Singular Integrals and weights: a modern introduction

Transcription

1 Singular Integrals and weights: a modern introduction Carlos Pérez University of Seville 2 Spring School on Function Spaces and Inequalities Paseky, June 203 The author would like to thank Professors Jaroslav Lukes, Lubos Pick and Petr Posta for the invitation to deliver these lectures. 2 The author would like to acknowledge the support of the Spanish grant MTM and the grant from the Junta de Andalucía, proyecto de excelencia FM-4745

2 Contents Improving the A p theorem: the mixed A p A approach 3. Improving the A p theorem: the mixed A p A approach Sharpenes of the exponents, Yano s condition and the Rubio de Francia s algorithm Two weight problem: sharp Sawyer s theorem Some applications 4 2. The factorization theorem with control on the bounds Commutators of operators with BMO functions: quadratic estimates A preliminary result: a sharp connection between the John-Nirenberg theorem and the A 2 class Results within the A p context Singular Integrals: The A theory 2 3. Estimates involving A weights The main lemma and the proof of the linear growth theorem Proof of the logarithmic growth theorem Singular Integrals: The A p theory 3 4. The dyadic A p theory of Singular Integrals

3 Chapter Improving the A p theorem: the mixed A p A approach. Improving the A p theorem: the mixed A p A approach Most of the basic definitions and notations can be found from the lectures delivered by Prof. Javier Duoandikoetxea Paseky-203, Lecture notes [3]. These lectures started from the celebrated paper [37] by B. Muckenhoupt where it is proved the fundamental result characterizing all the weights for which the Hardy Littlewood maximal operator is bounded on L p (w); the surprisingly simple necessary and sufficient condition is the celebrated A p condition of Muckenhoupt: ( ) ( p [w] Ap := sup w(x) dx w(x) dx) p. Of course the operator norm M L p (w) will depend on the A p condition of w but the first quantitative estimate was proved by S. Buckley [4] as part of his Ph.D. thesis: Let < p < and let w A p, then the Hardy-Littlewwod maximal function satisfies the following operator estimate: namely, M L p (w) c n p [w] p A p sup M w A p [w] Lp (w) c n p. (..) p A p Furthermore the result is sharp in the sense that: for any ɛ > 0 sup M w A p [w] p ɛ Lp (w) = (..2) A p 3

4 See the lecture notes [3] where a proof of this result can be found. Here we will present an improvement of this result based on an argument closely related to the original one of Muckenhoupt with better bounds. To do this we need to recall the A class of weights. This class of weights is defined in a natural way by the union of all A p class A := p> A p. The A class of weights shares a lot of interesting properties, we remit to [3] for more information. It is known that another possible way of defining a constant for this class is by means of the quantity ( A := sup w exp [w] exp ) log w, as can be found in for instance in [9]. This constant was introduced by Hruščev in [22]. This definition is natural because is obtained letting p in the definition of the A p constant. In fact we have by Jensen s inequality: [w] exp A [w] Ap. On the other hand in [25] the authors use a new A constant (which was originally introduced by Fujii in [7] and rediscovered later by Wilson in [49]) which is more suitable. Definition... that [w] A := sup M(wχ ) dx. w() Observe that [w] A by the Lebesgue differentiation theorem. It is not difficult to show using the logarithmic maximal function ( ) M 0 f := sup exp log f χ [w] A c n [w] exp A See [25] for details. In fact it is shown in the same paper with explicit examples that [w] A is much smaller than [w] exp A (actually exponentially smaller). We also refer the reader to the forthcoming work of Duoandikoetxea, Martin-Reyes and Ombrosi [4] for a discussion regarding different definitions of A classes (also [3]). The following result is an improvement of Buckley s estimate (..). 4

5 Theorem... Let < p < and let σ = w /(p ), then Using the duality relationship M L p (w) c n p ( [w] Ap [σ] A ) /p. (..3) [σ] p A p = [w] p A p we see immediately that ( [w]ap [σ] A ) /p [w] p A p yielding (..). This result and Theorem..2 below were first proved in [25] but the approaches we present here for both results are simpler and taken from [28]. Theorem..2 (An optimal reverse Hölder inequality). Define r w := + where c n is a dimensional constant. Note that r w [w] A. a) If w A, then ( ) /rw w rw 2 w. c n [w] A, b) Furthermore, the result is optimal up to a dimensional factor: If a weight w satisfies the RHI, i.e., there exists a constant K such that ( ) /r w r K w, then there exists a dimensional constant c = c n, such that [w] A c n K r. We mention that some similar results for the real line have been independently obtained by O. Beznosova and A. Reznikov in [3] by means of the Bellman function technique. We postpone the proof from now deriving first the following corollary which is usually called the open property of the A p condition. Corollary.. (The Precise Open property). Let < p < and let w A p. Then w A p ɛ where ɛ = p p = r(σ) + τ n [σ] A where as usual σ = w p. Furthermore [w] Ap ɛ 2 p [w] Ap 5

