] to the right hand side in the statement. ut Theorem 2.15. Let (G, )be a discrete group with a conditionally negative length giving rise to an n-dimensional cocycle ( ,↵ ,b ). Let ('j )j Z denote a standard radial Hn 2 Littlewood–Paley partition of unity in R . If 1 e0 for k Z.  + 2 Proof. Arguing as in Theorem B1 we may assume with no loss of generality that m vanishes where does, so that m(e) 0. According to Remark 2.11 and Lemma 2.14, = Lemmas 2.5 and 2.6 apply in our setting (`, ⌫, u). Moreover, according to our argument for Theorem B1, we ﬁnd

m(g) 3 j ( (g))('j m)(b (g)) = j X2Z for a certain smooth partition of unity j . If e 1 sup p`'j m ^ < , pu 1 j Z 2 2 the (generalized) proofs of Lemmas 2.5 and 2.6e imply that mj ('j m) b is a Riesz = transform m⇣,h with respect to the length function ⇣ ` b , which comes from a com- j = posite cocycle as in Lemma 2.6 thanks to the orthogonal invariance ofe⌫ (radial density). Since the families ' j ' j j satisfy the assumptions of Lemma 2.2, we combine 1 = 2 = this with Lemma 2.5 to obtain

m Mp(G) '1j ( ( ))'2j ( ( ))m⇣,hj k k = · · Mp(G) j Z X2 1 p c(p, n) sup hj ⇣ c(p, n) sup `'j m ^ .  k kH  pu j Z j Z 2 2 2 e Riesz transforms and Fourier multipliers 569

This proves the first estimate of the theorem. The second follows since 2 2 1 1 1 2 nk 2 f 'kf 'k f 2 wk 'k f . pu = pu ⇠ u(2k)k ⇤ k2 ⇠ k ⇤ k2 ut 2 k 2 k k X2Z X2Z X2Z Remark 2.16.b According to Remarkb 2.11, otherb limiting measures ⌫ applyb as long as n we know that 1 cos(2⇡ ⇠, ) belongs to L1(R ,⌫) and the measure ⌫ is invariant h ·i under the cocycle action ↵ . In particular, any radial measure d⌫(x) u(x) dx such that 2 = u(s)(s 1(0,1)(s) 1(1, )(s)) L1(R ,ds)satisfies these conditions. Note that any such + 1 2 + measure will provide an associated length of polynomial growth, so that the associated Sobolev-type space has the Schwartz class as a dense subspace satisfying the Fourier inversion formula, as demanded by Remark 2.11. If fact, it is conceivable that for slow- increasing lengths `, 2 j 2 'k p`'j m `(2 ) 'k ('j m) . ⇤ 2 ⇠ k ⇤ k2 Therefore, under this assumption we would finally get b e b je 1/2 nk `(2 ) 2 m Mp(G) .c(p,n) m(e) inf sup 2 k 'k ('j m) 2 . k k | |+m m b j u(2 )k ⇤ k = ⇢ Z✓k Z ◆ 2 X2 2 n Remark 2.17. The Besov space Bn/e2,1(R ) used in Seeger’sb result ise not dilation invariant. Thus, in order to compare his estimates with ours, we first need to dilate from j ' m(2 ) to 'j m. Using the elementary identity 0 · j jn/2 j ⇢ f(2 ) L ( n) 2 ⇢\(2 ) f L ( n) k ⇤ · k 2 R = k · ⇤ k 2 R and easy calculations, we obtain j b j nk/2 j '0m(2 ) B2 ( n) ('0m(2 )) 2 2 'k ('0m(2 )) 2 k · k n/2,1 R =k ⇤ · k + k ⇤ · k k 1 X 1/2 bjn/2 2 b nk/2 2 'k ('j m) 2 'k ('j m) . ⇠ k ⇤ k2 + k ⇤ k2 k j 0 k j 1 ⇣ +X ⌘ +X On the other hand, our estimate in Theoremb 2.15 gives b 1/2 1/2 nk 2 nk 2 2 2 'k p`'j m 2 k 'k p`'j m . ⇤ 2 + ⇤ 2 k 0 k 1 ⇣X ⌘ ⇣X ⌘ It seems we get betterb estimates fork min(0, j) andb worse for k>max (0, j). 

3. Analysis in free group branches In this section we prove Theorem C. We shall use the same terminology as in Section 1.4 for the natural cocycle of F associated to the word length . In particular, recall that the 1 || Hilbert space is R F /Re, with ↵ and b (g) g Re. The system H|| = [ 1] || = || = + ⇠g g g with g e forms an orthonormal basis of , where g is the word = 6= H|| which results after deleting the last generator on the right of g. By a branch of F , we 1 mean a subset B (gk)k 1 with gk gk 1. We will say that g1 is the root of B. = = + 570 Marius Junge et al.

Proof of Theorem C(i). Given t>0, let tj mt (j) tje m(j). = Our hypotheses on m imply that e tj e tj/2 tj/2 mt (j) mt (j 1) . te te . te . |e | + In particular, we find that e e 2 sup mt ( g ) g mt ( g ) g | | | | | | | | t>0 g B X2 p p e e 2 2 1 sup mt (j) mt (j 1) j mt (j 1) . | | +| | j t>0 j 1✓ ◆ X e2 ts e 2 2ets ds x 2x . sup t e sds (ts) e x(e e )dx < . t>0 R + R s = R + 1 ✓Z + Z + ◆ Z + Therefore, if we define ht g B ht ,⇠g ⇠g with = 2 h iH|| ht ,⇠g Pmt ( g ) g mt ( g ) g for g B, h iH|| = | | | | | | | | 2 it turns out that (ht )t>0 is uniformly boundedp in . Moreoverp e e H|| b (g), ht g ⇠h b (g) mt ( g ) h || iH|| for all g B. || = h g = ) | | = p g 2 X | | In other words, mt coincides one B with the Fourier symbol of the Riesz transform || R ,ht associated to the word length in the direction of ht . Assume (without loss of gen- || || erality) p 2. Now fix f Lp( (F )) with vanishing Fourier coefficients outside B. e 2 L 1 Recall that Tm ((g)) m( g )(g) and A generates a ‘noncommutative diffusion | | = | | || semigroup’ as defined in [32, Chapter 5], which satisfies the assumptions of [32, Corol- lary 7.7] by [e32, Propositione 5.4 and Theorem 10.12]. One side of [32, Corollary 7.7] 2 2z applied to x Tm f and x Tm f ⇤ with F (z) z e implies = | | = | | = Tm f p e e k | | k dt 1/2 e 2 2tA 2 2 2tA 2 .c(p) (tA ) e ||Tm f (tA ) e ||Tm f ⇤ || | | || | | R | | +| | t p ✓Z + ◆ e e dt 1/2 tA 2 tA 2 R ,ht (tA e ||)f R ,ht (tA e ||)f ⇤ || || || || = R | | +| | t p ✓Z + ◆ since Tmt R ,ht on B. By the integral version of Lemma 2.1 for p 2, together | | = || with the uniform boundedness of (ht )t>0, e 1/2 2 2 dt Tm f p .c(p) G(tA )f G(tA )f ⇤ | | || || k k R | | +| | t p ✓Z + ◆ for G(z) zee z. By the other side of [32, Corollary 7.7] applied to x f and x f , = = = ⇤ the right hand side is dominated by f p. k k ut Riesz transforms and Fourier multipliers 571

