The Schur Lemma

Appendix A The Schur Lemma

Schur’s lemma provides sufﬁcient conditions for linear operators to be bounded on Lp. Moreover, for positive operators it provides necessary and sufﬁcient such conditions. We discuss these situations.

A.1 The Classical Schur Lemma

We begin with an easy situation. Suppose that K(x,y) is a locally integrable function on a product of two σ-finite measure spaces (X, μ) × (Y,ν), and let T be a linear operator given by T( f )(x)= K(x,y) f (y)dν(y) Y when f is bounded and compactly supported. It is a simple consequence of Fubini’s theorem that for almost all x ∈ X the integral defining T converges absolutely. The following lemma provides a sufficient criterion for the Lp boundedness of T. Lemma. Suppose that a locally integrable function K(x,y) satisfies sup |K(x,y)|dν(y)=A < ∞, x∈X Y sup |K(x,y)|dμ(x)=B < ∞. y∈Y X

Then the operator T extends to a bounded operator from Lp(Y) to Lp(X) with norm − 1 1 A1 p B p for 1 ≤ p ≤ ∞.

Proof. The second condition gives that T maps L1 to L1 with bound B, while the ﬁrst condition gives that T maps L∞ to L∞ with bound A. It follows by the Riesz– − 1 1 Thorin interpolation theorem that T maps Lp to Lp with bound A1 p B p . This lemma can be improved signiﬁcantly when the operators are assumed to be positive.

A.2 Schur’s Lemma for Positive Operators

We have the following necessary and sufﬁcient condition for the Lp boundedness of positive operators.

L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics 250, 589 DOI 10.1007/978-1-4939-1230-8, © Springer Science+Business Media New York 2014 590 A The Schur Lemma

Lemma. Let (X, μ) and (Y,ν) be two σ-ﬁnite measure spaces, where μ and ν are positive measures, and suppose that K(x,y) is a nonnegative measurable function on X ×Y.Let1 < p < ∞ and 0 < A < ∞. Let T be the linear operator T( f )(x)= K(x,y) f (y)dν(y) Y and T t its transpose operator T t (g)(y)= K(x,y)g(x)dμ(x). X To avoid trivialities, we assume that there is a compactly supported, bounded, and positive ν-a.e. function h1 on Y such that T(h1) > 0 μ-a.e. Then the following are equivalent: (i) T maps Lp(Y) to Lp(X) with norm at most A. (ii) For all B > A there is a measurable function h on Y that satisﬁes 0 < h < ∞ ν-a.e., 0 < T(h) < ∞ μ-a.e., and such that

p p T t T(h) p ≤ Bp h p .

(iii) For all B > A there are measurable functions u on X and v on Y such that 0 < u < ∞ μ-a.e., 0 < v < ∞ ν-a.e., and such that

T(up ) ≤ Bvp , T t (vp) ≤ Bup.

Proof. First we assume (ii) and we prove (iii). Deﬁne u,v by the equations vp = T(h) and up = Bh and observe that (iii) holds for this choice of u and v. Moreover, observe that 0 < u,v < ∞ a.e. with respect to the measures μ and ν, respectively. Next we assume (iii) and we prove (i). For g in Lp (X) we have v(x) u(y) T( f )(x)g(x)dμ(x)= K(x,y) f (y)g(x) dν(y)dμ(x). X X Y u(y) v(x)

We now apply Holder’s¨ inequality with exponents p and p to the functions

v(x) u(y) f (y) and g(x) u(y) v(x) with respect to the measure K(x,y)dν(y)dμ(x) on X ×Y. Since p 1 v(x) p 1 ( )p ( , ) μ( ) ν( ) ≤ p f y p K x y d x d y B f Lp(Y) Y X u(y) A.2 Schur’s Lemma for Positive Operators 591 and 1 p 1 u(y) p ( )p ( , ) ν( ) μ( ) ≤ p , g x K x y d y d x B g Lp (X) X Y v(x)p we conclude that 1 1 + ( )( ) ( ) μ( ) ≤ p p . T f x g x d x B f Lp(Y) g Lp (X) X

Taking the supremum over all g with Lp (X) norm 1, we obtain ( ) ≤ . T f Lp(X) B f Lp(Y)

Since B was any number greater than A, we conclude that ≤ , T Lp(Y)→Lp(X) A which proves (i). We ﬁnally assume (i) and we prove (ii). Without loss of generality, take here A = 1 and B > 1. Deﬁne a map S : Lp(Y) → Lp(Y) by setting

p p S( f )(y)= T t T( f ) p p (y).

