Notes on Radial and Quasiradial Fourier Multipliers

NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS

ANDREAS SEEGER

This is an updated version of notes that were originally prepared for lectures at the Summer School on Nonlinear Analysis, Function Spaces and Applications (NAFSA 10), in Tˇreˇst’,Czech Republic. They are based on material in various joint papers ([1], [20], [26], [27], [34], [36], [37]). The notes could not have been written without the contribu- tions of my coauthors Jong-Guk Bak, Gustavo Garrig´os,Yaryong Heo, Sanghyuk Lee, Fedya Nazarov, and Keith Rogers. Any deﬁciencies and errors in these notes are my responsibility.

1. Fourier multipliers

Given a translation invariant operator T which is bounded from S(Rd) to some Lq(Rd) there is a tempered distribution K so that d q T f = K ∗ f for all f ∈ S(R ) (see [53]). We denote by Convp the space of all K ∈ S0(Rd) such that

kK ∗ fkq ≤ Ckfkp for all f ∈ S(Rd), and set

q kKkConvp = sup{kK ∗ fkq : f ∈ S, kfkp = 1} The convolution operators can also be written as a multiplier transfor- mation K ∗ f = F −1[mfb] where the Fourier transform of a Schwartz function in Rd is given by Z Ff(ξ) = fb(ξ) = f(y)e−ihy,ξidy and F −1[g](x) = (2π)−d R g(ξ)eihx,ξidξ is the inverse Fourier transform q of g. The space of Fourier transforms of distributions in Convp is q denoted by Mp and the Fourier transform is an isometric isomorphism q q q −1 q F : Convp → Mp , i.e. kmkMp = kF [m]kConvp . We also abbreviate

Date: July 1, 2018. 1991 Mathematics Subject Classiﬁcation. 42B15. Supported in part by an NSF grant. 1 2 ANDREAS SEEGER

p p Convp = Convp and Mp = Mp . Eﬀective characterizations of the spaces q q Convp or Mp are known only in a few cases, see [28], [53]: 1 • Conv1 is the space of ﬁnite Borel measures (with the total variation norm). q q • For 1 q, and Mp = Mq0 , p + p0 = q + q0 = 1.

Of particular interest are the radial Fourier multipliers on Rd m(ξ) = h(|ξ|) . We shall assume throughout that d ≥ 2. The radial multipliers cor- respond to operators commuting with translations and rotations, i.e. convolution operators with radial kernels, and also to spectral multiplier operators for the Laplace operator, via √ h( −∆)f = F −1[h(| · |)fb] . A great deal of research has been done on a class of radial model multipliers which are now called of Bochner-Riesz type. Given a smooth bump function φ supported in (−1/2, 1/2) we deﬁne 1−s (1.1) hδ(s) = φ( δ ) so that hδ(| · |) is supported on an annulus of width ∼ δ.

2d Conjecture I: For 1 < p < d+1 , one has −λ(p) (1.2) khδ(| · |)kMp . δ , λ(p) = d(1/p − 1/2) − 1/2

This is referred to as the Bochner-Riesz conjecture since a resolution would prove that the Bochner-Riesz means of the Fourier integral Z λ 1 |ξ|λ ihx,ξi R f(x) = d 1 − fb(ξ)e dξ . (2π) |ξ|≤t t converge to f in Lp(Rd) if λ > λ(p). Inequality (1.2) is known in two dimensions for the full range, and various partial results have been proved in higher dimensions (see [17],[13], [19], [5], [6], [61], [33], [8]). For the relationships of various related conjectures see [56], [57]. A somewhat more general problem involves classical function spaces 2 2 measuring regularity, in particular the L type Sobolev spaces Lα or NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 3

2 the Besov spaces Bα,p. The following estimate would be optimal within this class of spaces.

Conjecture II: Let h be supported in a compact subinterval J of (0, ∞). The estimate

(1.3) kh(| · |)k khk 2 , α = d(1/p − 1/2) Mp . Bα,p 2d holds for 1 ≤ p < d+1 . Partial results with the condition α > d(1/p − 1/2) and related square-function estimates are in [10], [12], [14], [34], and a proof of the 2 2(d+1) endpoint Bd(1/p−1/2),p bound, in the range 1 < p < d+3 can be found in [35]. Inequality (1.3) is still far from a characterization of radial multipliers (even those supported in (1/2, 2)). Consider the multiplier ht(s) = −its ∞ χ(s)e where χ ∈ Cc (1/2, 2). It is closely related to the solution operator for the wave equation. It is known (see [45]) that kht(|·|)kMp . (d−1)(1/p−1/2) α t for t > 1, 1 ≤ p ≤ 2, but kh k 2 t and thus the t Bα,q & optimal Mp bound cannot be derived from (1.3). An aﬃrmative answer to the following conjecture would close this gap.

2d Conjecture III: Let 1 < p < d+1 , and let h be supported in a compact subinterval J of (0, ∞). Then it −1 (1.4) kh(| · |)kMp ≈ F [h(| · |)] p (with the implicit constants depending on J)? Given that the weaker conjectures I and II are not completely re- solved, we may have, for Conjecture III, to settle for smaller ranges of 2(d−1) p. It has been shown to hold in [26] for the range d ≥ 4, p < d+1 , and no such result is currently known in dimensions 2 and 3. As in [26] the proof of this conjecture would likely lead to a simple characterization p 2d of all radial FL multipliers for p < d+1 , as in:

Conjecture IV: Let m = h(| · |) be a bounded radial function on Rd and d deﬁne the convolution operator Th on R by

Tdhf(ξ) = h(|ξ|)fb(ξ). 2d d Let 1 0 4 ANDREAS SEEGER

The resolution of this conjecture would imply that in the above range p p Th is bounded on L if and only if Th is bounded on the subspace Lrad of radial Lp functions. By Feﬀerman’s theorem [18] on the ball multiplier 2d (or some variant of it) this is clearly false for d+1 ≤ p ≤ 2. Similar 2d arguments show that the condition p < d+1 is necessary in the three previous conjectures.

Finally one can formulate a version of Conjecture IV for Lp → Lq estimates. Here we focus on the case q ≤ 2.

Conjecture V: Let m = h(| · |) be a radial function on Rd which is in 2 d 2d Lloc(R \{0}) and let Tdhf(ξ) = h(|ξ|)fb(ξ). Let 1 < p < d+1 , p ≤ q ≤ 2, d−1 0 d and q ≤ d+1 p . Let η be any nontrivial Schwartz function on R . Then p q Th is bounded from L to L if and only if

d/p sup t Th[η(t·)] q < ∞ t>0

d−1 0 2(d+1) 2d The condition q ≤ d+1 p is relevant for the range d+3 ≤ p < d+1 and cannot be dispensed with in general, by a Knapp example. One considers hδ(| · |) as in (1.1) and tests the operator√ on functions√ fδ, for which fbδ is a bump function supported in a δ × δ ×· · ·× δ rectangle addapted to the support of hδ(| · |).

These notes. In §2 we consider the case of general quasiradial multipli- 2 ers in Mp and discuss a characterization for the case when the under a decay assumption on the Fourier transform of the surface measure of Σρ. When ρ is the Minkowski functional of a set with smooth bound- q ary (then P = I) we examine further the M1 classes and express the condition F −1[m] ∈ Lq in terms of the one dimensional Fourier transform of h. This is done in §3. In §4 we discuss a sharp theorem from [20] on the Lp boundedness of convolution operators with radial kernels when acting on radial functions, and in §5 we prove the same result for radial multipliers acting on general functions. We shall focus on the case where the multipliers are supported in (1/2, 2). NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 5

2 2 2. L Fourier restriction theorems and quasiradial Mp multipliers

In this section we shall ﬁrst discuss a general version of the Stein- Tomas restriction theorem and then deduce as a consequence a straight- 2 forward characterization of radial and quasi radial multipliers in Mp , for the p-range of the L2 restriction theorem.

2.1. L2 Fourier restriction theorems. In the 1960’s Stein observed that the Fourier transform of an Lp function can, for a nontrivial range of p > 1 be restricted to some compact hypersurfaces with suitable curvature assumptions. An almost sharp theorem for the sphere was proved by Tomas [62], with an endpoint version due to Stein [51]. Further results are in Greenleaf [23] and many other papers. Here we present the rather general setup by Mockenhaupt [43] and by Mitsis [42], with the endpoint version in a joint paper with Bak [1]. We are given a compactly supported Borel measure µ satisfying the Fourier decay bound −β/2 (2.1) |µb(ξ)| . (1 + |ξ|) and the α-upper-regularity condition α (2.2) |µ(B(x, r))| . r for all balls B(x, r) with radius r.