We will say that a family Bk k 1 of branches forms a partition of the free T ={ : } group when F e k Bk and the Bk’s are pairwise disjoint. Given a branch B 1 ={ }[ 2 T let us write gB,1 for its root. We will say that B is a principal branch when its root S satisfies g , 1. If B is not a principal branch, there is a unique branch B in which | B 1|= T contains g . Given g F , define 5Bg to be the biggest element in B which is smaller B,1 2 1 than or equal to g. If there is no such element, set 5 g e. Now let us fix a standard B = Littlewood–Paley partition of unity in R . That is, given a smooth decreasing function + k ⌘ R R with 1(0,1) ⌘ 1(0,2), set (⇠) ⌘(⇠) ⌘(2⇠) and k(⇠) (2 ⇠) : + ! +   = = for k Z. Assume in addition that p is Lipschitz (as we assume in Theorem C). Then 2 construct ' j and 'j j for j 2, 1 = = j 1 X so that j 1 'j 1. Define 3j (g) 'j ( g ) (g) and = : 7! | | P p 'j ( 5Bk g ) 5Bk g 'j ( 5 g ) 5 g Bk Bk 3j,k (g) 5 g e | | | | | | | |(g) : 7! Bk 6= p g p | p| for any g F , with the convention that 5 g e if Bk is a principal branch. 2 1 Bk =

Lemma 3.1. If 1

2 2 2 hj,k . 'j ( g ) 'j ( g ) g 'j ( g ) p g g k kH|| | | | | | |+| | | | | | | | g Bk X2 p p p 2 1 'j (i) 'j (i 1) i 'j (i 1) . +| | i 2j 1 i 2j 1 1✓ ◆ X + + p p i 1 . . 1. 4j + i 2j 1 i 2j 1 1✓ ◆ X + +

In particular, the family (hj,k) is uniformly bounded in and H||

g ⇠h b (g) b (g), hj,k ⇠h,hj,k . = = || )h|| iH|| = h iH|| h g e h 5 g X 6= X Bk 572 Marius Junge et al.

By cancellation, the latter sum is

'j (5Bk g) 5Bk g 'j (5 g) 5 g when 5Bk g e. | | Bk | Bk | 6= Hence p p b (g), hj,k || || 3j,kf h iH f (g)(g) R ,hj,k f. = p g = || g F X2 1 | | h b h ,h Now, according to the definition of j,k it is easily checked that j,k j 0,k0 vanishes h iH|| unless k k and j j 1. This implies that each of the subsystems (h j,k) and = 0 | 0| 2 (h2j 1,k) is orthogonal and uniformly bounded. Once this is known and splitting into two systems,+ the assertion follows from Theorem A1 by standard considerations. ut Proof of Theorem C(ii). It clearly suffices to prove the result for B a principal branch. Let Bk k 1 form a partition of F which contains B1 B as a principal T ={ : } 1 = branch. Given f Lp( (F )) with vanishing Fourier coefficients outside B, it is then 2 L 1 easily checked that 3j f 3j, f because B is a principal branch and 3j,kf 0 for = 1 = other values of k. Therefore, the first estimate follows from Lemma 3.1. For the second estimate, we use the fact that 'j is a partition of unity together with the inequality in Remark 2.4(iv). Namely, we obtain

f 32f R (a b ) Lp(B) j ,hj,1 j j k k = Lp(B) = || + Lp(B) j 1 j 1 X X b / b1/2 1 2 b ( hj,1,hk,1 ) (` ) (aj⇤aj bj bj⇤) . kh i kB 2 + Lp( (F )) ⇣j 1 ⌘ L 1 X Now observe that hj, ,hk, vanishes when j k > 1, and it is bounded above other- h 1 1i | | wise. This shows that the matrix above is bounded since it is a band diagonal matrix of width 3 with uniformly bounded entries. We are done. ut Corollary 3.2. If B is any branch of F and 2 1/6 and the partial sums are chosen to lie in a sequence L | | of increasing balls with respect to word length. This may be regarded as some sort of Fef- ferman’s disc multiplier theorem [26] for the free group algebra, although discreteness might allow some room for Fourier summability near L2 in the spirit of Bochner–Riesz multipliers. This result indicates that we should not expect Littlewood–Paley estimates for nontrivial branches arising from sharp (nonsmooth) truncations in our partitions of unity, as it holds for R or Z. Riesz transforms and Fourier multipliers 573

Appendix A. Unimodular groups

Let (G,⌫)be a locally compact unimodular group with its Haar measure. Write G : ! (L (G)) for the left regular representation determined by (g)(⇢)(h) ⇢(g 1h) for B 2 = any ⇢ L (G). Recall in passing the definition of the convolution in G: 2 2 1 ⇢ ⌘(g) ⇢(h)⌘(h g) d⌫(h). ⇤ = ZG We say that ⇢ L (G) is left bounded if the map c(G) ⌘ ⇢ ⌘ L (G) 2 2 C 3 7! ⇤ 2 2 extends to a bounded operator on L2(G), denoted by (⇢). This operator defines the Fourier transform of ⇢. The weak operator closure of the linear span of (G) defines the group von Neumann algebra (G). It can also be described as the weak closure in L (L (G)) of the set of operators of the form B 2

f (f) f (g)(g) dµ(g) with f c(G). = = 2 C ZG The Plancherel weight ⌧ b (G) b 0, is given by b : L + ![ 1] 2 ⌧(f ⇤f) f (g) dµ(g) = | | ZG when f (f) for some left bounded f bL (G), and ⌧(f f) for any other = 2 2 ⇤ =1 f (G). After breaking into positive parts, this extends to a weight on a weak- dense 2 L ⇤ domain withinb the von Neumann algebrab (G). It is instrumental to observe that the L standard identity ⌧(f ) f (e) = holds for f c(G) c(G) (see [54, Section 7.2] and also [72, Section VII.3] for a 2 C ⇤ C detailed construction of the Plancherel weight).b Note that ⌧ is tracial precisely due to the unimodularityb of G, and it coincides with the ﬁnite trace ⌧(f ) fe,e for G =h i discrete. The pair ( (G), ⌧) is a semiﬁnite von Neumann algebra and we may construct L the noncommutative Lp-spaces

p ( c(G) c(G))kk for 1 p<2, Lp( (G), ⌧) Lp(G) C ⇤ Cp  L = = (( c(G))kk for 2 p< , C  1 b p 1/p where the norm is given by f p ⌧( f ) and the p-th power is calculated by func- k k = | | tional calculus applied to the (possibly unbounded) operator f (see Appendix B below for more details on the construction of noncommutative Lp-spaces). On the other hand, since left bounded functions are dense in L (G), the map ⇢ (⇢) extends to an 2 : 7! isometry from L2(G) to L2(G). Given m G C, set : ! b Tmf m(g)f (g)(g) dµ(g) for f c(G) c(G). = 2 C ⇤ C ZG Tm is called an Lp-Fourier multiplierb if it extends to a boundedb map on Lp(G).

b 574 Marius Junge et al.