We observe two things. First, f1 ≤ f2 implies S( f1) ≤ S( f2), which is an easy consequence of the fact that the same monotonicity is valid for T. Next, we observe that f Lp ≤ 1 implies that S( f )Lp ≤ 1 as a consequence of the boundedness of T on Lp (with norm at most 1). Construct a sequence hn, n = 1,2,..., by induction as follows. Pick h1 > 0on Y as in the hypothesis of the theorem such that T(h1) > 0 μ-a.e. and such that −p p h1Lp ≤ B (B − 1). (The last condition can be obtained by multiplying h1 by a small constant.) Assuming that hn has been deﬁned, we deﬁne 1 h + = h + S(h ). n 1 1 Bp n

We check easily by induction that we have the monotonicity property hn ≤ hn+1 and the fact that hnLp ≤ 1. We now deﬁne

h(x)=suph (x)= lim h (x). n →∞ n n n

Fatou’s lemma gives that hLp ≤ 1, from which it follows that h < ∞ ν-a.e. Since h ≥ h1 > 0 ν-a.e., we also obtain that h > 0 ν-a.e. 592 A The Schur Lemma

Next we use the Lebesgue dominated convergence theorem to obtain that hn → h p p p in L (Y). Since T is bounded on L ,itfollowsthatT(hn) → T(h) in L (X).It p p p t p follows that T(hn) p → T(h) p in L (X). Our hypothesis gives that T maps L (X) p p p t t p to L (Y) with norm at most 1. It follows T T(hn) p → T T(h) p in L (Y). p ( ) → ( ) p( ) Raising to the power p , we obtain that S hn S h in L Y . ( ) → ( ) It follows that for some subsequence nk of the integers we have S hnk S h a.e. in Y. Since the sequence S(hn) is increasing, we conclude that the entire sequence S(hn) converges almost everywhere to S(h). We use this information in conjunction 1 with hn+ = h + S(hn). Indeed, letting n → ∞ in this identity, we obtain 1 1 Bp 1 h = h + S(h). 1 Bp

p Since h1 > 0 ν-a.e. it follows that S(h) ≤ B h ν-a.e., which proves the required estimate in (ii). p It remains to prove that 0 < T(h) < ∞ μ-a.e. Since hLp ≤ 1 and T is L bounded, it follows that T(h)Lp ≤ 1, which implies that T(h) < ∞ μ-a.e. We also have T(h) ≥ T(h1) > 0 μ-a.e.

A.3 An Example

Consider the Hilbert operator ∞ f (y) T( f )(x)= dy, 0 x + y where x ∈ (0,∞). The operator T takes measurable functions on (0,∞) to measurable functions on (0,∞). We claim that T maps Lp(0,∞) to itself for 1 < p < ∞; precisely, we have the estimate ∞ π ( )( ) ( ) ≤ . T f x g x dx f Lp(0,∞) g Lp (0,∞) 0 sin(π/p) To see this we use Schur’s lemma. We take

− 1 u(x)=v(x)=x pp .

We have that

1 1 ∞ − ∞ − 1 y p − 1 t p 1 −1 1 − T(up )(x)= dy = x p dt = v(x)p (1 − s) p s p 1 ds, 0 x + y 0 1 +t 0 A.3 An Example 593 where last identity follows from the change of variables s =(1 +t)−1. Now an easy calculation yields 1 1 −1 1 − π ( − ) p p 1 = 1 , 1 = , 1 s s ds B p p 0 sin(π/p) π p→ p ≤ so the lemma in Appendix I.2 gives that T L L sin(π/p) . The sharpness of this constant follows by considering the sequence of functions

− 1+ε ( )= p χ ( ) hε t t (1,∞) t for ε > 0. To verify the last assertion notice that for x > 1 and 0 < ε < p−1wehave

− 1+ε − 1+ε ∞ t p 1 t p T(hε )(x)= dt − dt 0 x +t 0 x +t +ε +ε ∞ − 1 1 − 1 − 1+ε t p − 1+ε x t p = x p dt − x p dt 0 1 +t 0 1 +t +ε ∞ − 1 1 − 1+ε t p − 1+ε x x − 1+ε ≥ x p dt − x p t p dt 0 1 +t x + 1 0 +ε ∞ − 1 − 1+ε t p 1 p = x p dt − . 0 1 +t x + 1 p − 1 − ε Notice that the expression directly after the ≥ sign is nonnegative, and so is the last expression. It follows that

− 1+ε ∞ p t p 1 ( ε ) ≥ ε p − . T h Lp(1,∞) dt h L (1,∞) (·)+1 Lp(1,∞) 0 1 +t p − 1 − ε

= = ε−1/p Dividing both sides of this inequality by hε Lp(0,∞) hε Lp(1,∞) , and letting ε → 0 we obtain

∞ − 1 ( ε ) p p π T h L (0,∞) t 1 1 liminf ≥ dt = B , = . ε→ + p p (π/ ) 0 hε Lp(0,∞) 0 1 t sin p