Remark. For measures satisfying (2.2) the Hausdorff dimension of the support is at least α. For measures satisfying (2.1) the Hausdorff dimension of the support is at least β. The latter holds since for γ < β the γ-energy ZZ Z −γ γ−d 2 Iγ(µ) = |x − y| dµ(x)dµ(y) = C |ξ| |µb(ξ)| dξ is finite and for the Hausdorff dimension of supp(µ) we have dim(supp(µ)) = sup{γ : Iγ(µ) < ∞} (see [40], [64]). For this reason the number β/2 dimF (µ) = sup{β : sup(1 + |ξ|) |µb(ξ)| < ∞} ξ is often called the Fourier dimension of µ. Theorem 2.1. Let µ be as in (2.1), (2.2). Then Z 2 2 4(d − α) + 2β (2.3) |fb(ξ)| dµ kfk , p ≤ pcr = . . p 4(d − α) + β 6 ANDREAS SEEGER

The Lp norm on the right can be replaced by the smaller Lp,2 Lorentz- norm.

Proof. Let T denote the Fourier transform, as an operator mapping Lp to L2(dµ) (and as such a priori deﬁned for L1 functions). The proofs of Mockenhaupt and Mitsis, for the open range (1, pcr), follow Tomas’ argument in [62]. One ﬁrst observes that ∗ (2.4) T T f = f ∗ [µb(−·)] . Indeed, Z ∗ hT T f, gi = fb(ξ)gb(ξ)dµ(ξ) Z = fb(ξ)bg(−ξ)dµ(ξ) Z Z = g(y) fb(ξ)eihξ,yidµ(ξ) dy

d −1 = (2π) hF [fb dµ], gi = f ∗ [µb(−·)] . Thus p,q 2 p,q p0,q0 (2.5) F : L → L (dµ) ⇐⇒ µb∗ : L → L . P∞ Following Tomas split µb = j=0 µbj, where µb0 is supported in {|ξ| ≤ 2} and, for j ≥ 1 j−1 j+1 supp(µbj) ⊂ {2 ≤ |ξ| ≤ 2 }. This decomposition can be arranged such that µ0 = µ ∗ φ0, and, for jd j d j ≥ 1, µj = µ ∗ 2 φ(2 (x)), for suitable φ0, φ ∈ S(R ). It is easy to see that kf ∗ µ0kp0 . kfkp for 1 ≤ p ≤ 2. By (2.1), −jβ/2 kµbjk∞ . 2 for j ≥ 1, and hence −jβ/2 (2.6) kf ∗ µbjk∞ . 2 kfk1 . By (2.2) we get, with N d, Z 2jd |µ (x)| dµ(y) j . (1 + 2j|x − y|)N j X −j+l jd −lN j(d−N) . µ(B(x, 2 )2 2 + CN 2 l=0 j X j(d−α) −l(N−α) j(d−N) j(d−α) . 2 2 + CN 2 . 2 l=0 NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 7 and hence

j(d−α) (2.7) kf ∗ µbjk2 . kµjk∞kfk2 . 2 kfk2 . Interpolating (2.6) and (2.7) we get

−j(2(d−α)+ β − 2(d−α)+β ) kf ∗ µbjkp0 . 2 2 p kfkp where the exponent is negative when p < pcr. Sum in j to get the asserted inequality (2.3) for p < pcr. For the endpoint inequality the familiar analytic families interpolation argument (see e.g. [51] does not seem to work in the generality considered here, moreover it does not seem to yield the better Lorentz space bound. Instead one uses real methods (see [29], [24] for related arguments). A familiar argument by Bourgain [4] yields the restricted weak type estimate

(2.8) f ∗ µ 0 kfk pcr,1 b Lpcr,∞ . L or equivalently, for any f with |f| ≤ χE

1/p 0 |E| cr pcr meas({x : |f ∗ µ| > λ}) . b . λ To see this observe that for any R > 1

X X −jβ/2 0 −β/2 f ∗ µbj(x) ≤ C0 2 kfk1 ≤ C0R |E|. 2j >R 2j >R

0 −β/2 2/β Choose R = Rλ so that C0Rλ |E| = λ/2 (and thus Rλ ≈ (|E|/λ) ). Then meas {x : |f ∗ µb| > λ} X . meas {x : | f ∗ µbj(x)| > λ/2} j 2 ≤Rλ 2 −2 X −2 (d−a) 2 . λ f ∗ µj . λ Rλ kfk2 b 2 j 2 ≤Rλ 4(d−a)+β 4(d−a)+2β −2 4(d−a)/β − . λ (|E|/λ) |E| . |E| β λ β

0 which is (|E|1/pcr /λ)pcr . The restricted weak type estimate implies, by (2.5), (2.9) F : Lpcr,1 → L2(dµ) . 8 ANDREAS SEEGER

To upgrade this result to a strong type Lpcr → L2(dµ) (or even Lpcr,2 → L2(dµ)) bound we ﬁrst prove an Lpcr,1 → L2 estimates for the convo- lutions f ∗ µbj. We claim that (2.9) implies j(d−α)/2 (2.10) kf ∗ µbjk2 . 2 kfkLpcr,1 . j(d−α) Indeed, since kµjk∞ = O(2 ), Z 1/2 j(d−α)/2 2 kf ∗ µbjk2 . kfb µjk2 . 2 |fb(ξ)| |µj|dξ and, from (2.9), Z 1/2 2 |fb(ξ)| |µj|dξ

Z jd Z 1/2 2 2 |fb(η + ξ)| dµ(η)dξ kfkLpcr,1 . . (1 + 2j|ξ|)N . 0 Thus (2.10) follows. By duality we also get the L2 → Lpcr,∞ estimate, and then, by the Marcinkiewicz interpolation theorem, j(d−α)/2 (2.11) kf ∗ µbjkq˜ . 2 kfkp˜ for (1/p,˜ 1/q˜) on the open line segment joining the points (1/pcr, 1/2) 0 and (1/2, 1/pcr). We now repeat Bourgain’s argument above, interpolating the bounds (2.11) with the estimate (2.6). This gives the bound

(2.12) f ∗ µ kfk p,1 b Lq,∞ . L for (1/p, 1/q) on an open line segment which is parallel to the diagonal 0 {x, x}, and has midpoint (1/pcr, 1/pcr). Using the general Marcinkiewicz interpolation theorem again (on this line segment) we get

(2.13) f ∗ µ 0 kfk pcr,r b Lpcr,r . L for all r. Applying this for r = 2, and then applying (2.5) one more time yields F : Lpcr,2 → L2(dµ) .

2 2.2. Quasiradial multipliers in Mp . Some theorems for radial multipliers can be generalized for quasiradial multipliers, which do not possess any group invariance properties. They are given by m(ξ) = h(ρ(ξ)) where ρ is a suitable P -homogeneous distance function. NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 9

To deﬁne this notion let P be a real d × d matrix whose eigenvalues have positive real part and let tP = exp(P log t). ρ is a P -homogeneous distance function if ρ(tP ξ) = tρ(ξ) for all t > 0, and ρ is continuous in Rd and ρ(ξ) > 0 for ξ 6= 0. In addition we shall throughout assume that ρ belongs to C∞(Rd \{0}. Let

(2.14) Σρ = {ξ : ρ(ξ) = 1} and let σ be the surface measure on Σρ. It will be convenient to use P 0 0 generalized polar coordinates ξ = s ξ with ρ(ξ) = s, ξ ∈ Σρ; then Z Z ∞ Z 0 0 dσ(ξ ) ν−1 g(x)dx = g(sξ ) 0 0 s ds 0 Σρ hP ξ , n(ξ )i 0 0 where n(ξ ) is the outer normal vector at ξ ∈ Σρ and ν = trace(P ). Euler’s homogeneity relation in this setting becomes ρ(ξ) = hP ξ, ∇ρ(ξ)i 0 0 so that hP ξ , n(ξ )i is bounded above and below on Σρ. 2 We now discuss a simple characterization of quasiradial Mp multipliers in the case that the Fourier dimension of Σρ is positive. Let σ be surface measure on Σρ. We assume β/2 (2.15) sup |ξ| |σb(ξ)| < ∞ ξ∈Rd for some β > 0. As Σρ is a hypersurface we can apply Theorem 2.1 with α = d − 1 and get Z 1/2 0 2 0 2β + 4 (2.16) |g(ξ )| dσ(ξ ) kfk p d , 1 ≤ p ≤ b . L (R ) Σρ β + 4

(cf. [23]). For example, (2.15) holds with β/2 = (d − 1)/m if Σρ is convex and all tangent lines to Σρ have contact order of at least m with Σρ (see e.g. [9]). One version of the following observation was stated in [20] although it had been known as a ”folk result” for quite some time. 2β+4 Theorem 2.2. Suppose that (2.15) holds and 1 0 t ρ 10 ANDREAS SEEGER

2 Proof. If h is supported in (1, 4) then h ◦ ρ ∈ Mp immediately implies that h ◦ ρ ∈ L2(Rd), by testing the operator on suitable Schwartz func- 2 P 2 tions. Thus, by polar coordinates, h ∈ L . By scaling, km(t ·)kMp = ν(1/p−1/2) 2 t kmkMp and the necessity of the condition (2.17) follows.