6 Proof. Since w A p, σ A p A, and hence ( w σ r(σ) ) p r(σ) ( w 2 p. σ) Choose ɛ so that p p = p ɛ, namely ɛ = and observe that ɛ > 0 and p ɛ >. r(σ) r(σ) This yields that w A p ɛ. Proof of Theorem... We will use M L q, (w) [w] q A q < q < (with dimensional constant) and {x R n : Mf(x) > 2t} {x R n : Mf t (x) > t} where f t = f χ f>t. Now, since w A p we use the The Precise Open property above: w A p ɛ, ɛ = p and [w] r(σ) Ap ɛ 2 p [w] Ap. Mf(y) p w(y)dy = p t p w{y R n : Mf(x) > t} dt R n 0 t = p2 p 0 t p w{y R n : Mf(x) > 2t} dt t p 2p p c p ɛ n [w] Ap ɛ 2 p t p p c p ɛ n 2 p [w] Ap 2 p R n 0 0 f t (y) p ɛ R t n p ɛ f(y) 0 t p w{y R n : Mf t (x) > t} dt t w(y)dy dt t t ɛ dt t f(y)p ɛ w(y)dy = ɛ 22p p c p ɛ n [w] Ap f(y) p w(y)dy = p r(σ) 2 2p cn p ɛ [w] p f(y) R R p w(y)dy n n and this yields (..3). = p ( + τ n [σ] A ) p 2 2p c p ɛ n [w] p R n f(y) p w(y)dy To prove Theorem..2 we begin with the following lemma. It it interesting on its own, since it can be viewed as a self-improving property of the maximal function when restricted to A weights. 6

7 Lemma..3. Let w be any A weight and let 0 be a cube. Then for any 0 < ε 2 n+ [w] A, we have that (M(wχ 0 )) +ε dx 2[w] A 0 ( ) +ε w dx, (..4) 0 where M denotes the dyadic maximal function associated to the cube 0. Proof. Let w = wχ 0 and define Ω λ := 0 {Mw > λ}. We start with the layer cake formula: (Mw) +ε dx = ελ ε Mw(Ω λ ) dλ 0 0 = w0 0 ελ ε 0 Mw dλ + w 0 ελ ε Mw(Ω λ ) dλ Now, for λ w 0, by the Calderón-Zygmund decomposition (see [3]) there is a family of maximal nonoverlapping dyadic cubes { j } j for which Ω λ = j and w dx > λ. j j Therefore, by using this decomposition and the definition of the A constant, we can write (Mw) +ε dx w ε 0 [w] A w( 0 ) + ελ ε Mw dxdλ. (..5) 0 w 0 j j By maximality of the cubes in { j } j, it follows that the dyadic maximal function M can be localized: Mw(x) = M(wχ j )(x) for any x j, for all j N. Now, if we denote by to the dyadic parent of a given cube, we have that Mw dx = M(wχ j ) [w] A w( j ) [w] A w( j ) j j = [w] A w j j [w] A λ2 n j Therefore, j j Mw dx j [w] A λ2 n j [w] A λ2 n Ω λ, 7

8 and then (..5) becomes 0 (Mw) +ε dx w ε 0 [w] A w( 0 ) + ε[w] A 2 n w 0 λ ε Ω λ dλ. Averaging over 0, we obtain that (Mw) +ε dx w +ε [w] A + ε2n [w] A (Mw) +ε dx. (..6) 0 + ε 0 So the desired inequality follows for any 0 < ε 2 n+ [w] A by absorbing the last term into the left. Proof of Theorem..2. Denoting again w = wχ 0 we have that w +ε dx 0 (Mw) ε wdx. 0 Now we argue in a similar way as in the previous lemma to obtain that (Mw) ε wdx = ελ ε w(ω λ ) dλ 0 0 = w0 0 w ε 0 w( 0 ) + ελ ε w( 0 ) dλ + w 0 ελ ε j w 0 ελ ε w(ω λ ) dλ w( j ) dλ, where the cubes { j } j are from the decomposition of Ω λ above. Therefore, (Mw) ε wdx w ε 0 w( 0 ) + ε2 n λ ε j dλ 0 w 0 j w ε 0 w( 0 ) + ε2 n λ ε Ω λ dλ w 0 w ε 0 w( 0 ) + ε2n (Mw) +ε dx. + ε 0 Averaging over 0 we obtain w +ε dx w +ε 0 + ε2n 0 + ε (Mw) +ε dx. 0 8