Since T(hε ) p π L (0,∞) ≤ limsup (π/ ) ε→0 hε Lp(0,∞) sin p as already shown, it follows that ( ) T hε Lp(0,∞) π lim = . ε→ (π/ ) 0 hε Lp(0,∞) sin p 594 A The Schur Lemma A.4 Historical Remarks

We make some comments related to the history of Schur’s lemma. Schur [312]ﬁrst proved a matrix version of the lemma in Appendix I.1 when p = 2. Precisely, Schur’s original version was the following: If K(x,y) is a positive decreasing function in both variables and satisﬁes

sup∑K(m,n)+sup∑K(m,n) < ∞, m n n m then ∑∑amK(m,n)bn ≤ C{am}m2 {bn}n2 . m n Hardy–Littlewood and Polya´ [185] extended this result to Lp for 1 < p < ∞ and disposed of the condition that K be a decreasing function. Aronszajn, Mulla, and Szeptycki [6] proved that (iii) implies (i) in the lemma of Appendix I.2. Gagliardo in [149] proved the converse direction that (i) implies (iii) in the same lemma. The case p = 2 was previously obtained by Karlin [208]. Condition (ii) was introduced by Howard and Schep [197], who showed that it is equivalent to (i) and (iii). A multilinear analogue of the lemma in Appendix I.2 was obtained by Grafakos and Torres [174]; the easy direction (iii) implies (i) was independently observed by Bekolle,´ Bonami, Peloso, and Ricci [20]. See also Cwikel and Kerman [107] for an alterna- tive multilinear formulation of the Schur lemma. The case p = p = 2 of the application in Appendix I.3 is a continuous version of Hilbert’s double series theorem. The discrete version was ﬁrst proved by Hilbert in his lectures on integral equations (published by Weyl [367]) without a determination of the exact constant. This exact constant turns out to be π, as discovered by Schur [312]. The extension to other p’s (with sharp constants) is due to Hardy and M. Riesz and published by Hardy [181]. Appendix B Smoothness and Vanishing Moments

B.1 The Case of No Cancellation

Let a,b ∈ Rn, μ,ν ∈ R, and M,N > n.Set

μn νn ( , μ, ,ν, )= 2 2 . I a M;b N μ M ν N dx Rn (1 + 2 |x − a|) (1 + 2 |x − b|) Then we have

2min(μ,ν)n I(a, μ,M;b,ν,N) ≤ C , 0 (M,N) 1 + 2min(μ,ν)|a − b| min where M4N N4M C = v + 0 n M − n N − n n and vn is the volume of the unit ball in R . To prove this estimate, ﬁrst observe that dx ≤ vnM . M Rn (1 + |x|) M − n

Without loss of generality, assume that ν ≤ μ. Consider the cases 2ν |a−b|≤1 and 2ν |a − b|≥1. In the case 2ν |a − b|≤1weusetheestimate

νn νn min(M,N) 2 ν 2 2 ≤ 2 n ≤ , (1 + 2ν |x − b|)N (1 + 2ν |a − b|)min(M,N) and the claimed inequality is a consequence of the estimate νn min(M,N) μn ( , μ, ,ν, ) ≤ 2 2 2 . I a M;b N ν ( , ) μ M dx (1 + 2 |a − b|)min M N Rn (1 + 2 |x − a|)

ν In the case 2 |a−b|≥1letHa and Hb be the two half-spaces, containing the points a and b, respectively, formed by the hyperplane perpendicular to the line segment n [a,b] at its midpoint. Split the integral over R as the integral over Ha and the integral ∈ | − |≥ 1 | − | ∈ over Hb.Forx Ha use that x b 2 a b .Forx Hb use a similar inequality and the fact that 2ν |a − b|≥1 to obtain

L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics 250, 595 DOI 10.1007/978-1-4939-1230-8, © Springer Science+Business Media New York 2014 596 B Smoothness and Vanishing Moments

μn μn M (ν−μ)(M−n) νn 2 ≤ 2 ≤ 4 2 2 . (1 + 2μ |x − a|)M ( μ 1 | − |)M (1 + 2ν |a − b|)M 2 2 a b The claimed estimate follows.