For the sufficiency assume first that ht is supported in [t, 4t]. Then using polar coordinates and the restriction theorem (2.16) we see that Z 2t Z 0 1/2 2 0 2 dσ(ξ ) ν−1 kht ◦ρ fbk2 . |ht(r)| |fb(rξ )| 0 0 r dr t Σρ hP ξ , n(ξ )i Z 2t 1 2 1 · 2 ν−1 2 . |ht(r)| k rν f( r )kpr dr t Z 2t 1 2 2ν( 1 − 1 ) dr 2 = kfkp |ht(r)| r p 2 . t r P Now, dropping the support assumption we decompose h = h k k∈Z 2 k k+1 where h2k is supported in [2 , 2 ]. Let k+1 Z 2 1/2 kν( 1 − 1 ) 2 dρ A = sup 2 p 2 |h(ρ)| . k 2k−1 ρ Let ϕ ∈ C∞ supported in (1/4, 4) so that ϕ(s) = 1 on [1/2, 2]. Define −k Lk by Ldkf(ξ) = ϕ(2 ρ(ξ))fb(ξ). By Littlewood-Paley theory (adapted to the geometry determined by P ) we know that the operator f 7→ p p 2 {Lkf}k∈Z is bounded from L to L (` ). The above shows −1 2 X 2 kF [h(ρ(·))fb]k2 . kh2k ◦ρ fb]k2 k 1/2 2 2 X 2 2 X 2 2 2 . A kLkfkp . A |Lkf| . A kfkp. p k k Remark 2.3. It can be shown that for ρ(ξ) = |ξ|a, and the operator acting on radial functions only we have the same inequality in a larger p 2 2d range, namely Lrad → L boundedness for 1 1. See [20].

3. An FLq result

In this section h is assumed to be supported in a compact subinterval q J of (0, ∞). In order for h ◦ ρ to belong to Mp it is necessary for F −1[h ◦ ρ] to belong to Lq; this is immediate because on can test the convolution operator on Schwartz-functions whose Fourier transform is NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 11 equal to one on the support of h ◦ ρ. It is sometimes useful to express this condition in terms of a condition on the inverse Fourier transform of h (considered as a function on (−∞, ∞)), i.e. Z (3.1) κ(r) = h(ρ)e−iρrdρ .

Such a characterization is possible in the isotropic case, when ρ is the Minkowski functional of an open set Ω with smooth boundary, starlike with respect to the origin in its interior, so that ρ is homogeneous of degree one and Σρ is the boundary of Ω. The following result can be found for ρ(ξ) = |ξ| in [20]. The proof for general ρ is taken from our joint paper with Lee [37]. Theorem 3.1. Let 1 ≤ q ≤ 2, Ω as above, let ρ be the Minkowski functional of Ω and let J be a compact subinterval of (0, ∞). Then for h with support in J ,

Z 1/q −1 q (d−1)(1− q ) (3.2) F [h ◦ ρ] ≈ |κ(r)| (1 + |r|) 2 dr . Lq(Rd) The implicit constant depends on J.

Proof. Let χ ∈ S(R) so that χ is compactly supported in (0, ∞) and χ(s) = 1 on supp(h). The inequality ”.” in (3.2) follows by showing Z (3.3) F −1χ(ρ(·)) κ(r)eirρ(·)dt Lq(Rd) Z 1/q q (d−1)(1− q ) . |κ(r)| (1 + |r|) 2 dr for the cases q = 2 and q = 1, and using analytic interpolation. For q = 2 we have Z 2 (2π)d F −1[χ(ρ(·)) κ(r)eirρ(·)dr] L2(Rd) Z Z 2 irρ(ξ) = χ(ρ(ξ)) κ(r)e dr dξ Z Z 2 irρ d−1 = c χ(ρ) κ(r)e dr ρ dρ Z 2 . |κ(r)| dr where we have ﬁrst used Plancherel in Rd, applied generalized polar coordinates, and then used Plancherel on the real line. 12 ANDREAS SEEGER

In order to prove (3.3) for q = 1 we use the fact

−1 irρ(·) d−1 (3.4) F [χ(ρ(·))e ] . (1 + |r|) 2 . L1(Rd) This is a rescaled version of an inequality in [49]; the assumption that ρ is homogeneous of degree one is crucial here. (3.4) and Minkowski’s integral inequality immediately give Z Z −1 irρ(·) d−1 F [χ(ρ(·)) κ(r)e dt] . |κ(r)|(1 + |r|) 2 dr 1 which is (3.3) for q = 1. We now show the converse inequality Z 1/q q (d−1)(1−q/2) −1 (3.5) |κ(r)| (1 + |r|) dr . kF [h ◦ ρ]kLq(Rd) . for h smooth. This a priori assumption implies that the left hand side in (3.5) is ﬁnite; it is easy to remove by an approximation argument.

Let ξ0 ∈ Σρ be a point where the Gaussian curvature does not vanish (e.g. a point where |ξ| is maximal on Σρ). Let U be a small neighborhood of ξ0 on which the Gauss map is injective and the curvature is bounded below. Let γ be homogeneous of degree zero, γ(ξ0) 6= 0 and supported on the closure of the cone generated by U. Then −1 −1 F [γ h◦ρ] Lq . F [h◦ρ] Lq . Use polar coordinates (with respect to ρ) to write Z ∞ Z 0 d −1 d−1 0 iρhξ0,xi dσ(ξ ) (3.6) (2π) F [γ h◦ρ](x) = h(ρ)ρ γ(ξ )e 0 dρ . 0 Σρ |∇ρ(ξ )| d x Let n(ξ0) the outer normal at ξ0, let Γ = {x ∈ R : |x| − n(ξ0) ≤ ε}, with ε small and let, for large R 1, ΓR = {x ∈ Γ: |x| ≥ R}. We may assume that for each x ∈ Γ there is a unique ξ = Ξ(x) ∈ Σρ, so that γ(Ξ(x)) 6= 0 and so that x is normal to Σρ at Ξ(x). Clearly x 7→ Ξ(x) is homogeneous of degree zero on Γ. By the method of stationary phase we have for x ∈ ΓR N −1 X (3.7) F [γh(ρ(·))](x) = I0(x) + IIj(x) + III(x) j=1 where Z ∞ −1 d−1 iρhΞ(x),xi γ(Ξ(x))|∇ρ(Ξ(x))| I0(x) = c h(ρ)ρ e dρ d−1 0 (ρhΞ(x), xi) 2 |K(Ξ(x))|1/2 NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 13 where K(Ξ(x)) is the Gaussian curvature at Ξ(x) and |c| = (2π)−d. There are similar formulas for the higher order terms IIj(x), with the − d−1 − d−1 −j main term (ρhΞ(x), xi) 2 replaced by (ρhΞ(x), xi) 2 . Finally

−N |III(x)| .N khk1|x| , x ∈ ΓR .

d−1 −j −1 Let h (ρ) = h(ρ)ρ 2 and let κ = F [h ], then j j R j κ (hΞ(x), xi) |I (x)| ≈ 0 , x ∈ Γ . 0 d−1 R hΞ(x), xi 2 We also have |x| ≈ hΞ(x), xi, x ∈ Γ, which is a consequence of ρ(ξ) = hξ, ∇ρ(ξ)i (Euler’s homogeneity relation) and the positivity of ρ on Σρ. It follows

Z ∞ 1/q q (d−1)(1−q/2) q (3.8) kI0kL (ΓR) & |κ0(r)| (1 + |r|) dr C0R for some C0 > 1. Similarly Z ∞ 1/q −j q (d−1)(1−q/2) q (3.9) kIIjkL (ΓR) . R |κj(r)| (1 + |r|) dr . 0 We will need to use the straightforward bound

Z ∞ 1/q Z ∞ 1/q q a q a (3.10) |ζ ∗ g(r)| (1 + |r|) dr . |g(r)| (1 + |r|) dr 0 0 whenever ζ ∈ S(R). Since κj = ζj ∗ κ0 this shows that we can replace κj in (3.9) by κ0. There are also the trivial inequalities

−1 q kIIIkL (ΓR) . R khk1 and −1 khk kh◦ρk q0 d kF [h◦ρ]k q d . 1 . L (R ) . L (R ) Thus

−1 −1 q q d (3.11) kIIIkL (ΓR) . R kF [h◦ρ]kL (R ). Moreover, we estimate crudely (3.12) Z C0R 1/q q (d−1)(1−q/2) d/q d/q |κ(r)| (1 + |r|) dr . R kκ0k∞ . R khk1 . 0 14 ANDREAS SEEGER

We now combine the estimates and get Z 1/q 2 (d−1)(1−q/2) |κ0(r)| (1 + |r|) dr

d/q −1 q q d . kI0kL (ΓR) + R kF [h◦ρ]kL (R ) d/q −1 −1 X (3.13) . R kF [h◦ρ]kLq(Rd) + R kIIjkLq(Rd) j=1 here we have used (3.7), (3.11)) for the second inequality, and (3.8), (3.12)) for the third. To estimate the second term in (3.13) we use (3.9) and then, using (3.10) replace κj with κ0 to get Z ∞ 1/q −1 q (d−1)(1−q/2) q kIIjkL (ΓR) . R |κ0(r)| (1 + |r|) dr . 0 Thus, choosing R suﬃciently large (and using the ﬁniteness of the right hand side of the last display) we get Z 1/q 2 (d−1)(1−q/2) −1 |κ0(r)| (1 + |r|) dr . kF [h◦ρ]kLq(Rd) .