Our goal in this appendix is to study the validity of our main results in this paper— Theorems A1, A2, B1 and B2—in the context of not necessarily discrete unimodular groups. More precisely, given G unimodular we shall be working with continuous conditionally negative lengths G R which are in one-to-one correspondence with : ! + continuous affine representations ( ,↵ ,b ), as follows from Schoenberg’s theorem. H Precise definitions of these notions as well as the construction of crossed products of von Neumann algebras with locally compact unimodular groups will be given in Appendix B below. Most of the time, our results in the discrete case must be modified by the simple replacement of sums over G (discrete case) with integrals over G (unimodular case) with respect to the Haar measure ⌫. We should also replace finite sums (trigonometric polynomials) by elements in ( c(G)) or ( c(G) c(G)) depending on whether p 2 or C C ⇤ C p<2. Notice that the proof of Theorem A1 for discrete groups rests on the validity of Theorem A2 for this class of groups. In the unimodular case, we may follow verbatim the proof (replacing sums by integrals as indicated above) and conclude that The- orem A1 is valid for arbitrary unimodular groups as long as the same holds for its ‘Meyer form’ in Theorem A2. Thus, we shall only prove this latter result for unimodular groups. The crucial difference from the original argument is that the group homomorphism G g b (g) o g o G is no longer continuous when we take the 3 7! 2 H discrete topology on and the product topology on the semidirect product o G. H H Recall that the discrete topology was crucial in our argument to obtain a trace preserving -homomorphism ⇤ ⇡ (G) (g) (b (g) o g) ( o G) ( ) o G. : L 3 7! o 2 L H ' L H

Here we use the isometric isomorphism (b (g) o g) exp(2⇡i b (g), ) o o 7! h ·iH (g) between ( o G) and ( ) o G. Thus, our argument at this point requires an L H L H explanation or a modification. We would like to thank the referee for the proof presented below. Although in the same line as ours (even longer), it is a bit more natural, it avoids this subtle point and improves certain estimates obtained along our argument. Of course, according to Theorem 1.1—whose proof extends trivially to unimodular groups—and as we did in our proof of Theorem A2, it suffices to show that 1/2 f c(p) A f G ( ) G. (A.1) k kLp (G) ⇠ k k p C o Proof of (A.1). Let denote the standard gaussian measure in the real Hilbert space b H (recall that dim is admissible). Let L ( ,)o↵ G and consider the H =1 M := 1 H map J (G) (g) 1 o (g) , : L 3 7! 2 M which extends trivially to a -homomorphism. Let E (G) (G) stand for the ⇤ L : M ! L corresponding conditional expectation. Construct the -derivation D , densely defined in ⇤ the weak- topology of as ⇤ M

D fg o (g) d⌫(g) 2⇡i fg b (g), o (g) d⌫(g) = h ·i ✓ZG ◆ ZG F | {z } Riesz transforms and Fourier multipliers 575 where the function g fg is continuous and compactly supported on G with values in 7! 1/2 L ( ,), so that D J . If A , the crucial identity is 1 H = R = i dt tD f Q p.v. e J (f ) for f L(G) ( c(G)), (A.2) R = p ⇡ t 2 2 \ C 2 ✓ ZR ◆ b where Q is deﬁnedb as in Theorem 1.1. Before justifying it, we shall complete the proof of (A.1). We claim that b dt p2 tD (F ) Lp( ) p.v. e (F ) . F Lp( ) (A.3) kU k M := t p 1k k M ZR Lp( ) M for 1

f (g) f (g) d⌫(g) R = p (g) ✓ZG ◆ b 1 2 2 1 t b (g) f (g) e 2 | | dt b (g), o (g) d⌫(g) = p ⇡ h ·i 2 ZG ✓ZR ◆ 1 b it b (g),y f (g) e h i d (y) b (g), o (g) dt d⌫(g) = p2⇡ G h ·i Z ZR✓ZH ◆ b for f L (G) ( c(G)). Integrating by parts coordinatewise yields 2 2 \ C

it b b(g),y it b (g),y e h i d (y) b (g), b (g), ej e h i d (y) ej , h ·i= h i h ·i ✓ ◆ j 1✓ ◆ ZH X ZH @ 1 it b (g),y e h i d (y) ej , = it @y h ·i j 1✓ j ◆ X ZH 1 it b (g),y e h iyj d (y) ej , = it h ·i j 1✓ ◆ X ZH 1 it b (g),y 1 it b (g), e h i y, d (y) Q(e h ·i). = it h ·i = it ZH 576 Marius Junge et al.

Combining this identity with our expression above for f , we obtain R i dt it b (g),y f f (g) e h i y, d (y) o (g) d⌫(g). R = p2⇡ G h ·i t Z ZR✓ZH ◆ Truncating the integralb over R to the compact set N," N, N ( ", "), it is clear = [ ]\ that we can apply Fubini. In particular, we may rewrite the term in square brackets above as dt it b (g),y lim AN,"(g, y) y, d (y) with AN,"(g, y) e h i . " 0 h ·i = ," t N ! ZH Z N !1 By the symmetry of N,", we may replace the imaginary exponential in AN,"(g, y) by sin(t b (g), y ). Thus, A ,"(g, y) is uniformly bounded in N," for g ﬁxed and h i N b (g), y 0. In particular, by the dominated convergence theorem, h i6= dt it b (g),y lim AN,"(g, y) y, d (y) p.v. e h i y, d (y). " 0 h ·i = t h ·i N ! ZH ZH ✓ ZR ◆ !1 Now, since f c(G), we obtain 2 C i dt it b (g),y f b f (g) p.v. e h i y, d (y) o (g) d⌫(g) R = p2⇡ G t h ·i Z ZH ✓ ZR ◆ i dt b it b (g), f (g)Q p.v. e h ·i o (g) d⌫(g) = p ⇡ t 2 ZG ✓ ZR ◆ i dt b it b (g), Q f (g) p.v. e h ·i o (g) d⌫(g) . = p ⇡ t 2 ZG ZR This reduces the proofb ofb (A.2) to showing that the term in square brackets is J (f ). U This follows by applying Fubini, which in turn can be justiﬁed as above. Note that it b (g), tD e h ·i o (g) e J ((g)) follows from D idL ( , ) o and the fact = = 1 H that is a derivation because ( ,↵ ,b ) is a cocycle. (A.3) follows from the fact that H (exp(tD ))t is a one-parameter group of isometries of Lp( ). Indeed, each of the 2R M maps exp(tD ) is a -automorphism since ( ,↵ ,b ) is a cocycle. On the other hand, ⇤ H both exp(tD ) and exp( tD ) are trace preserving, hence isometries of L ( ). By in- 1 M terpolation, we obtain a one-parameter group of isometries of Lp( ) for 1 p . M  1 According to [5, Theorems 5.12 and 5.16], we deduce (A.3) as a consequence of the fact that Lp( ) is UMD for 1

1 3 f f . ,'j j .c(p,dim ) Lp(G) (A.4) ⌦ RCp( (G)) H k k j 1 L X= b The problem here is again that we use the map

⇡ (G) (g) (b (g), g) ( o G) ( ) o G : L 3 7! o 2 L H ' L H n (where is some R and G is discrete) in a crucial way for [34, Theorem 4.3]. H disc Since the argument used above for Theorem A2 seems to be very speciﬁc to the Riesz transform, we need an alternative approach. The strategy is to use the fact that (A.4) holds for arbitrary discrete groups, as proved in [34]. In particular, if we let Gdisc denote the group G equipped with the discrete topology and consider the linear map