B.2 One Function has Cancellation

Fix a,b ∈ Rn, M > 0, μ,ν ∈ R, and L ∈ Z+. Assume that ν ≥ μ and that N > L + M + n. Given a function Ψ on Rn and another function Φ ∈ C L(Rn) consider the quan- tities

M,L μ M β Kμ,a (Φ)= sup sup (1 + 2 |x − a|) |∂ Φ(x)|, |β|=L x∈Rn N (Ψ)= ( + ν | − |)N|Ψ( )| Kν,b sup 1 2 x b x x∈Rn and assume they are both ﬁnite. Suppose, moreover, that β Ψ(x)x dx = 0 for all |β|≤L − 1. Rn

Then there is a constant CM,N,L,n such that −νL−νn M,L N 2 Φ( )Ψ( ) ≤ , , , μ, (Φ) (Ψ) . x x dx CM N L n K a Kν,b μ M Rn (1 + 2 |a − b|)

To prove this claim, we subtract the Taylor polynomial of order L − 1ofΦ at the point a from the function Φ using the cancellation of Ψ. Then we write Φ(x)Ψ(x)dx Rn γ ∂ Φ(b) γ = Φ(x) − ∑ (x − b) Ψ(x)dx n γ R |γ|≤L−1 ! ∂ β Φ(ξ ) b,x β = ∑ (x − b) Ψ(x)dx n β R |β|=L ! | − |L ≤ M,L(Φ) N (Ψ) 1 x b 1 Kμ,a Kν,b ∑ μ ν dx β n ( + |ξ , − |)M ( + | − |)N |β|=L ! R 1 2 b x a 1 2 x b −νL ≤ M,L(Φ) N (Ψ) 1 2 1 Kμ,a Kν,b ∑ μ ν − dx β n ( + |ξ , − |)M ( + | − |)N L |β|=L ! R 1 2 b x a 1 2 x b where ξb,x lies on the open segment joining b to x. B.3 One Function has Cancellation: An Example 597

Now since ν ≥ μ, the triangle inequality and the fact μ ≤ ν give

μ μ μ 1 + 2 |a − b|≤1 + 2 |a − ξb,x| + 2 |ξb,x − b| μ ν ≤ 1 + 2 |a − ξb,x| + 2 |x − b| μ ν ≤ (1 + 2 |ξb,x − a|)(1 + 2 |x − b|), hence + ν | − | 1 ≤ 1 2 x b . μ μ 1 + 2 |ξb,x − a| 1 + 2 |a − b| Thus we obtain the estimate Φ(x)Ψ(x)dx n R −νL ≤ M,L(Φ) N (Ψ) 2 1 1 Kμ,a Kν, μ ∑ ν − − dx b ( + | − |)M β n ( + | − |)N L M 1 2 a b |β|=L ! R 1 2 x b −νn −νL M,L N 2 2 = Kμ, (Φ)Kν, (Ψ) C , , , , a b (1 + 2μ |a − b|)M M N L n since the last integral produces a constant in view of the assumption N > L+M +n.

B.3 One Function has Cancellation: An Example

Fix L ∈ Z+, A,B,M,N > 0, and a,b ∈ Rn satisfy N > M + L + n and ν ≥ μ.Let Φ ∈ C L(Rn) and Ψ be another function on Rn.Let

α A = sup sup |∂ Φ(x)|(1 + |x|)M |α|=L x∈Rn and α B = sup |∂ Ψ(x)|(1 + |x|)N x∈Rn and suppose that A + B < ∞. Suppose moreover that β Ψ(x)x dx = 0 for all |β|≤L − 1. Rn Then there is a constant CM,N,L,n such that μn −(ν−μ)L 2 2 Φ −μ ( − )Ψ −ν ( − ) ≤ . 2 x a 2 x b dx CM,N,L,n AB μ M Rn (1 + 2 |a − b|) 598 B Smoothness and Vanishing Moments

In particular, we have

μn −(ν−μ)L 2 2 (Φ −μ ∗Ψ −ν )(x) ≤ C , , , AB 2 2 M N L n (1 + 2μ |x|)M

−n −1 −n −1 −μ Let Φt (x)=t Φ(t x) and Ψs(x)=s Ψ(s x) for t,s > 0. Set 2 = t and 2−ν = s. The assumption ν ≥ μ can be equivalently stated as s ≥ t. The preceding inequalities can also be written equivalently as t−n s L Φ ( − )Ψ ( − ) ≤ t . t x a s x b dx CM,N,L,n AB −1 M Rn (1 +t |a − b|) and −n s L t t (Φ ∗Ψ )(x) ≤ C , , , AB t s M N L n (1 + 2μ |x|)M for all x ∈ Rn. These results are easy consequences of the inequality in Appendix B.2.IfΨ has no cancellation (i.e., L = 0), then the estimate reduces to that in Appendix B.1.