Finally observe that κ = κ0 ∗ζ0 for some Schwartz function ζ0 and thus, by (3.10) we may replace κ0 by κ. This ﬁnishes the proof of (3.5). Remark. An application of the Hausdorﬀ-Young inequality recovers necessary conditions in [22],[47], namely for h supported in a compact interval J ⊂ (0, ∞) −1 (3.14) khk q0 . kF [h◦ρ]kLq( d) . B 1 1 J R (d−1)( q − 2 ,q In [47] inequality (3.14) was proved for more general P -homogeneous distance functions ρ, but we don’t have a simple analogue of Theorem 3.1 in this case.

4. Radial Mp multipliers acting on radial functions

Let g be a radial Schwartz-function, g = g0(|x|) then the Fourier transform of g is given by gb(ξ) = Bdg0(ρ) where Bd denotes the Fourier- Bessel transform (or modiﬁed Hankel transform) of g0. It is given by Z ∞ d−1 (4.1) Bdf(ρ) = f(s)J(sρ)s ds 0 where − d−2 (4.2) J(ρ) = ρ 2 J d−2 (ρ) 2 and Jα denotes the standard Bessel function. NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 15

The convolution with a radial kernel maps radial functions to radial functions. Let h(|ξ|) be a radial multiplier and g = g0(| · |) Then −1 (4.3) F [h(| · |)gb](x) = c Bd[hBdg0](|x|) . p p We denote by Lrad the space of radial L functions. The following p characterization of Lrad boundedness was proved in joint work with G. Garrig´os[20]. Let φ be a nontrivial bump function which is compactly supported in (0, ∞).

2d −1 Theorem 4.1. Let 1 < p < d+1 . Deﬁne κt(τ) = F [φh(t·)](τ) . Then −1 kF [h(| · |)f]k p kfk p b Lrad . Lrad for all radial Schwartz functions holds if and only if the condition Z 1/p p (d−1)(1−p/2) (4.4) sup |κt(τ)| (1 + |τ|) dτ < ∞ t>0 is satisﬁed.

The necessity is easily checked by testing the operator on functions of the form td/pη(t·) where η is a suitable Schwartz function. For the full proof of Theorem 4.1 we refer to [20]. Here we consider only the special case where h is compactly supported on a compact subinterval J of (0, ∞). In this case the condition (4.4) reduces to the ﬁniteness of −1 kF [h(|·|)]kp and thus, by Theorem 3.1, we need to prove the estimate

(4.5) Bd[hBdg] Lp(rd−1dr) Z ∞ 1/p p (d−1)(1−p/2) . |bh(τ)| (1 + |τ|) dτ kgkLp(rd−1dr) . −∞ Proof of (4.5). Here we give the simple idea for the proof of this special case. We need to use the standard asymptotics for Bessel functions (see [16], 7.13.1(3)), namely for |x| ≥ 1,

M X d−1 d−1 −2ν− 2 (4.6) J(x) = cν,d cos(x − 4 π)x ν=0 M X d−1 −2ν− d+1 −M + c sin(x − π)x 2 + x E (x) eν,d 4 eM,d ν=0 1/2 with c0,d = (2/π) , and the derivatives of EeM,d are bounded. Note that Z d−1 (4.7) Bd[hBdg](r) = K(r, s)s g(s)ds 16 ANDREAS SEEGER where Z ∞ (4.8) K(r, s) = h(ρ)J(ρr)J(ρs)ρd−1dρ. 0 Using the above asymptotic expansion in this integral, for both J(ρr) and J(ρs) one derives the following pointwise bound Z X − d−1 − d−1 |κ(±r ± s − u)| (4.9) |K(r, s)| (1 + r) 2 (1 + s) 2 du. . (1 + |u|)N (±,±) where κ = F −1[h]. If we incorporate the weights into the operator, set R f(s) = s(d−1)/pg(s), and use (4.9), then (4.5) reduces to (4.10) d−1 d−1 Z h |r| p ZZ |s| p0 |κ(±r ± s − u)| ip 1/p d−1 d−1 N |f(s)| dsdu dr (1 + |r|) 2 (1 + |s|) 2 (1 + |u|)

. kκkLp((1+|r|)(d−1)(1−p/2)dr)kfkLp(dr) . Let (d−1)( 1 − 1 ) 1 + |r| p 2 K(r, s) = |κ(±r ± s)|. 1 + |s| By an application of Minkowski’s inequality and a straightforward estimate we see that (4.10) follows from Z Z p 1/p

(4.11) K(r, s)f(s)ds dr

. kκkLp((1+|τ|)(d−1)(1−p/2)dτ)kfkLp(ds) . 1 2 We split the kernel into K (r, s) = K(r, s)χ|s|≤|r|/2 and K (r, s) = K(r, s)χ|s|≥|r|/2. The main contribution comes from the ﬁrst kernel and we get Z Z p 1/p 1 K (r, s)f(s)ds dr Z Z 1/p |f(s)| (d−1)(1− p ) p (1 + |r|) 2 |κ(±r ± s)| dr ds . (d−1)( 1 − 1 ) (1 + |s|) p 2 |r|≥2|s| Z Z 1/p |f(s)| (d−1)(1− p ) p ds (1 + |r|) 2 |κ(r)| dr . . (d−1)( 1 − 1 ) (1 + |s|) p 2 2d For p < d+1 H¨older’sinequality yields Z −(d−1)( 1 − 1 ) (1 + |s|) 2 p |f(s)|ds . kfkp and we get the analogue of (4.11) for the kernel K1. NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 17

For the contribution with |s| ≥ |r|/2 we can drop the term 1 + |r|(d−1)(1/p−1/2) 1 1 + |s| . and estimate Z Z p 1/p 2 K (r, s)f(s)ds dr . kκk1kfkp .

2d Since by H¨older’sinequality and p < d+1 Z 1/p (d−1)(1− p ) p kκk1 . (1 + |r|) 2 |κ(r)| dr , we get the analogue of (4.11) for the kernel K2. This ﬁnishes the proof of (4.5).

p p Open problem: Is there an eﬀective characterization of Lrad → Lrad 2d boundedness in the range d+1 ≤ p < 2?

5. Radial Fourier multipliers with compact support

We now give the proof of the characterization of radial multipliers with compact support away from the origin, which was obtained in the paper with Heo and Nazarov [26].

2d−2 Theorem 5.1. Let d ≥ 4, 1 < p < pd := d+1 , and let m be radial, m = h(| · |)

1 −1 with h supported in ( 2 , 2). Let κ as in (3.1), i.e. κ(r) = (2π)F [h](r). The following are equivalent: (i) m ∈ M p(Rd) 1/p R ∞ p (d−1)(1−p/2) (ii) −∞ |κ(r)| (1 + |r|) dr < ∞ (iii) F −1[m] ∈ Lp(Rd).

For the equivalence of (ii) and (iii) see §3. Clearly (i) implies (iii), in view of the compact support of m. In this section we prove that (iii) implies (i).

Let φ0 be a radial Schwartz function, compactly supported in {|x| ≤ 1 2M 2 } such that φb0(ξ) > 0 on {ξ : 1/2 ≤ |ξ| ≤ 2}. Take ψ0 = ∆ φ0 for 18 ANDREAS SEEGER

M > 5d. Then ψ0 is a radial Schwartz function, compactly supported 1 in {|x| ≤ 2 } such that ψb0(ξ) > 0 on {ξ : 1/2 ≤ |ξ| ≤ 2} and such that Z ψ0(x)P (x)dx = 0 for all polynomials of degree ≤ 4M. Let

ψ = ψ0 ∗ ψ0.