1 L f 3 ,' f j , = j ⌦ j 1 X= our goal is to prove the following inequality for 1

Lp(0k) Lp(G ) ⇢ disc isometrically for all k 1. In particular, this immediately yields b b sup L Lp(0k) RCp( (0k)) L Lp(Gdisc) RCp( (Gdisc)) , k 1 k |0k : ! L kk : ! L k b sup L ⇤ RCpb( (0k)) Lp(0k) L ⇤ RCpb( (Gdisc)) Lp(Gdisc) . k 1 k |0k : L ! kk : L ! k Indeed, bothb inequalities are obviousb for p>2, and the case 1

L Lp(G) RCp( (G)) sup L Lp(0k) RCp( (0k)) , k : ! L kk 1 k |0k : ! L k b L ⇤ RCpb( (G)) Lp(G) sup L ⇤ RCpb( (0k)) Lp(0k) . k : L ! kk 1 k |0k : L ! k This is a Hilbert space valued formb of the noncommutativeb extension of Igari’sb lattice approximation theorem [13, Theorem C]. Following the notation used in the proof of [13, Theorem C], deﬁne

p p p p0 S 8 3 ,' 9 where 9 (8 )⇤. jk k j 0 k k k = | k = p p It is worth mentioning that although the maps 8k and 9k depend on p, the operator Sjk does not. This will be relevant below. We shall also use the operators

1 1 1 A ,kf S f j and B ,k fj j S fj . = jk ⌦ ⌦ = jk j 1 j 1 j 1 X= ⇣X= ⌘ X= p Since 8 Lp(0k) Lp(G) is a complete contraction [13], we deduce k : ! A ,k Lp(G) RCp( (G)) sup L Lp(0k) RCp( (0k)) , b b 0 k : ! L kk 1 k | k : ! L k b B ,k RCpb( (G)) Lp(G) sup L ⇤ RCpb( (0k)) Lp(0k) . k : L ! kk 1 k |0k : L ! k Therefore, the assertion will followb if we can showb that b

L f w-RCp( (G))- lim A ,kf, L ⇤ f w-Lp(G)- lim B ,kf, = L k = k !1 !1 on a dense class in Lp(G), RCp( (G)) respectively. To justify theb first limit, we argue as L in [13, Theorem C] and reduce it to proving strong L -convergence for f (f)with 2 = f c(G) c(G). Indeed,b if (qj )j 1 is a sequence of projections in `r (Lr (G)) with 2 C ⇤ C 1/p 1/2 1/r, then b = + b b 1 1 q (S f 3 f) q (S f 3 f) j jk ,'j j j jk ,'j Lp(G) ⌦ RCp( (G))  k k j 1 L j 1 X= X= b 1/r 1/2 1 r 1 2 qj Sjkf 3 ,'j f .  k kLr (G) k kL2(G) j 1 j 1 ⇣X= ⌘ ⇣X= ⌘ b b Since f is compactly supported, only finitely many elements in (Sjkf 3 ,'j f)j 1 do not vanish. As in [13, Theorem C], this implies that strong L2-convergence implies weak Lp-convergence.b On the other hand, since only finitely many j’s give nonzero terms, it suffices to show that

L2- lim S f 3 ,' f (A.6) k jk = j !1 for each j 1 and every f ( c(G) c(G)). The proof of this is exactly the same as 2 C ⇤ C in [13, Theorem C] and we shall not reproduce it here. This justiﬁes the ﬁrst limit. Riesz transforms and Fourier multipliers 579

The second one is very similar. Again we may reduce it to proving strong L2- convergence, this time with the exact same argument as in [13]. Moreover, we may pick f fj j with fj 0 for ﬁnitely many j’s. Then the problem reduces once more = j ⌦ 6= to justifying (A.6) as indicated above. P ut Finally, we conclude by analyzing Theorem B2. In this case, the proof is completely parallel to that of Theorem B1 with the only difference that we use Lemma 2.3 instead of Lemma 2.2. The Littlewood–Paley estimate used in that lemma follows from [32] and holds for any semiﬁnite von Neumann algebra . In particular, it holds with (G) M M = L for every unimodular group G.

Appendix B. Operator-algebraic tools

Along this paper we have used some concepts from noncommutative integration which include noncommutative Lp-spaces and sums of independent noncommuting random variables. In the context of group von Neumann algebras, we have also used crossed products, length functions and cocycles. In this section we brieﬂy review these notions for the read- ers who are not familiar with them. Noncommutative integration. Part of von Neumann algebra theory has evolved as the noncommutative form of measure theory and integration. A von Neumann algebra [41, 72] is a unital weak-operator closed C⇤-algebra; and, according to the Gelfand–Naimark– Segal theorem, any such algebra embeds in the algebra ( ) of bounded linear oper- M B H ators on a Hilbert space . We write 1 for the unit. The positive cone is the set of H M M+ positive operators in , and a trace ⌧ 0, is a linear map satisfying M : M+ ![ 1] ⌧(f ⇤f) ⌧(ff ⇤). = It is normal if sup↵ ⌧(f↵) ⌧(sup↵ f↵) for bounded increasing nets (f↵) in ; it is = M+ semiﬁnite if for any nonzero f there exists 0

p p p 1/p We have f f + ⌧( f )< . If we set f p ⌧( f ) , we 2 SM )| | 2 S ) | | 1 k k = | | obtain a norm in for 1 p

1 p 1/p 1/2 1/2 fj fj p E(fj⇤fj ) E(fj fj⇤) . p p ⇠ k k + p + p j ⇣j ⌘ ⇣j ⌘ ⇣j ⌘ X2J X2J X2J X2J (B.1) Group von Neumann algebras. Let G be a discrete group with left regular representation G (` (G)) given by (g)h gh, where the g’s form the unit vector basis : ! B 2 = of ` (G). Write (G) for its group von Neumann algebra, the weak operator closure of 2 L the linear span of (G) in (`2(G)). Consider the standard trace ⌧((g)) g e where e B = = denotes the identity of G. Any f (G) has a Fourier series expansion of the form 2 L f (g)(g) with ⌧(f ) f (e). = g G X2 b b Deﬁne

p 1/p Lp(G) Lp( (G), ⌧) closure of (G) in the norm f (⌧ f ) , = L ⌘ L k kLp(G) = | | the natural Lp-space over the noncommutative measure space ( (Gb), ⌧). Note that when b L G is abelian we get the Lp-space on the dual group equipped with its normalized Haar Riesz transforms and Fourier multipliers 581 measure, after identifying (g) with the character g. The group von Neumann algebra (G) associated to a locally compact unimodular group G is deﬁned similarly; L we refer to Appendix A above for the details. Let G be a locally compact unimodular group. Given another noncommutative measure space ( ,⌫) with ( ) assume there exists a trace preserving continuous M M ⇢ B H action ↵ G Aut( ). Deﬁne the crossed product algebra o↵ G as the weak oper- : ! M M ator closure of the -algebra generated by 1 (G) and ⇢( ) in (L2(G )). The ⇤ M ⌦ M B ; H -representation ⇢ (L (G )) is determined by the identity ⇤ : M ! B 2 ; H

⇢(x) (')(g) ↵ 1 (x)('(g)). [ ] := g When G is discrete, the operator ⇢(x) takes the form