B.4 Both Functions have Cancellation: An Example

Let L ∈ Z+, A,B,N > 0 and μ,ν ∈ R. Suppose that N > L + n.LetΩ,Ψ be C L functions on Rn such that

γ A = sup sup |∂ Ω(x)|(1 + |x|)N < ∞ |γ|≤L x∈Rn

γ B = sup sup |∂ Ψ(x)|(1 + |x|)N < ∞ |γ|≤L x∈Rn and moreover, for all multi-indices β with |β|≤L − 1wehave β β Ω(x)x dx = Ψ(x)x dx = 0. Rn Rn > < − − Then given M 0 satisfying M N L n there is a constant CN,M,L,n such that for all x,a,b ∈ Rn we have μn νn −|ν−μ|L min(2 ,2 )2 Ω −μ ( − )Ψ −ν ( − ) ≤ . 2 x a 2 x b dx CN,M,L,n AB μ ν M Rn (1 + min(2 ,2 )|a − b|) B.5 The Case of Three Factors with No Cancellation 599

In particular, we have

μn νn −|ν−μ|L min(2 ,2 )2 (Ω −μ ∗Ψ −ν )(x) ≤ C , , , AB 2 2 N M L n (1 + min(2μ ,2ν )|x|)M for all x ∈ Rn and for all μ,ν ∈ R. −n −1 −n −1 −μ Let Ωt (x)=t Ω(t x) and Ψs(x)=s Ψ(s x) for t,s > 0. Then if 2 = t and 2−ν = s, the preceding statements can also be written as L max(t,s)−n min s , t Ω ( − )Ψ ( − ) ≤ t s . t x a s x b dx CN,M,L,n AB −1 M Rn (1 + max(t,s) |a − b|) and ( , )−n s , t L max t s min t s (Ω ∗Ψ )(x) ≤ C , , , AB t s N M L n (1 + max(t,s)−1|x|)M for all x ∈ Rn and for all t,s > 0. These assertions follow from the results in Appendix B.3 by interchanging the roles of Ω and Ψ, noting that

γ A ≥ sup sup |∂ Ω(x)|(1 + |x|)M |γ|≤L x∈Rn

γ B ≥ sup sup |∂ Ψ(x)|(1 + |x|)M |γ|≤L x∈Rn since M < N.

B.5 The Case of Three Factors with No Cancellation

Given three numbers a,b,c we denote by med(a,b,c) the number with the property min(a,b,c) ≤ med(a,b,c) ≤ max(a,b,c). n n Let xν ,xμ ,xλ ∈ R . Suppose that ψν , ψμ , ψλ are functions deﬁned on R such that for some N > n and some Aν ,Aμ ,Aλ < ∞ we have

νn/2 |ψ ( )|≤ 2 , ν x Aν ν (1 + 2 |x − xν |)N μn/2 |ψ ( )|≤ 2 , μ x Aμ μ (1 + 2 |x − xμ |)N λn/2 |ψ ( )|≤ 2 , λ x Aλ λ (1 + 2 |x − xλ |)N 600 B Smoothness and Vanishing Moments for all x ∈ Rn. Then the following estimate is valid: |ψν (x)||ψμ (x)||ψλ (x)|dx Rn −max(μ,ν,λ)n/2 med(μ,ν,λ)n/2 min(μ,ν,λ)n/2 ≤ CN,n Aν Aμ Aλ 2 2 2 (ν,μ) (μ,λ) (λ,ν) ((1 + 2min |xν − xμ |)(1 + 2min |xμ − xλ |)(1 + 2min |xλ − xν |))N for some constant CN,n > 0 independent of the remaining parameters. Analogous estimates hold if some of these three factors are assumed to have cancellation and the others vanishing moments; see Grafakos and Torres [176]for precise statements and applications. Similar estimates with m factors, m ∈ Z+,are studied in Benyi´ and Tzirakis [26]. Glossary

A ⊆ BAis a subset of B (also denoted by A ⊆ B) A BAis a proper subset of B A ⊃ BBis a proper subset of A Ac the complement of a set A

χE the characteristic function of the set E d f the distribution function of a function f f ∗ the decreasing rearrangement of a function f fn ↑ ffn increases monotonically to a function f Z the set of all integers Z+ the set of all positive integers {1,2,3,...} Zn the n-fold product of the integers R the set of real numbers R+ the set of positive real numbers Rn the Euclidean n-space Q the set of rationals Qn the set of n-tuples with rational coordinates C the set of complex numbers Cn the n-fold product of complex numbers T the unit circle identiﬁed with the interval [0,1] Tn the n-dimensional torus [0,1]n,

2 2 n |x| |x1| + ···+ |xn| when x =(x1,...,xn) ∈ R

L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics 250, 601 DOI 10.1007/978-1-4939-1230-8, © Springer Science+Business Media New York 2014 602 Glossary