−1 Let η be a radial Schwartz function such that ηb(ξ) = [ψb(ξ)] for 1/2 ≤ |ξ| ≤ 2. Then h(|ξ|)ψb(ξ)ηb(ξ) = m(ξ) and thus −1 F [m] = ψ ∗ K = ψ0 ∗ ψ0 ∗ K where K = η ∗ F −1[m]. Note that K is radial and Kb is compactly supported in {ξ : 1/2 ≤ |ξ| ≤ 2}. Clearly

−1 kKkp . kF [m]kp . Thus we need to show

(5.1) kψ ∗ K ∗ fkp . kKkpkfkp. Let K(r) be deﬁned on (0, ∞) so that K(|x|) = K(x). If g ∈ S we have, by polar coordinates, Z Z ∞ Z K(x)g(x)dx = K(r)rd−1 g(rx0)dσ dr 0 Sd−1 and thus Z ∞ (5.2) K = K(r)σrdr 0 where σr is the surface measure on the sphere of radius r and (5.2) is understood in the sense of distributions. Note that

d−1 kσrkM = O(r ). d−1 −1 −1 Also hσr, gi = r hσ, g(r·)i , i.e. σr = r σ(r ·). Lemma 5.2. We have rd−1 |σ (ξ)| ≤ br d−1 (1 + |ξ|r) 2 and

d−1 (5.3) kψ\0 ∗ σrk∞ . (1 + r) 2 NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 19

Proof. The ﬁrst formula follows from stationary phase or the explicit formula ([53])

(5.4) σb1(|ξ|) = J(|ξ|), with J as in (4.2).

Now (5.3) follows because ψb vanishes at the origin to order 20d.

The contribution of the integral in (5.2) for r ≤ A is trivial for our purposes since Z A Z A d−1 K(r)σrdr . |K(r)|r dr 0 1 0 Z A 1/p d/p0 p d−1 . A |K(r)| r dr . 0

Let f ∈ Lp(Rd). We need to prove Z ∞ Z

(5.5) K(r) ψ ∗ σr(x − y)f(y)dy dr 2 p Z ∞ 1/p p d−1 . |K(r)| r dr kfkp. 2 Let

(5.6) Fy,r(x) := ψ ∗ σr(x − y). We replace the tensor product function f(y)K(r) with a more general function g in the Lebesgue space Lp(Rd × (0, ∞); dy rd−1dr). We will then prove the more general inequality Z ∞ Z Z ∞ Z 1/p p d−1 (5.7) Fy,r(x)g(y, r)dy dr . |g(y, r)| dy r dr 2 p 2

It is convenient for notation to discretize the above inequality. This is natural in view of the compact support of the Fourier transform of K 0 d and (w.l.o.g.) the Fourier transform of f. For u = (u , ud+1) ∈ [0, 1] × [0, 1] let Zu be the half-lattice consisting of those (y, r) = (y1, . . . , yd, r) such that yi = zi + ui for some integer zi and r = n + ud+1 for some integer n ≥ 2. It suﬃces to prove the inequality

1/p X X p d−1 (5.8) Fy,r(x)g(y, r) . |g(y, r)| r p (y,r)∈Zu (y,r)∈Zu 20 ANDREAS SEEGER with a bound uniform in u ∈ [0, 1]d+1. Henceforth it is assumed that all (y, r) sums are taken over Zu. Later we shall also use the notation

k k+1 (5.9) Zk,u = {(y, r) ∈ Zu : 2 ≤ r < 2 } .

Inequality (5.7) (with the r-integration over [1, ∞)) is an immediate consequence of (5.8), by averaging and H¨older’sinequality. Indeed, we have Z ∞ Z

ψ ∗ σr(x − y)g(y, r)dy dr 1 p Z X . ψ ∗ σr(x − y)g(y, r) du [0,1]d+1 p (y,r)∈Zu Z X . ψ ∗ σr(x − y)g(y, r) du [0,1]d+1 p (y,r)∈Zu Z p 1/p X . Fy,r(x)g(y, r) du [0,1]d+1 p (y,r)∈Zu and, once we prove (5.8), the right hand side of the previous display is bounded by a constant times Z X 1/p Z ∞ Z 1/p |g(y, r)|p rd−1 du = |g(y, r)|pdy rd−1dr . [0,1]d+1 1 (y,r)∈Zu

Support properties. Recall that the support of ψ is in the unit ball and thus

(5.10) supp(Fy,r) ⊂ {x : |x − y| − r ≤ 1},

d−1 which has measure O(r ) if r ≥ 2. Thus if E is a subset of Zu, Zk,u as in (5.9), then we have the trivial consequence

X X X k(d−1) (5.11) meas supp Fy,r . 2 #(E ∩ Zk,u) . k (y,r)∈E∩Zk,u k

This estimate may be improved if the set E ⊂ Zk,u is concentrated on a k ball of radius R0 ∈ (1, 2 ). Observe that the cardinality of E is always d+1 O(R0 ) in this case and an improvement over (5.11) happens if this cardinality is substantially larger than R0. NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 21

k Lemma 5.3. Let 1 ≤ R0 ≤ 2 , Ek ⊂ Zk,u and suppose that Ek is contained in a ball B0 of radius R0. Then

X k(d−1) (5.12) meas supp |Fy,r| . 2 R0 . (y,r)∈Ek

Proof. If (y0, r0) is the center of B0 and if Fy,r(x) 6= 0 then |x−y|−r ≤ 1 and consequently

|x − y0| − r0 ≤ |x − y| − r + |y − y0| + |r − r0| ≤ 1 + 2R0.

Thus

X supp Fy,r ⊂ {(y, r) ∈ Zk,u : |x − y0| − r0 ≤ 2R0 + 1}

(y,r)∈Ek

k(d−1) and the measure of this set is O(2 R0).

A weak orthogonality property. Inequality (5.8) trivially holds for d−1 2 d−1 p = 1 since kFy,rk1 = O(r ). We also know that kFy,rk2 = O(r ) and thus if the functions Fy,r were orthogonal, or almost orthogonal in a strong sense then one would get the inequality for p = 2. However, this would imply that we get L2 boundedness for convolution operators for which the compactly supported Fourier multiplier is merely in L2, but of course boundedness of the Fourier multiplier is a necessary 2 and suﬃcient condition for L boundedness. Consequently the Fy,r are not “almost orthogonal” enough. Nevertheless there is a weak orthogonality which will be crucial in our estimates. It is expressed in the following

Lemma 5.4.

0 d−1 (rr ) 2 0 0 (5.13) Fy,r,Fy ,r . d−1 . (1 + |y − y0| + |r − r0|) 2

Proof. By Parsevals’s identity

−d Fy,r,Fy0,r0 = (2π) Fby,r, Fby0,r0 . 22 ANDREAS SEEGER

Recall that σb1(|ξ|) = J(|ξ|)(cf. (5.4)) and thus, with ψb(ξ) = β(|ξ|), we have d (2π) hFby,r, Fby0,r0 i Z 2 ihy0−y,ξi = σbr(ξ)σbr0 (ξ)|ψb(ξ)| e dξ Z Z = (rr0)d−1 |β(ρ)|2J(rρ)J(r0ρ) eiρhy0−y,ξ0idσ(ξ0) ρd−1dρ Sd−1 Z = (rr0)d−1 |β(ρ)|2ρd−1J(rρ)J(r0ρ)J(ρ|y − y0|) dρ.

− d−1 Now |J(s)| . (1 + |s|) 2 and if we take into account that |β(s)| . 20d −N |s| for |s| ≤ 1 and |β(s)| . |s| for |s| ≥ 1, we obtain 0 d−1 (rr ) 2 0 0 Fy,r,Fy ,r . d−1 . (1 + |y − y0|) 2 This gives the asserted bound when |r − r0| ≤ C(1 + |y − y0|). But from

supp(Fy,r) ⊂ {x : |x − y| − r| ≤ 1}, 0 0 supp(Fy0,r0 ) ⊂ {x : |x − y | − r | ≤ 1} 0 0 we see that for |r − r | 1 + |y − y |, the supports of Fy,r and Fy0,r0 are disjoint. Hence hFy,r,Fy0,r0 i = 0 in this case. We note that one can prove a ﬁner estimate (5.14) 0 d−1 0 − d−1 X 0 0 −N |hFy,r,Fy0,r0 i| ≤ CN (rr ) 2 (1+|y−y |) 2 1+ r±r ±|y−y | ±,± if one uses the oscillations of the Bessel functions (as in (4.6). We will not have to use (5.14).

Reduction to a restricted weak type inequality. We need to 2(d−1) prove the inequality (5.8) for p in the open range (1, d+1 ). Consider the operator T acting on functions {g(y, r)} on the lattice Zu, deﬁned by X T g(x) = Fy,r(x) g(y, r) . (y,r) By the generalized Marcinkiewicz interpolation theorem it suﬃces to p,1 d−1 p,∞ d prove that T maps the weighted Lorentz space ` (Zu, r ) to L (R ). For k = 1, 2,... let

(5.15) Ek ⊂ Zk,u NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 23 and let cy,r be constants satisfying

(5.16) sup |cy,r| ≤ 1 . y,r p,1 d−1 p,∞ d This ` (Zu, r ) → L (R ) bound follows if we can show the restricted weak type inequality (5.17) d X X −p X k(d−1) meas x ∈ R : cy,rFy,r > λ . λ 2 #Ek . k (y,r)∈Ek k We note that (5.17) is true for p = 1, and thus implies (5.17) for p > 1 when λ ≤ 1. Hence in what follows we will assume that λ > 1. We may also assume that #Ek < ∞ for all k.