⇢(x) ↵ 1 (x) eh,h = h ⌦ h G X2 with eg,h the matrix units in (`2(G)). A generic element of o↵ G has the form B M fg o↵ (g) with fg . By playing with and ⇢, it is easy to see that o↵ G sits g 2 M M in (` (G)): PM ⌦¯ B 2

fg o↵ (g) ⇢(fg)(1 (g)) (↵h 1 (fg) eh,h)(1 egh ,h ) M M 0 0 g = g ⌦ = g,h,h ⌦ ⌦ X X X0 ↵ 1 (fg) e 1 ↵ 1 (f 1 ) eg,h. h h,g h g gh = g,h ⌦ = g,h ⌦ X X When G is unimodular, the expression for ⇢(x) is replaced by a direct integral with respect to the Haar measure on G, and a generic element in o↵ G has the form fg o M G (g) dµ(g). Similar computations lead to the following formulae for the basic operations R in the crossed product algebra: 1 (f o↵ (g))⇤ ↵ 1 (f ⇤) o↵ (g ), • = g (f o↵ (g))(f 0 o↵ (g0)) f↵g(f 0) o↵ (gg0), • = Moreover, if ⌫ denotes the trace in we consider the trace M

⌫ o↵ ⌧ fg o↵ (g) dµ(g) ⌫(fe). = ✓ZG ◆ Since ↵ will be ﬁxed, we relax the notation and write fg o (g) dµ(g) o G. G 2 M Let us now consider semigroups ( ,t )t 0 of operators on (G) which act S = S R L t (g) diagonally on the trigonometric system. In other words, ,t (g) e (g) for S : 7! some function G R . The semigroup ( ,t )t 0 deﬁnes a noncommutative : ! + S = S Markov semigroup when:

(i) ,t (1 (G)) 1 (G) for all t 0. S L = L (ii) Each ,t is normal and completely positive on (G). S L (iii) Each ,t is self-adjoint, i.e. ⌧(( ,t f) g) ⌧(f ( ,t g)) for f, g (G). S S ⇤ = ⇤ S 2 L (iv) ,t f f as t 0 in the weak- topology of (G). S ! ! + ⇤ L 582 Marius Junge et al.

These conditions are reminiscent of Stein’s notion of diffusion semigroup [68]. They imply that ,t is completely contractive, trace preserving and also extends to a semigroup S of contractions on Lp( (G)) for any 1 p . As in the classical case, always L  1 S admits an inﬁnitesimal generator

,t id (G) A lim S L with ,t exp( tA ). = t 0 t S = ! In the L2 setting, A is an unbounded operator deﬁned on

,t f f dom2(A ) f L2(G) lim S L2(G) . = 2 t 0 t 2 ⇢ ! b b As an operator in L2(G), A is positive and so we may define the subordinated Poisson semigroup ( ,t )t 0 by ,t exp( t A ). This is again a Markov semigroup. P = P P2 = Note that Pt is chosenb so that (@ A ) ,t 0. In general, we let A ,p denote the t P p= generator of the realization of ( ,t )t 0 on Lp( (G)). It should be noticed that S = S L ker A ,p is a complemented subspace of Lp( (G)). Let Ep denote the corresponding L projection and Jp id Ep. Consider the complemented subspaces = Lp(G) b L (G) Jp(Lp(G)) f Lp(G) lim ,t f 0 . p = = 2 t S = n !1 o The associated gradientb form or “carrb e´ du champs”b is defined as 1 0 (f ,f ) A (f ⇤)f f ⇤A (f ) A (f ⇤f ) . 1 2 = 2 1 2 + 1 2 1 2 Since ( ,t )t 0 is a Fourier multiplier, we get A ((g)) (g)(g) and S = S = 1 1 (g ) (h) (g h) 1 0 (f1,f2) f1(g)f2(h) + (g h) dµ(g) dµ(h). = G G 2 Z ⇥ The crucial condition 0b (f, fb ) 0 is characterized in the following subsection. Length functions and cocycles. A left cocycle ( , ↵, b) for the unimodular group G is a H triple given by a Hilbert space , a continuous isometric action ↵ G Aut( ) and a H : ! H continuous map b G such that : ! H ↵g(b(h)) b(gh) b(g). = 1 1 A right cocycle satisfies the relation ↵g(b(h)) b(hg ) b(g ) instead. In this paper, = we say that G R is a length function if it vanishes at the identity e, (g) 1 : ! + = (g ) and 1 g 0 gh (g h) 0 g = ) g,h  X X for any finite family of coefficients g. Functions satisfying the last condition are called conditionally negative. It is straightforward to show that length functions take values in R . In what follows, we only consider cocycles with values in real Hilbert spaces. Any + Riesz transforms and Fourier multipliers 583 cocycle ( , ↵, b) gives rise to an associated length function b(g) b(g), b(g) , as H =h iH can be checked by the reader. Conversely, any length function gives rise to a left and a right cocycle. This is a standard application of the ideas around Schoenberg’s theorem [64], which states that G R is a length function if and only if the associated : ! + semigroup ( ,t )t 0 given by S ,t (g) exp( t (g))(g) is Markovian S = S : 7! on (G). Let us collect these well-known results. L Lemma B.1. If G R is a continuous length, then: : ! + (i) The Gromov forms (g) (h) (g 1h) (g) (h) (gh 1) K1 (g, h) + ,K2 (g, h) + = 2 = 2 define positive matrices on G G and lead to ⇥ j f1,f2 ,j f1(g)K (g, h)f2(h) h i = g,h X b b on the group subalgebra R G of (G) given by [ ] L R G (f) f G R finitely supported . [ ]={ | : ! } j (ii) Let be the Hilbert space completion of H b b j j (R G /N , , ,j ) with N null space of , ,j . [ ] h· ·i = h· ·i j j j The mappings b G g (g) (e) N form left/right cocycles : 3 7! + 2 H together with

1 1 1 2 2 1 2 ↵ (u N ) (g)u N ,↵(u N ) u(g ) N , ,g + = + ,g + = + j j j which determine isometric actions ↵ G Aut( ) of G on . j : ! H H j (iii) If is endowed with the discrete topology, then the semidirect product G jH = o G becomes a unimodular group and we ﬁnd the group homomorphisms H 1 1 1 2 2 1 2 ⇡ G g b (g) o g G ,⇡G g b (g ) o g G . : 3 7! 2 : 3 7! 2 The lemma allows one to introduce two pseudo-metrics on the unimodular group G in terms of the length function . Indeed, a short calculation leads to the crucial identities

1 1 1 1 1 1 1 2 (g h) b (g) b (h), b (g) b (h) ,1 b (g) b (h) 1 , =h i =k kH 1 2 2 2 2 2 2 2 (gh ) b (g) b (h), b (g) b (h) ,2 b (g) b (h) 2 . =h i =k kH In particular, 1 1 1 dist1(g, h) (g h) b (g) b (h) 1 = =k kH p 584 Marius Junge et al. deﬁnes a pseudo-metric on G, which becomes a metric when the cocycle map is injective. Similarly, we may work with dist (g, h) (gh 1). When the cocycle map is not 2 = injective, the inverse image of 0, p G g G (g) 0 , 0 ={ 2 | = } is a subgroup. The following elementary observation is relevant.