Sn−1 the unit sphere {x ∈ Rn : |x| = 1} e j the vector (0,...,0,1,0,...,0) with1inthe jth entry and 0 elsewhere logt the logarithm to base e of t > 0 > = > loga t the logarithm to base a of t 0(1 a 0) log+ t max(0,logt) for t > 0 [t] the integer part of the real number t · n =( ,..., ) =( ,..., ) x y the quantity ∑ j=1 x jy j when x x1 xn and y y1 yn B(x,R) the ball of radius R centered at x in Rn n−1 ωn−1 the surface area of the unit sphere S n vn the volume of the unit ball {x ∈ R : |x| < 1} |A| the Lebesgue measure of the set A ⊆ Rn dx Lebesgue measure

1 ( ) AvgB f the average |B| B f x dx of f over the set B , ( ) ( ) f g the real inner product Rn f x g x dx f |g the complex inner product n f (x)g(x)dx R u, f the action of a distribution u on a function f p the number p/(p − 1), whenever 0 < p = 1 < ∞ 1 the number ∞ ∞ the number 1 f = O(g) means | f (x)|≤M|g(x)| for some M for x near x0 −1 f = o(g) means | f (x)||g(x)| → 0asx → x0 At the transpose of the matrix A A∗ the conjugate transpose of a complex matrix A A−1 the inverse of the matrix A O(n) the space of real matrices satisfying A−1 = At

TX→Y the norm of the (bounded) operator T : X → Y ≈ > −1 ≤ B ≤ A B means that there exists a c 0 such that c A c

|α| indicates the size |α1| + ···+ |αn| of a multi-index α =(α1,...,αn) ∂ m ( ,..., ) j f the mth partial derivative of f x1 xn with respect to x j α ∂ α ∂ 1 ···∂ αn f 1 n f Glossary 603

C k the space of functions f with ∂ α f continuous for all |α|≤k

C0 space of continuous functions with compact support

C00 the space of continuous functions that vanish at infinity C ∞ 0 the space of smooth functions with compact support D the space of smooth functions with compact support S the space of Schwartz functions C ∞ ∞ C k the space of smooth functions k=1 D(Rn) the space of distributions on Rn S (Rn) the space of tempered distributions on Rn E (Rn) the space of distributions with compact support on Rn P the set of all complex-valued polynomials of n real variables S (Rn)/P the space of tempered distributions on Rn modulo polynomials (Q) the side length of a cube Q in Rn ∂Q the boundary of a cube Q in Rn Lp(X, μ) the Lebesgue space over the measure space (X, μ) Lp(Rn) the space Lp(Rn,|·|) Lp,q(X, μ) the Lorentz space over the measure space (X, μ) p ( n) p( ) n Lloc R the space of functions that lie in L K for any compact set K in R |dμ| the total variation of a finite Borel measure μ on Rn M (Rn) the space of all finite Borel measures on Rn n p Mp(R ) the space of L Fourier multipliers, 1 ≤ p ≤ ∞ M p,q(Rn) the space of translation-invariant operators that map Lp(Rn) to Lq(Rn) μ | μ| μ n M Rn d the norm of a finite Borel measure on R M the centered Hardy–Littlewood maximal operator with respect to balls M the uncentered Hardy–Littlewood maximal operator with respect to balls

Mc the centered Hardy–Littlewood maximal operator with respect to cubes

Mc the uncentered Hardy–Littlewood maximal operator with respect to cubes

Mμ the centered maximal operator with respect to a measure μ

Mμ the uncentered maximal operator with respect to a measure μ 604 Glossary

Ms the strong maximal operator

Md the dyadic maximal operator M# the sharp maximal operator M the grand maximal operator p n p Ls (R ) the inhomogeneous L Sobolev space p n p L˙ s (R ) the homogeneous L Sobolev space

Λα (Rn) the inhomogeneous Lipschitz space

Λ˙α (Rn) the homogeneous Lipschitz space H p(Rn) the real Hardy space on Rn p n n Bs,q(R ) the inhomogeneous Besov space on R p n n B˙s,q(R ) the homogeneous Besov space on R p n n B˙s,q(R ) the homogeneous Besov space on R p n n Fs,q(R ) the inhomogeneous Triebel–Lizorkin space on R p n n F˙s,q(R ) the homogeneous Triebel–Lizorkin space on R BMO(Rn) the space of functions of bounded mean oscillation on Rn References