Modiﬁed Calder´on-Zygmund decompositions. The following proposition is motivated by the support estimate in Lemma 5.3. In what follows λ > 1.

Proposition 5.5. For every k there is ﬁnite collection Bk of disjoint balls so that k (i) Each ball B ∈ Bk has radius rad(B) ≤ 2 and p #(B ∩ Ek) ≥ λ diam(B) .

∗ (ii) For each B ∈ Bk denote by B the ball with same center, and radius equal to five times the radius of B. Define the sparse set (or low sp density set) Ek as sp [ ∗ Ek = Ek \ B . B∈Bk k+1 Then for every subset D of Zk,u with diameter ≤ 2 we have sp p #(E ∩ D) . λ diam(D) . Proof. This is analogous to the proof of the usual Vitali type covering k lemma. We set B0,k = ∅. If there are no balls of radius at most 2 with the property that the cardinality of the intersection is at least λp times sp the diameter of the ball then we set Ek = Ek and properties (i) and (ii) are satisfied with Bk = B0,k = ∅. Otherwise we choose a maximal k p ball B1,k of radius at most 2 such that #(B1,k ∩ Ek) ≥ λ diam(B1,k) .

At stage ` we are given a collection B`−1,k = {B1,k,...,B`−1,k} of `−1 p disjoint balls such that #(Ek ∩ Bi,k) ≥ λ diam(Bi,k) for i = 1, . . . , ` − 1 and such that the radii of Bi,k do not increase if i increases. If there k `−1 are no balls of radius at most 2 in the complement of ∪i=1 Bi,k such p that the cardinality of the intersection with Ek is at least λ times the 24 ANDREAS SEEGER

sp `−1 ∗ diameter of the ball then we set Ek = Ek\∪i=1 B1,k and the construction k stops. Otherwise choose a maximal ball B`,k of radius at most 2 in the `−1 p complement of ∪i=1 Bi,k such that #(B`,k ∩ Ek) ≥ λ diam(B`,k). Note that diam(B`,k) ≤ diam(Bi.k) for i = 1, . . . , ` − 1 (since otherwise B`,k would have been selected before). We set Bk,` = {B1,k,...,B`,k}. The construction stops at some stage, after having selected disjoint balls Bi,k, for i = 1,...,Nk. Then we set

Bk := BNk,k = {B1,k,...,BNk,k} ,

sp Nk ∗ Ek = Ek \ ∪i=1Bi,k .

p If B is any ball satisfying #(B ∩ Ek) ≥ λ diam(B) then B must be ∗ contained in Bi,k for at least one of the balls Bi,k in Bk. Hence this ball sp is a subset of the complement of Ek . Thus if D is any set of diameter k sp p ≤ 2 we get card(E ∩ D) . λ diam(D). We use the construction from Proposition 5.5 to build an exceptional set.

For each k ∈ N and for each ball B ∈ Bk, with center (yB, rB), we deﬁne the subset of Rd d ∗ VB = {x ∈ R : |x − yB| − rB ≤ 2(diam(B ) + 1)} .

Observe that for B ∈ Bk, X supp cy,rFy,r ⊂ VB ∗ (y,r)∈Ek∩B and thus, if we deﬁne [ [ (5.18) V = VB

k∈N B∈Bk then X X (5.19) supp cy,rFy,r ⊂ V . k sp (y,r)∈E/ k Proposition 5.6. The Lebesgue measure of V satisﬁes the estimate

−p X k(d−1) meas(V) . λ 2 card(Ek) . k

Proof. Recall that for each B ∈ Bk we have by property (i) in Propo- sition 5.5, −p diam(B) ≤ λ card(Ek ∩ B). NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 25

Thus

X X k(d−1) ∗ meas(V) . 2 diam(B ) k B∈Bk X X k(d−1) . 2 diam(B) k B∈Bk X X k(d−1) −p . 2 λ card(Ek ∩ B) k B∈Bk −p X k(d−1) . λ 2 #Ek . k

L2-estimates. The purpose of this section is to prove

sp Proposition 5.7. For k = 1, 2,... let Ek be subsets of Zk,u with the property that for any ball B of diameter ≤ 2k we have the sparseness assumption

sp p card(Ek ∩ B) . λ (diamB). Then

2 X X 2p X k(d−1) sp d−1 cy,rFy,r . λ log(2 + λ) 2 #Ek . 2 k sp k (y,r)∈Ek

The proof is a combination of three lemmata. We begin with a straightforward L2 estimate which does not rely on our weak orthogonality argument. Recall that always sup |cy,r| ≤ 1.

Lemma 5.8. Let 1 ≤ L ≤ 2k and I be a subinterval of [2k, 2k+1] of length L. Let EI be a subset of Zk,u such that for all (y, r) ∈ EI we have r ∈ I. Then 2 X k(d−1) cy,rFy,r . L2 #EI . 2 (y,r)∈EI

Proof. We have

X 2 X X 2 cy,rFy,r = ψ0 ∗ σr ∗ cy,rψ0(· − y) 2 b 2 y,r r∈I y:(y,r)∈EI 26 ANDREAS SEEGER and, by the Cauchy-Schwarz inequality, this is estimated by a constant times

X X 2 L ψ0 ∗ σr ∗ cy,rψ0(· − y) b 2 r∈I y:(y,r)∈EI 2 X d−1 X . L r cy,rψ0(· − y) 2 r∈I y:(y,r)∈EI X d−1 X 2 . L r kψ0(· − y)k2 r∈I y:(y,r)∈EI X d−1 . L r #{y :(y, r) ∈ EI } r∈I where we have of course used that the y and r’s are 1-separated. The k(d−1) last displayed quantity is bounded by L2 #(EI ).

We now let λ > 1 and prove an L2 estimate for the set E sp constructed in the previous section. We thus assume that every ball of radius ≤ 2k p sp contains no more than λ diam(B) points in Ek . Lemma 5.9. Let 1 ≤ L ≤ 2k and let I be a collection of disjoint subintervals of [2k−1, 2k+2] which are of length L. Let sp Ek,I = {(y, r) ∈ Ek : r ∈ I}. Then 2 X X p −` d−3 k(d−1) sp cy,rFy,r . L + λ L 2 2 #E . 2 I∈I (y,r)∈Ek,I

Proof. We may assume that the intervals are 10L-separated (after split- ting the collection I into eleven subcollections with this property). Let X X Gk,I = cy,rFy,r. I∈I sp (y,r)∈Ek,I Then X 2 X 2 X Gk,I . Gk,I + hGk,I0 ,Gk,I i . 2 2 I I I06=I By Lemma 5.8,

X 2 X k(d−1) k(d−1) sp (5.20) Gk,I 2 . L 2 #(Ek,I ) . L2 #(Ek ) . I I NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 27

We claim that X p − d−3 k(d−1) (5.21) hGk,I ,Gk,I0 i . λ L 2 2 #Ek. I06=I

Let (y, r) ∈ Ek,I . Then by Lemma 5.2, part (i), X X hFy,r,Fy0,r0 i 0 0 0 I 6=I (y ,r )∈Ek,I0 k(d−1) X 0 0 − d−1 . 2 |(y , r ) − (y, r)| 2 0 0 sp (y ,r )∈Ek L≤|(y0,r0)−(y,r)|≤2k+3 k(d−1) X −` d−1 p ` . 2 2 2 (λ 2 ) 2`≥L/2 k(d−1) p − d−3 . 2 λ L 2 . where we have used that each ball of diameter 2` contains no more than ` p sp 2 λ points in Ek . We also used d > 3 to sum the geometric series. Now we sum in (y, r) and get X hGk,I ,Gk,I0 i I06=I X X X X ≤ hFy,r,Fy0,r0 i 0 0 0 I (y,r)∈Ek,I I 6=I (y ,r )∈Ek,I0 X X k(d−1) p −` d−3 . 2 λ L 2 I (y,r)∈Ek,I d−3 sp p −` 2 k(d−1) . λ L 2 #Ek . This proves (5.21). The lemma follows if we combine (5.20) and (5.21). Corollary 5.10. 2 X 2p k(d−1) sp c F λ d−1 2 #E . y,r y,r . k sp 2 (y,r)∈Ek

p 2 p 2 Proof. If 2k ≤ λ d−1 this follows from Lemma 5.8. If 2k > λ d−1 this 2 follows from Lemma 5.9 if we choose L = λ d−1 .