Remark B.2. Let ( ,↵ ,b ) and ( ,↵ ,b ) be a left and a right cocycle on G. H1 1 1 H2 2 2 Assume that the associated length functions b1 and b2 coincide. Then we find an isometric isomorphism 1 3 b (g) b (g ) . 12 : H1 3 1 7! 2 2 H2 In particular, given a length function we see that 1 2 via b1 (g) b2 (g 1). H ' H 7! Remark B.3. According to Schoenberg’s theorem, Markov semigroups of Fourier multipliers in (G) are in one-to-one correspondence with conditionally negative length func- L tions G R . Lemma B.1 automatically gives : ! + 1 0 (f, f ) A (f ⇤)f f ⇤A (f ) A (f ⇤f) = 2 + 1 f (g)f (h)K (g, h)(g h) dµ(g) dµ(h) 0. = G G Z ⇥ Theorem B.4. Let 5 denoteb the spaceb of trigonometric polynomials in (G) whose 0 L Fourier coefficients have vanishing sum, as defined in the Introduction. A given function G R defines a conditionally negative length if and only if there exists a positive : ! + linear functional ⌧ 50 C satisfying the identity : ! 1 (g) ⌧ 2(e) (g) (g ) . = Proof. Assume first that G R satisfies the given identity for some positive linear : ! + functional ⌧ 50 C. To show that is a conditionally negative length it suffices to : ! construct a cocycle ( ,↵ ,b ) so that (g) b (g), b (g) . Since 50 is a - H =h iH ⇤ subalgebra of (G), f1,f2 ⌧ (f ⇤f2) is well-defined on 50 50. If we quotient L h iH = 1 ⇥ out the null space of this bracket, we may define as the completion of such a quotient. H As usual, we interpret as a real Hilbert space by decomposing every element in 5 H 0 into its real and imaginary parts. If N denotes the null space, let

↵ ,g(f N ) (g)f N and b (g) (g) (e) N . + = + = + It is easily checked that ( ,↵ ,b ) defines a left cocycle on G. Moreover, since H 2(e) (g) (g 1) (g) (e) 2, our assumption can be rewritten in the form =| | (g) b (g), b (g) as expected. =h iH Let us now prove the converse. Assume that G R defines a conditionally : ! + negative length and define

1 ⌧ ((g) (e)) (g). =2 Riesz transforms and Fourier multipliers 585

Since the polynomials (g) (e) span 50, ⌧ extends to a linear functional on 50 1 1 1 which satisﬁes ⌧ (2(e) (g) (g ) ( (g) (g )) (g). Therefore, it = 2 + = just remains to show that ⌧ 50 C is positive. Let f ag(g) 50 so that : ! = g 2 ag 0. In particular, we also have f ag((g) (e)). By the conditional g = = g negativity of we ﬁnd P P P 2 ⌧ ( f ) agah⌧ ((g) (e))⇤((h) (e)) | | = g,h G X2 1 1 agah⌧ (g h) (g ) (h) (e) = + g,h G X2 1 1 (g ) (h) (g h) 1 1 agah + agah (g h) 0. = = g,h G 2 2 g,h G X2 X2 This shows that our functional ⌧ 50 C is positive. : ! ut

Appendix C. A geometric perspective

In this appendix we will describe tangent modules associated with a given length function, and how they can be combined with Riesz transform estimates. Recall that a Hilbert module over an algebra is a vector space X with a bilinear map m X (⇢, a) A : ⇥ A 3 7! ⇢a X and a sesquilinear form , X X such that ⇢,⌘a ⇢,⌘ a, 2 h i: ⇥ ! A h i=h i ⇢a,⌘ a ⇢,⌘ and ⇢,⇢ 0. We refer to Lance’s book [42] for more information. h i= ⇤h i h i Deﬁne (X) as the C -algebra of right-module maps T which admit an adjoint. That is, L ⇤ there exists a linear map S X X such that S⇢, ⌘ ⇢,T⌘ .AHilbert bimodule : ! h i=h i is additionally equipped with a -homomorphism ⇡ (X), and a derivation ⇤ : A ! L X is a linear map which satisﬁes the Leibniz rule : A ! (ab) ⇡(a)(b) (a)b. = + A typical example for such a derivation is given by an inclusion into some von A ⇢ M Neumann algebra , a conditional expectation E , and a vector ⇢ such that M : M ! A00 (a) a⇢ ⇢a. We have seen above that for a conditionally negative length function , = we can construct an associated (left) cocycle ( ,↵ ,b ). In the following, we will H assume that the R-linear span of b (G) is . H We recall the Brownian functor B L () which comes with an extended : H ! 2 action ↵ of G on L (). This construction is usually called the gaussian measure space construction. Here1 the derivation is given by

(g) B(b (g)) o (g). : 7! We have already encountered the corresponding bimodule in the context of the Khintchine inequality. The proof of the following lemma is obvious. In fact, here left and right actions are induced by the actions on L () o G. M = 1 586 Marius Junge et al.

Lemma C.1. Let G be discrete. The Hilbert bimodule

(G) (C G )(1 o (G)) with ⇢,⌘ E (G)(⇢⇤⌘) = [ ] h i= L is exactly given by the vector space X B(⇠g) o (g) ⇠g . ={ g | 2 H } 1 Proof. For ⇠ b (g) we consider ((g))(P1 o (g )) B(⇠). Since is the real = = H linear span of such ⇠’s, we deduce that X is contained in (C G )(1 o (G)). The [ ] converse is obvious. Moreover, since is a derivation, it is easy to see that the space (C G )(1 o (G)) is invariant under the left action. [ ] ut Of course, the gaussian functor B is not really necessary to describe the bimodule (G) o G. Note that the product of two elements ⇢,⌘ in L2(L () o G) ' H 1 is well-defined as an element of L1(L () o G), so E (G)(⇢⇤⌘) ⇢,⌘ makes perfect 1 L =h i sense. The following proposition shows that our previous results extend to the tangent module and not only to differential forms with ‘constant’ coefficients given by elements in (G). Given ⇢ B(⇠h) o (h) (G), define the extended Riesz H ⇢ = h G 2 transform in the direction of ⇢ as follows:2 P b (g), ⇠ 1 h 1 R ,⇢f (h )R ,⇠ f 2⇡i h iH f (g)(h g) = h = p (g) h G g,h G X2 X2 b ⇤ f (g) 2⇡iE (G) B(⇠h) o (h) B(b (g)) o (g) = L p (g)  h G ✓g G ◆ ⇣X2 ⌘ X2 b 1/2 2⇡iE (G)(⇢⇤ A f) (C.1) = L since

B(⇠)B(⇠ 0)dµ ⇠,⇠0 . =h iH Z Note that we recover the Riesz transforms R ,h for ⇢ B(h) o (e). =

Proposition C.2. Given ⇢,⇢j (G), and 2 p< : 2  1 (i) R ,⇢ Lp(G) Lp(G) c(p) ⇢ ( ), k : ! k . k k G 1/2 (f ) 2 ( ⇢ ,⇢ ) 1/2 f (ii) R ,⇢jb b .c(p) j k (` ) (G) p, | | p k h i kB 2 ⌦¯ L k k ⇣j 1 ⌘ X 1/2 1/2 2 1/2 2 (iii) R ,⇢j (fj ) .c(p) sup E (G)(⇢j⇤⇢j ) (G) fj . | | p j 1 k L kL | | p ⇣j 1 ⌘ ⇣j 1 ⌘ X X Proof. Assertion (i) follows from (ii) or (iii). The second and third assertions follow from the well-known Cauchy–Schwarz inequality for conditional expectations, whose noncommutative form follows from Hilbert module theory [30]:

1/2 1/2 E (G) id (` )(xy) p E (G) id (` )(xx⇤) (E (G) id (` )(y⇤y)) p. k L ⌦ B 2 k k L ⌦ B 2 k1 k L ⌦ B 2 k Riesz transforms and Fourier multipliers 587

Indeed, for (ii) we use (C.1) and pick 1/2 x ⇢⇤ ej and y A f e . = j ⌦ 1 = ⌦ 11 j 1 X Now, the inequality follows from Theorem A2. On the other hand, for (iii) we take 1/2 x ⇢⇤ ejj and y A fj ej . = j ⌦ = ⌦ 1 j 1 j 1 X X Then the result follows from the cb-extension of Theorem A2 in Remark 1.8. ut Remark C.3. The ‘adjoint’ of R ,⇢ given by † R (f ) R ,⇢(f ⇤)⇤ ,⇢ = † 1/2 1/2 can be written as R ,⇢(f ) E (G)(⇢⇤ A (f ⇤))⇤ E (G)( A (f )⇢). Hence = L = L 1/2 † (f ) 2 E (⇢ ⇢ ) 1/2 f 2 . R ,⇢j j ⇤ .c(p) sup (G) j⇤ j (G) j⇤ | | p j 1 k L kL | | p ⇣j 1 ⌘ ⇣j 1 ⌘ X X † It is however, in general, difficult to find an element ⌘ such that R (f ) R ,⇢(f ) ,⌘ = unless G is commutative and the action is trivial. This is a particular challenge if we want to extend the results from above literally to p<2, because then we need both a row and a column bound to accommodate the decomposition R(f ) a b in the tangent = + module (G). Let us now indicate how to construct the corresponding real spectral triple. We first recall that (G) is a quotient of the universal object (G) C G C G spanned • ⇢ [ ]⌦ [ ] by (a)b, where the universal derivation is (a) a 1 1 a. In view of J (a b) • • = ⌦ ⌦ ⌦ b a , the universal object becomes a real bimodule. In other words, the left and = ⇤ ⌦ ⇤ right representations ⇡`(a)(⇢) (a 1)⇢ and ⇡r (a)(⇢) ⇢(1 a) are related via = ⌦ = ⌦ J⇡r (a )J ⇡`(a). This implies in particular that a,JbJ 0, and hence we find a ⇤ = [ ]= real spectral triple. In our concrete situation, we have a natural isometry J (x) x on = ⇤ M L () o G, which leaves the subspace (G) L2(M) invariant. Hence (G) = 1 ⇢ is a quotient of (G). The Dirac operator for this spectral triple is easy to construct. The underlying Hilbert• space is (G) L (G) H = 2 where (G) denotes the closure of (G) in L2(M), and b 0 D . = ⇤ 0 ✓ ◆ Note that ⇤ (B(⇠) o (g)) ⇠,b (g) (g) is densely defined. Using the diagonal =h i representation for C G , we find that Tg , (g) is a right module map from L2(G) [ ] =[ ] to (G) such that Tg(x) b (g) o ((g)x), and hence = 1 b x,T ⇤Tgy b (h), b (g) x, (h g)y (x0 ((h), (g))y) h h i=h ih i= 588 Marius Junge et al. for all x,y L (G). It then follows that 2 2 ⇡(f ) 0 1/2 1/2 D , b max 0 (f, f ) , 0 (f ⇤,f⇤) . 0 f = {k k (G) k k (G)}  ✓ ◆ ( ) L L B H Here ⇡((g))((h)) ((g)(h)) ((g))(h). This gives precisely the Lip-norm = considered in [33]. The drawback, however, is that we have replaced the natural candidate L2(L () o G) by the much ‘smaller’ module . 1 H Replacing the gaussian construction by the corresponding free analogue, it is possible to work with a larger object. As in the gaussian category, given any Hilbert space , we K have a function s 0 ( ) : K ! 0 K into the von Neumann algebra generated by free semicircular random variables and a representation ↵ O( ) Aut(00( )) with s(o(h)) ↵o(s(h)). This allows us to define : K ! K = ((g)) s(b (g)) o (g) 00( ) o G. The boundedness of the corresponding free = 2 H Riesz transforms 1/2 f A f 7! free follows from the corresponding Khintchine inequality for x s(⇠)o(g). Namely, = ⇠,g we find 1/2 P 1/2 x p C max E (G)(x⇤x) p, E (G)(xx⇤) p k k ⇠ {k L k k L k } for 2 p< , and  1 x (x x )1/2 (x x )1/2 . p0 C inf E (G) 1⇤ 1 p E (G) 2 2⇤ p k k ⇠ x x1 x2 k L k +k L k = + In fact, it turns out that 2 2 E (G)( xfree ) E (G)( xgauss ) L | | = L | | for xfree s(⇠) o (g) and xgauss B(⇠) o (g). This means the bimodule = ⇠,g = ⇠,g X can be realized either with independent gaussian or free semicircular variables, where free P P X s( ) o G 00( ) o G. Recall the natural inclusion of 00( ) o G into the = H ⇢ H H Hilbert space L2(00( ) o G) L2(00( )) `2(G). More formally, we may denote H ' H ⌦ by 1⌧ the separating vector in the GNS construction and then find (g)1⌧ eg. The map = D is densely defined on L2(00( ) o G) as follows: free H 1 D (a eg) s(b (g ))a eg. free ⌦ = ⌦ Proposition C.4. The tuple

(C G ,L2(00( ) o G), D ,J) [ ] H free is a real spectral triple satisfying the following identities for f f (g)(g) in = g C G 00( ) o G: [ ]⇢ H P b D , (g) s(b (g)) o (g), [ free ]= D ,f 0 (f, f ) 1/2 , 0 (f ,f ) 1/2 . free 00( )oG max (G) ⇤ ⇤ (G) k[ ]k H ⇠ {k kL k kL } Riesz transforms and Fourier multipliers 589

Proof. If g G, we ﬁnd 2 1 D , (g) (a eh) D (a egh) (g)( s(b (h )a eh) [ free ] ⌦ = free ⌦ ⌦ 1 1 s(b ((gh) ))(a egh) s(b (h ))(a egh) = ⌦ + ⌦ 1 ↵ (s(b (g)))a egh ⇢(s(b (g))) (g)(a eh), = ,gh ⌦ = ⌦ 1 where ⇢(b)(c eg) ↵ (b)c eg on the tensor product. After the correspond- ⌦ = ,g ⌦ ing identiﬁcations in the inclusion 00( ) (G) L2(00( ) o G), this implies H ⌦ L ⇢ H D , (g) ((g)). The operation J is the adjoint for the crossed product, and [ free ]= free hence (g), J ((h))J 0 shows that we have obtained a real spectral triple (we ignore [ ]= further compatibility properties for Dfree,J at this point). By linearity we deduce that D ,f (f ). [ free ]= free m Now, we use a central limit procedure. Consider the crossed product 00(`2 (H )) o G. Then the copies ⇡j (00(H ) o G) given by the j-th coordinate are freely independent over (G). Thus Voiculescu’s inequality from [31] applies and yields, for any ! L = ⇠,g a⇠,gs(⇠) o (g) and the sum of independent copies,