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jth transpose symbol of a symbol, 534 Bochner–Riesz means, 339 m-linear Riesz transform, 540 boundedness in higher dimensions, 381 m-linear convolution operator, 480, 526 Bochner–Riesz operator m-linear multiplier, 480 boundedness result, 342 maximal, 356, 392 accretive function, 297 unboundedness result, 340 accretivity condition on matrices, 303 bounded distribution, 56 adjoint kernel, 211 bounded mean oscillation, 154 adjoint of an operator, 210 admissible growth, 513 Calderon´ commutator, 215, 254 almost everywhere convergence Calderon´ commutator kernel, 215 of Fourier integrals Calderon´ reproducing formula, 7 of L2 functions, 415 Calderon–Vaillancourt´ theorem, 279 of Lp functions, 450 Calderon–Zygmund´ kernel almost orthogonality lemma, 269 m-linear, 538 analytic family Calderon–Zygmund´ decomposition, 225, 226, of multilinear operators, 513 462 antisymmetric kernel, 213, 540 with bounded overlap, 235 arms of a sprouted ﬁgure, 329 Calderon–Zygmund´ operator atom CZO(δ,A,B),215 for H p(Rn),120 boundedness of, 226 smooth, 109 deﬁnition on L∞, 222 atomic decomposition Calderon–Zygmund´ singular integral operator, nonsmooth, 114 221 of Hardy space, 120 Calderon–Zygmund´ theorem smooth, 109 multilinear, 541 auxiliary maximal function, 79 Carleson function, 174 average of a function, 154 Carleson measure, 174 Carleson measures and BMO,178 bad function, 83, 149 Carleson operator, 415 basic estimate over a single tree, 430 maximal, 473 Besov–Lipschitz space one-sided, 417 homogeneous, 92 Carleson’s theorem, 177, 415 inhomogeneous, 92 Carleson–Sjolin¨ theorem, 342 Bessel potential operator Jz,13 Cauchy integral along a Lipschitz curve, 284 bilinear fractional integral, 482 Cauchy–Riemann equations bilinear Hilbert transform, 493 generalized, 143

L. Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics 250, 621 DOI 10.1007/978-1-4939-1230-8, © Springer Science+Business Media New York 2014 622 Index

Coifman-Meyer multiplier theorem, 561 Gagliardo–Nirenberg inequality, 33 commutator of a singular integral, 196 generalized Cauchy–Riemann equations, 143 commutator of operators, 305 good function, 83, 149 complex inner product, 210 good lambda inequality conical tent, 182 for the Carleson operator, 477 continuous paraproduct, 268 for the sharp maximal function, 186 continuous wave packet, 449 gradient, 1 continuously differentiable function grand maximal function, 59, 79 of order N,1 Cotlar’s inequality, 228 H1-BMO duality, 169 Cotlar–Knapp–Stein lemma, 269 Hormander-Mihlin¨ multiplier theorem counting function, 447 multilinear, 564 Hardy space, 56 δ-separated tubes, 371 vector-valued, 80 derivative Hardy space characterizations, 59, 80 of a function (partial), 1 Hardy–Littlewood–Sobolev theorem, 11 difference operator, 35 hemispherical tent, 182 dilation operator, 417 Holder’s¨ inequality directional Carleson operators, 472 for Orlicz spaces, 205 discrete fractional integral, 20 homogeneous Besov–Lipschitz space, 92 discrete maximal operator, 19 homogeneous Lipschitz space, 36 discretization homogeneous singular integral of the Carleson operator, 421 multilinear, 554 distributional inequality, 182 homogeneous Sobolev space, 29 homogeneous Triebel–Lizorkin space, 92 for the sharp maximal function, 186 Hormander’s¨ condition, 234 distributional kernel, 213 Hunt’s theorem, 450 distributions modulo polynomials, 2 divergence form operator, 303 inﬁnitely differentiable function, 1 doubling measure, 194 inhomogeneous Besov–Lipschitz space, 92 dual inhomogeneous Triebel–Lizorkin space, 92 of multilinear convolution operator, 534 inner product duality H1-BMO,169 complex, 210 dyadic BMO,195 real, 209 dyadic maximal function, 185 integral representation of CZO(δ,A,B),216 dyadic tile, 417 interpolation between H1 and Lp,148 ellipticity condition on matrices, 303 using BMO,190 energy lemma, 430 energy of a function John–Nirenberg theorem, 160 with respect to a set of tiles, 429 exponential integrability, 33 Kakeya maximal function, 357 exponentially integrable, 164 Kakeya maximal function theorem, 359 extension operator, 385 Kakeya maximal operator, 349, 367 extension theorem without dilations, 349, 366 for a hypersurface, 376 Kato’s square root operator, 303 kernel Fefferman’s theorem, 169 of a multilinear operator, 486 ﬁnitely simple functions, 493 fractional integral Laplacian, 9 discrete, 20 powers of, 9 fractional integration theorem, 11 linearization fractional maximal function, 18 of the Carleson operator, 425 frequency projection of a tile, 417 of the Kakeya maximal operator, 369 Index 623