Finally we show some almost orthogonality for the sums in which we allow to vary k. 28 ANDREAS SEEGER

Lemma 5.11. 2 X X 2p X k(d−1) sp cy,rFy,r . λ d−1 log(2 + λ) 2 #E . 2 k sp k (y,r)∈Ek

p Proof. Let N(λ) be the smallest integer larger than 10 log2(2+λ ). Let X Gk = cy,rFy,r. sp (y,r)∈Ek We write

X 2 X 2 X 2 (5.22) Gk ≤ 2 Gk + Gk 2 2 2 k k≤N(λ) k>N(λ) and estimate X 2 X 2 Gk . N(λ) Gk 2 2 k≤N(λ) k≤N(λ) 2p X k(d−1) sp (5.23) . λ d−1 log(2 + λ) 2 #E , k≤N(λ) by Lemma 5.9. 2 P Now, in order to estimate the L norm of k>N(λ) Gk we may assume the sum in k is taken over a 10-separed subsets of integers ≥ N(λ) (just split the original sum in eleven diﬀerent sums with this property). We then have 2 X X 2 X (5.24) Gk ≤ kGkk2 + 2 hGk,Gk0 i . 2 k>N(λ) k>N(λ) k>k0>N(λ) Again, by Lemma 5.9, X 2p X 2 d−1 k(d−1) sp (5.25) kGkk2 . λ 2 #E . k>N(λ) k>N(λ)

To estimate the second term in (5.24) we ﬁx sp (y, r) ∈ Ek . 0 0 0 sp Let k < k−10 and let (y , r ) ∈ Ek0 . Then Fy0,r0 is supported on a ball of 0 0 radius r +1 centered at y and Fy,r is supported on the 1 neighborhood of the sphere of radius r centered at y. This means hFy,r,Fy0,r0 i can be diﬀerent from zero only if r − 2k0+3 ≤ |y − y0| ≤ r + 2k0+3. NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 29

0 0 k0 k0+1 The set of (y , r ) ∈ Zu satisfying 2 ≤ r ≤ 2 and satisfying the displayed inequality can be covered with O(2(k−k0)(d−1)) balls of radius k0 p k0 sp 2 and each of them contains no more than O(λ 2 ) points of Ek0 . For 0 0 k those points the distance of (y, r) and (y , r ) is & 2 and therefore, by Lemma 5.13,

0 d−1 −k d−1 k0 d−1 |hFy,r,Fy0,r0 i| . (rr ) 2 2 2 . 2 2 .

Then also

X (k−k0)(d−1) p k0 k0 d−1 p k(d−1) −k0 d−3 |hFy,r,Fy0,r0 i| . 2 (λ 2 )2 2 . λ 2 2 2 . (y0,r0)∈Esp k0

Hence X X |hFy,r,Fy0,r0 i| k0 k−10 (y0,r0)∈Esp N(λ)<2 <2 k0 p k(d−1) X −k0 d−3 . λ 2 2 2 N(λ)<2k0 <2k−10 p −N(λ) d−3 k(d−1) k(d−1) . λ 2 2 2 . 2 since d > 3. sp Finally we sum over all (y, r) ∈ Ek and get from the last display

X hGk,Gk0 i k>k0>N(λ) X X X X . |hFy,r,Fy0,r0 i| k>N(λ) (y,r)∈Esp k0 k−10 (y0,r0)∈Esp k N(λ)<2 <2 k0 X X k(d−1) k(d−1) sp (5.26) . 2 . 2 #Ek . sp k>N(λ) (y,r)∈Ek

If we combine the last estimate with (5.25) we obtain

2 X p k(d−1) sp Gk . λ 2 #Ek . 2 k>N(λ) and this, together with (5.23), proves the lemma. 30 ANDREAS SEEGER

Conclusion. We prove the restricted weak type inequality (5.17), for λ > 1. Let V be as in (5.18). Using (5.19) we get

d X X meas {x ∈ R : cy,rFy,r > λ} k (y,r)∈Ek

d X X . meas(V) + meas {x ∈ R : cy,rFy,r > λ} k sp (y,r)∈Ek 2 −2 X X . meas(V) + λ cy,rFy,r , 2 k sp (y,r)∈Ek by Tshebyshev’s inequality. By Proposition 5.6 and Proposition 5.7 we get

d X X meas {x ∈ R : cy,rFy,r > λ} k (y,r)∈Ek −p 2p −2 p X k(d−1) −p . λ + λ d−1 log(2 + λ ) 2 λ #Ek , k and the proof is concluded by observing that for λ > 1

2p −2 p −p 2(d − 1) λ d−1 log(2 + λ ) λ if p < . . d + 1

6. Characterization of radial Fourier multipliers and further results

One can generalize the theorem in the previous section to obtain a characterization of all Lp bounded operators which commute with 2(d−1) rotations and translations, in the range 1 < p < d+1 . The following is proved in [27]. Theorem 6.1. Let m = h(| · |) be a bounded radial function on Rd and d deﬁne the convolution operator Th on R by

(6.1) Tdhf(ξ) = h(|ξ|)fb(ξ). 2d Let 1 < p1 < d+1 and assume that (5.7) holds for p1. Let 1 < p < d p1, and let η be any nontrivial Schwartz function on R and let φ be ∞ any nonzero Cc function compactly supported on (0, ∞). Then the following statements are equivalent. p (i) Th is bounded on L . p p (ii) Th maps Lrad to Lrad. d/p (iii) supt>0 t Th[η(t·)] p < ∞ NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 31

(iv) The functions κ = F −1[φh(t·)] belong to Lp((1+|r|)(d−1)(1−p/2)dr) t R with norm uniformly bounded independently of t.

The proof is somewhat technical and uses atomic decompositions of Lp spaces. By the results of the previous sections we have the following corollary (proved ﬁrst in [26]).

2(d−1) Corollary 6.2. Let d ≥ 4 and 1 < p < d+1 . Then statements (i) - (iv) are equivalent.

It would be very interesting to get a sharp inequality in two and three dimensions, for some nontrivial range of p. It is easy to see (by H¨older’sinequality on dyadic intervals and Plancherel’s theorem) that

Z 1/p (d−1)(1−p/2) (6.2) κ(r)(1 + |r|) dr kκk 2 . . b Bd/p−d/2,p

2(d−1) Thus Conjecture II, for d ≥ 4 in the range p < d+1 follows from Conjecture III . However one can modify the proofs of Theorems 5.1 and 6.1 to get Conjecture II and its global analogue in the range of the Tomas restriction theorem for the sphere, see [35].

2(d+1) Theorem 6.3. For d ≥ 2, 1 < p < d+3 ,

−1 F [h(| · |)f] sup kϕh(t·)k 2 kfk . b p . B 1 1 p t>0 d( p − 2 ),p

Finally, the upper bounds in Corollary 6.2 and Theorem 2.2 can be interpolated (using Calder´on’s[·, ·]θ-method, see [3]) to get the follow- s ing Mp theorem, for s between p and 2. Theorem 6.4. Let d ≥ 4, and let m = h(| · |) be a bounded radial d function on R and let Th be as in (6.1). Let either 2(d−1) (a) 1 ≤ p < d+1 and p ≤ s ≤ 2, or 2(d−1) 2(d+1) 1 1 d+1 1 1 (b) d+1 ≤ p < d+3 and 2 < s < 2 ( p − 2 ). Then the following are equivalent: p s (i) Th maps L to L . p s (ii) Th maps Lrad to Lrad. d/p (iii) supt>0 t Th[η(t·)] s < ∞. 32 ANDREAS SEEGER

(iv) Let κ = F −1[φh(t·)]; then the function td/p−d/sκ belong to t R t Ls((1 + |r|)(d−1)(1−s/2)dr) with norm uniformly bounded independently of t.