P m a⇠,gs(⇠ ej ) o (g) ⌦ 00oG j 1 X= p 1/2 p 1/2 ! 00oG m E (G)(!⇤!) (G) m E (G)(!!⇤) (G). k k + k L kL + k L kL 1/2 m Dividing by pm and observing that ! and m a⇠,gs(⇠ ej ) o (g) are equal in j 1 ⌦ distribution, we ﬁnd indeed, letting m , = !1 P 1/2 1/2 ! 00oG max E (G)(!⇤!) (G), E (G)(!!⇤) (G) . k k ⇠ {k L kL k L kL } Thus for a differential form ! Xfree(G), we get 2

! free max ! X , !⇤ X . k kX ⇠ {k k k k } In particular, we conclude that

1/2 1/2 free(f ) 00oG max 0 (f, f ) (G), 0 (f ⇤,f⇤) (G) . k k ⇠ {k kL k kL } This expression is certainly ﬁnite for f C G , and the proof is complete. 2 [ ] ut

It turns out that in the free case Dfree cannot be extended to a global derivation on 00( ) o G. On the other hand, for the gaussian case ((g)) B(b (g)) o (g) H = does not belong to L () o G, and hence both models for generalized tangent spaces have their advantages1 and disadvantages. Let us now return to the gaussian spectral triple on . As in [15], we have to deal with H the fact that this spectral triple might have some degenerate parts, but in many calculations 590 Marius Junge et al. of the ⇣-function of D the kernel is usually ignored. We recall that for a self-adjoint | | 1 operator D, the signature is deﬁned as sgn(D) D D . In our particular case, if 1 = | | 1 A (eg) (1/ (g))eg is a compact operator and F D D is the corresponding = = | | signature, then it is well-known [15] that ⇡(a) 0 F, 0 a  ✓ ◆ is compact for all a C G . This follows from the boundedness of 2 [ ] ⇡(a) 0 D , 0 a  ✓ ◆ [15, Proposition 2.4], and then [15, Proposition 2.7] applies. In our situation, 1/2 = R A , and hence vanishes on span eg (g) 0 . Clearly, A is H0 = { | = } ⇤ = the generator of our semigroup which also vanishes on . On the other hand, H0 ⇤ R A R⇤ , = and hence the range of is given by the ﬁrst Hodge projection 5Hdg R R . This ⇤ = ⇤ can be described explicitly. Indeed, for g with (g) 0 we denote by Qg the projection 6= onto the span of B(b (g)) L () and get 2 2 0 R R R Q e F . ⇤ g gg R 0 = g ⌦ ) = ⇤ X ✓ ◆ Problem C.5. Show that F (G) (G) Lp(L () o G), where the closure is : + L ! 1 taken in Lp, admits dimension free estimates.

Appendix D. Meyer’s problem for Poisson

n Let 1 @2 be the laplacian operator on R . The classical theory of semigroups of op- = x erators shows that the fractional laplacians ( 1) with 0 <<1 are closed densely n deﬁned operators on Lp(R ) [74, Chapter 9, Section 11], and regarding them as convolution operators we see that the Schwartz class lies in the domain of any of them. Moreover, n they generate Markov semigroups on L (R ). When 1/2, we get the Poisson semi- 1 = groups Pt exp( tp 1). = In this appendix we shall show that Meyer’s problem (MP) fails for this generator when p 2n/(n 1). Recall that Theorem A2 conﬁrms that (MP) holds for p 2 and  + provides a substitute for 1

1 2 0 / (f, f ) Pt Pt f dt 1 2 = |r | Z0 where g(x, t) (@x g,...,@x g,@t g) includes spatial and time variables. r = 1 n Riesz transforms and Fourier multipliers 591

2 2 Proof. Let 't Pt f and Ft (@t Pt )('t ) Pt (@t 't ). Since @ Pt 1Pt 0, =| | = t + = 2 2 2 @t Ft (@ Pt )('t ) Pt (@ 't ) 1Pt ('t ) Pt (@ 't ). = t t = t On the other hand, we may calculate

2 2 2 2 @ 't 2 @t Pt f (Pt f ⇤)(@ Pt f) (@ Pt f ⇤)(Pt f) t = | | + t + t 2 2 @t Pt f (Pt f ⇤)(1Pt f) (1Pt f ⇤)(Pt f ). = | | Therefore, we get 2 @t Ft 2Pt ( Pt f ). = |r | Note that F0 limt 0 Ft 201/2(f, f ) by the deﬁnition of carre´ du champ, and Ft 0 = ! = ! as t . We get !1

1 1 2 20 / (f, f ) @t Ft dt 2 Pt Pt f dt. 1 2 = = |r | ut Z0 Z0 tp 1 Proposition D.2. The equivalence (MP) fails for the Poisson semigroup Pt e n = on R for any 1

1/2 n 2n 01/2(f, f ) / Lp(R ) for any p . 2  n 1 + n Proof. We follow an argument from [25]. Fix a nonzero f Lp(R ) and x > 4. Then 2 | |

1 2 0 / (f, f )(x) Pt ( Pt f )(x) dt 1 2 = |r | Z0 1 t 2 cn Pt f (y) dy dt = n ( x y 2 t2)(n 1)/2 |r | Z0 ZR | | + + 2 1 1 c P f (y) 2 dy dt c c n n 1 t n f n 1 1 y <1 x + |r | = x + Z Z| | | | | | 2 2 for cf 1 y <1 Pt f (y) dy dt > 0 (since f 0) and any x > 4. Then = | | |r | 6= | | R R 1/2 n p 2n 01/2(f, f ) Lp(R ) (n 1)>n p> . 2 ) 2 + ) n 1 + We then conclude that

1/2 1/4 0 / (f, f ) p while ( 1) f p < k 1 2 k =1 k k 1 for any nonzero Schwartz function f with p 2n/(n 1). Therefore, (MP) fails.  + ut Thus, our revision of (MP) in this paper is needed even for commutative semigroups. 592 Marius Junge et al.

n Remark D.3. According to the proof of Proposition D.2, there exists f (R ) such 2 S 1/4 n n that ( 1) f Lp(R ) but 01/2(f, f ) E (R )(1/2f ⇤1/2f) does not belong to n 2 = L n Lp/2(R ). This should be compared with Theorem A2 for G R , which states that = n there is a decomposition 1/2f 1 2 with 1,2 Lp(L () oR ) and such = + 2 1 n n n that E (R )(1⇤1) and E (R )(22⇤) belong to Lp/2(R ). On the other hand, by [30, PropositionL 2.8] we knowL that

1/2 1/2 1/2 01/2(f, f ) p E ( n)(1⇤1) p E ( n)(2⇤2) p. k k k L R k +k L R k This implies that n E ( n)(2⇤2)/Lp/2(R ), L R 2 n even if we know that 2 Lp(L () oR ). We recover the known fact: for p/2 < 1, 2 1 2 E n ( ) n n n . (R ) 2⇤ 2 Lp/2(R ) ⇥ 2⇤ 2 Lp/2(L ()oR ) 2 Lp(L ()oR ) k L k k k 1 =k k 1 Acknowledgements. We are indebted to Anthony Carbery, Andreas Seeger and Jim Wright for interesting comments and references. We also want to thank the referee for a very careful reading of the paper and quite precise comments, including the proof of Theorem A2 presented in Appendix A. Junge is partially supported by the NSF DMS-1201886, Mei by the NSF DMS-1266042 and Parcet by the ERC StG-256997-CZOSQP and the Spanish grant MTM2010-16518. Junge and Parcet are also supported in part by ICMAT Severo Ochoa Grant SEV-2011-0087 (Spain).

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