Lipschitz function, 34 multilinear symbol, 527 Lipschitz space, 35 multiplier problem for the ball, 337 homogeneous, 36 inhomogeneous, 35 nonsmooth atomic decomposition, 114 Littlewood–Paley characterization nontangential maximal function, 59, 79, 173 of Hardy space, 101 norm of homogeneous Lipschitz spaces, 39 of a multilinear operator, 485 of homogeneous Sobolev space, 29 norm of a Schwartz function, 76 of inhomogeneous Lipschitz spaces, 45 normalized bump, 236 of inhomogeneous Sobolev space, 25 Littlewood–Paley operator, 5, 258 Littlewood–Paley theorem operator associated with a standard kernel, 215 for intervals with equal length, 350 Orlicz maximal operator, 198 Orlicz norm, 197 Marcinkiewicz function, 86 Orlicz space, 197 mass lemma, 430 oscillation of a function, 154 mass of a set of tiles, 429 maximal Bochner–Riesz operator, 356, 392 para-accretive function, 302 maximal Carleson operator, 473 paraproduct, 257, 556 maximal directional Carleson operator, 476 paraproduct operator, 258 maximal function paraproducts associated with a set of directions, 357 boundedness of, 260 ∗∗ auxiliary M ,59 partial derivative, 1 b∗∗ auxiliary Mb ,79 partial order of tiles, 428 directional along a vector, 373 partial sum discrete, 19 of Littlewood–Paley operator, 258 dyadic, 185 Peetre maximal function, 99 fractional, 18 Peetre’s theorem, 99 grand, 59, 79 Peetre–Spanne–Stein theorem, 233 Kakeya, 357, 367 Poisson kernel, 56 Kakeya without dilations, 366 Poisson maximal function, 56, 80 nontangential, 59, 79, 173 potential Peetre, 99 Bessel Jz,13 Poisson, 56, 80 Riesz Is,10 sharp, 184 powers of Laplacian, 9 with respect to a measure, 194 pseudo-Haar basis, 302 smooth, 59, 79 pseudodifferential operator, 279 strong, 357 multilinear, 556 with respect to cubes, 357 pyramidal tent, 182 maximal operator associated with a cube, 198 quadratic T(1) type theorem, 293 of Orlicz type, 198 T(b) maximal singular integrals quadratic theorem, 297 boundedness of, 228 maximal singular operator, 218 real inner product, 209 mean of a function, 154 resolution of an operator, 289 mean oscillation of a function, 154 resolution of the Cauchy integral, 292 modulation operator, 417 restricted weak type, 482 multi-index, 1 restriction condition, 375 multilinear complex interpolation, 516 restriction of the Fourier transform multilinear convolution operator, 486 on a hypersurface, 375 multilinear interpolation restriction theorem for analytic families of operators, 514 in R2, 385 multilinear multiplier, 527 Riesz potential operator Is,10 624 Index

Schur Lemma, 589 T(1) reduced theorem, 273 Schwartz kernel, 213 T(1) theorem, 237, 275 Schwartz kernel theorem, 212 T(b) theorem, 302 selection of trees, 428 tempered distributions modulo polynomials, 2 self-adjoint operator, 210 tensor product, 480 self-transpose operator, 210 tent semitile, 417, 428 conical, 182 sharp maximal function, 184 cylindrical, 173 controls singular integrals, 191 hemispherical, 182 with respect to a measure, 194 over a set, 174 singular integral characterization of H1,141 pyramidal, 182 singular integrals on function spaces, 127 tile, 417 singular integrals on Hardy spaces, 81 of a given scale, 418 SK(δ,A),211 time projection of a tile, 417 smooth atom, 109 total order of differentiation, 1 smooth atomic decomposition, 109 translation operator, 417 smooth function, 1 transpose smooth function with compact support, 1 of a multilinear operator, 486 smooth maximal function, 79 transpose kernel, 211 smoothing operators, 11 transpose of an operator, 210 smoothly truncated singular integral, 132 tree of tiles, 428 Sobolev embedding theorem, 24 1-tree of tiles, 428 Sobolev space, 20 2-tree of tiles, 428 homogeneous, 29 Triebel–Lizorkin space inhomogeneous, 21 homogeneous, 92 space inhomogeneous, 92 BMO,154 truncated kernel, 217 BMOd ,195 α,q tube, 369 Bp ,92 α,q Fp ,92 H p(Rn),56 vector-valued Hardy space, 80 n vector-valued inequalities Λγ (R ),35 n for half-plane multipliers, 335 D0(R ),221 S /P,2 for the Carleson operator, 478 .α,q Bp ,92 .α,q wave packet, 449 F.p ,92 Λγ ,36 WBP weak boundedness property, 237 C N ,1 weak type Orlicz estimate, 206 C ∞,1 weighted estimates C ∞ for the Carleson operator, 472 0 ,1 sprouting of a triangle, 329 Whitney decomposition, 175 square root operator, 303, 322 standard kernel, 211 Young’s function, 197 0 symbol of class S1,0, 282 Young’s inequality m symbol of class Sρ,δ ,279 for Orlicz spaces, 205