s Similarly, the Mp analogue of Theorem 6.3 is

2(d+1) Theorem 6.5. For d ≥ 2, 1 0 d( s − 2 ),s

A Hardy space bound. The following Hardy space result is more ele- ∞ mentary than Theorem 6.4. We let ϕ be a C0 function supported in (1/2, 2) such that P ϕ(2k·) = 1. k∈Z Theorem 6.6. Let d ≥ 2. For f ∈ S(Rd) deﬁne T by Tc f(ξ) = −k h(|ξ|)fb(ξ) and Tk by Tdkf(ξ) = ϕ(2 |ξ|)Tc f(ξ). Then

kT kH1→L1 ≈ sup kTkkL1→L1 . k

1 1 1 1 Proof. Clearly if T : H → L then Tk : L → L with uniform bounds. k Vice versa let Kk be the radial kernel with Kbk = ϕ(|ξ|)h(2 |ξ|). R ∞ Then, as before we can write Kk = 0 Kk(r)ψ ∗ σrdr where ψ = ψ◦ ∗ψ◦ ∗ψ◦ ∗ψ◦, ψ◦ is supported in a ball of radius 1/4 and all moments of order ≤ 5d of ψ◦ vanish. Let Z ∞ d−1 B = sup |Kk(r)|(1 + r) dr ≈ sup kTkkL1→L1 . k∈Z 0 k We use the atomic decomposition of H1 (see e.g. [52]). By translation invariance and by dilation invariance of the hypothesis we need to prove (6.3) Z ∞ Z ∞ X kd k d−1 Kk(r)2 [ψ ∗ σr](2 ·) ∗ a dr . sup |Kk(r)|(1 + r) dr 0 1 k∈Z 0 k∈Z R where supp(a) ⊂ {y : |y| ≤ 1}, kak∞ ≤ 1 and a(x)dx = 0. For k < 0 we use the cancellation condition. We have kd k d−1 kd k k2 [ψ ∗ σr](2 ·) ∗ ak1 . (1 + r) k2 ψ(2 ·) ∗ ak1 kd k k and since k2 ψ(2 ·)∗ak1 = O(2 ) we see using Minkowski’s inequality Z ∞ X kd k X k Kk(r)2 [ψ ∗ σr](2 ·) ∗ a dr . B 2 . B. 1 k≤0 0 k≤0 NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 33

Next we consider the corresponding sum for k > 0. Note that the expression Z 2k+10 kd k Kk(r)2 [ψ ∗ σr](2 ·) ∗ a dr 0 is supported in {x : |x| ≤ C} and for every n ≥ 210 the expression Z 2k(n+1) kd k Kk(r)2 [ψ ∗ σr](2 ·) ∗ a dr 2kn is supported in an annulus centered at 0 with radius ≈ n and width ≈ 1. kd k We set ψ◦,k = 2 ψ◦(2 ·). Apply the Cauchy-Schwarz inequality on {|x| ≤ C} and (5.3) to get Z 2k+10 X kd k Kk(r)2 [ψ ∗ σr](2 ·) ∗ a dr 1 k≥0 0 Z 2k+10 X kd k . ψ◦,k ∗ Kk(r)2 [ψ◦ ∗ ψ◦ ∗ σr](2 ·) ∗ ψ◦,k ∗ a dr 2 k≥0 0 k+10 Z 2 2 1/2 X kd k . Kk(r)2 [ψ◦ ∗ ψ◦ ∗ σr](2 ·) ∗ ψ◦,k ∗ a dr 2 k≥0 0 k+10 X h Z 2 i21/2 . |Kk(r)|kψ\◦ ∗ σrk∞kψ◦,k ∗ ak2 dr 2 k≥0 0 X 2 1/2 . B( kψ◦,k ∗ ak2) . B. k Finally, applying Cauchy-Schwarz on {x : ||x| − n| ≤ C}, and using (5.3) again yields Z 2k(n+1) X X kd k Kk(r)2 [ψ ∗ σr](2 ·) ∗ a dr k 1 k≥0 n≥210 2 n Z 2k(n+1) X X d−1 kd k . n 2 Kk(r)2 [ψ ∗ σr](2 ·) ∗ a dr k 2 k≥0 n≥210 2 n Z 2k(n+1) X X d−1 d−1 . n 2 |Kk(r)|r 2 dr kak2 k k≥0 n≥210 2 n Z 2k(n+1) X −k d−1 X d−1 . 2 2 |Kk(r)|r dr kak2 . B. k k≥0 n≥210 2 n 34 ANDREAS SEEGER

Remark. There is no equivalent of Theorem 6.6 in the case d = 1. Consider the even multipliers on the real line

2N X i2k|ξ| k mN (ξ) = e χ(|ξ| − 2 ) k=N ∞ where χ ∈ C0 (R). Then the dyadic pieces represent multipliers with√ L1 norm uniform in k, but the H1(R) → L1(R) operator norm is ≈ N 1/p−1/2 for large N; moreover for 1 < p ≤ 2, kmN kMp ≈ N . See [39] and [55]. This example is in some sense optimal, see [11], [48].

An Lp space time estimate for the wave equation. For the proof of our Lp multiplier theorems we need the main inequality (5.7) only for tensor product functions of the form g(y, r) = f(y)κ(r) . However using duality, the more general inequality can be used to prove sharp Lq-Sobolev space time estimates for the wave equation (see [26], [27]).

2(d−1) Theorem 6.7. For d ≥ 4, q > d−3 ,

Z 1 √ 1/q it −∆ q 1 1 1 (6.4) e f dt kfk q , α = d( − ) − . q . Lα −1 2 q 2

This can be compared with the fixed-time bound (Peral [45]) √ it −∆ q q 1 1 e f q . kfkLq , β = (d − 1)( − ) . β 2 q 2d Sogge’s ”local smoothing” conjecture [50] says that for q > d−1 the inequality (6.4) should be true with α > β−1/q. That is the t integration gains almost 1/q derivatives over the fixed time estimate. Hence Theo- rem 6.7 realizes an endpoint version of Sogge’s conjecture in a restricted q-range. For a variable coefficient analogue and other extensions see 1 1 1 [36]. The non-endpoint version of Theorem 6.7, for α > d( 2 − q ) − 2 follows from Wolff’s inequality for plate decompositions of cone multipliers ([65], [32]); these were recently established by Bourgain and 2(d+1) Demeter [7] in the full range p ≥ d−1 . However the extension of 2(d+1) 2(d−1) Theorem 6.7 to the range d−1 ≤ p ≤ d−3 remains open. Finally, q 2 Lee and Vargas [38] proved that the wave operator maps Lε(R ) to Lq(R2 × [−1, 1]), for any ε > 0 and 2 < q ≤ 3. Their method is differ- ent and relies on multilinear adjoint restriction estimates by Bennett, Carbery and Tao [2] and the method by Bourgain and Guth [8]. NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 35

7. Further notes

p 7.1. On maximal operators on Lrad. We describe an extension of Theo- rem 4.1 which covers maximal functions, and consequently some results on pointwise convergence. Theorem 7.1. ([21]) Let h be supported in a compact subinterval of −1 2d (0, ∞) and let Kt = F [h(t| · |)]. Let 1 0 |F [h(t| · |)fb] extends to a bounded operator on p d Lrad(R ) if and only if the condition

k sup |Ktkp < ∞ t∈I is satisﬁed for some open interval I with compact closure in (0, ∞).

One should note that the result does not extend to the full Lp space. λ ∞ Consider hλ(s) = χ(s)(1 − s) where χ ∈ Cc (0, ∞) and χ(1) = 1, and let mλ = hλ(| · |), the case of a truncated Bochner-Riesz multiplier. p Theorem 7.1 yields Lrad boundedness of the maximal operator for p < 2d d+1 and λ > d(1/p − 1/2) − 1/2. However Tao [56] showed that the condition α ≥ d(1/p − 1/2) − 1/2p is a necessary condition for the Lp boundedness of the operator. The problem of ﬁnding the right suﬃcient conditions is largely open (see Tao [58] for some improvements in dimension two of Stein’s classical results presented e.g. in [53]).

7.2. Further results in three and four dimensions. Laura Cladek [15] obtained further results on the characterization of radial Fourier multipliers in three and four dimensions, by using (5.14) and geometric- combinatorial estimates in addition to the L2 methods presented above. She was able to show that for h supported in a compact interval away from the origin, the function m(ξ) = h(|ξ|) is a Fourier h(|ξ|) is a 4 Fourier multiplier in Mp(R ), 1 ≤ p < 36/29 if and only if its Fourier inverse belongs to Lp(R4). Note that 6/5 < 36/29 and 1 < p < 6/5 is the range covered by Theorem 5.1. Moreover, under the same support assumption on h consider the convolution operator Th associated with the multiplier h(| · |). Then Cladek proved that 1 < p < 13/12 the op- p,1 3 p 3 erator Th maps L (R ) → L (R ). It remains currently open whether the restricted strong type estimate can be upgraded to an Lp → Lp estimate.

7.3. Versions for quasiradial multipliers. Jongchon Kim [30] proved an extension of Theorem to quasiradial multipliers of the form h(ρ(ξ)) as considered in Theorem 3.1. Here ρ is a distance function, homogeneous 36 ANDREAS SEEGER of degree one, ρ(tξ) = tρ(ξ), ρ ∈ C∞(Rd \{0}), ρ(ξ) > 0 for ξ 6= 0, with the property that the unit sphere Σρ = {ξ : ρ(ξ) = 1} has nonvanishing Gaussian curvature everywhere. In [31] he also proved a result involving the condition in Conjecture II (cf. (1.3)) for functions of the Laplacian on a compact manifold, extending some results in [35].

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Department of Mathematics, University of Wisconsin-Madison, Madi- son, WI 53706, USA