9 Now we use Lemma..3 to conclude with the proof: w +ε dx w +ε 0 + ε2n 0 + ε (Mw) +ε dx 0 ( w +ε 0 + ε2n+ [w] A w dx + ε 0 ( ) +ε 2 w dx 0 since, by hypothesis, ε2n [w] A +ε 2. ) +ε.2 Sharpenes of the exponents, Yano s condition and the Rubio de Francia s algorithm. In this section we will prove the sharpness of the exponent in (..2). However we will do it from a very general point of view showing that the sharpness is intimately related to the failure of the boundedness of the operator when p =. More precisely is related to the way it blows up the bound norm of the operator as p which in turn is related to the well known Yano s classical condition. In fact the method is so general that also shows that the exponents in Theorem.. are sharp but we will leave these details aside and remit to the work [34]. T will denote in this section an operator (not necessarily linear) bounded on L p (R n ), < p <. We will assume that for some 0 < α < T L p (R n ) = O( (p ) ) p α and we will denote by α T to the smallest of these exponents α > 0, namely for any ɛ > 0 lim sup (p ) α T ɛ T L p (R n ) = (.2.) p This is satisfied in the usual cases with appropriate exponents by: maximal operators, singular integrals (α T = ), commutators of singular integrals with BMO functions α T = 2, generalized commutators α T = k ((see Chapter 2) for definitions). Theorem.2.. Let T be an operator as above with exponent α T (0, ). Suppose further that for some < p 0 < α T p T L p 0 (w) c p0,α T [w] 0 A p0 w A p0. (.2.2) Then the inequality is sharp in the sense the exponent α T cannot be replaced by α T ɛ, ɛ > 0. 9

10 This result can be applied to many operators: T = M k, T = [b, H], b BMO, or more generally Tb k. In each case an explicit example was built to show the sharpness of the exponent. Corollary.2.. Let M be Hardy-Littlewood maximal function and let < p < and w A p. Then M k L p (w) c n,,k,p [w] k p A p. (.2.3) and the exponent is sharp. Te proof of the theorem is based on the Rubio de Francia s algorithm which is the main tool to prove the extrapolation theorem and in particular some of the new ideas from [0] (also [2]) Proof of Theorem.2.. To prove the theorem we let α > 0 be a positive parameter satisfying (.2.2) instead of α T : α p T L p 0 (w) c p0,α [w] 0 A p0 w A p0. (.2.4) First we claim first the following estimate which follows from (.2.4) and is the key inequality: T L p (R n ) 2 +α c p0,α M α p p 0 p 0 L p (R n ) < p < p 0. (.2.5) Suppose then < p < p 0 and perform the iteration technique R as follows: R(h) = k=0 2 k M k (h) M k L p (R n ) Then we have (A) h R(h) (B) R(h) L p (R n ) 2 h L p (R n ) (C) [R(h)] A 2 M L p (R n ) Then, combining Holder s inequality, the key hypothesis (.2.4) together with properties of (A) and (B) ( ) T (f) L p (R n ) = (T f) p (Rf) (p p 0 p) p 0 (Rf) (p 0 p) p /p p 0 dx R n ( ) T f p 0 (Rf) (p0 p) /p0 ( ) p 0 p dx (Rf) p pp dx 0 R n R n 2 c p0,α [R(f) (p0 p) α ( ) p ] 0 /p0 f p 0 p f p 0 (Rf) (p0 p) dx R n A p0 p 0 L p (R n ) 0

11 2 c p0,α [R(f) (p 0 p) ] = 2 c p0,α [R(f) (p 0 p) ] α p 0 A p0 α p 0 A p0 ( ) /p0 f f p p dx R n p 0 L p (R n ) f L p (R n ) = 2 c p0,α [R(f) p0 p p 0 ] α A p 0 f L p (R n ) q since [w] q A q = [w q ] A. Now, since p 0 p q p 0 continue with < we can use Jensen s inequality to 2 c p0,α [R(f)] α p 0 p p 0 A p 0 f L p (R n ) 2 c p0,α [R(f)] α p 0 p p 0 A f L p (R n ) 2 +α c p0,α M α p 0 p p 0 L p (R n ) f L p (R n ) by making use of property (C). Since this holds for any f we get the key inequality T L p (R n ) 2 +α c p0,α M α p 0 p p 0 L p (R n ) < p < p 0. To finish the proof we let α = α T in this argument and assume that (.2.4) holds with α T replace by α T ɛ, for a tiny ɛ. Then (.2.5) holds with α T ɛ: T L p (R n ) 2 +α T ɛ c p0,α T ɛ M (α T ɛ) p 0 p p 0 L p (R n ) < p < p 0. and distributing the exponents: M (α T ɛ) p p 0 L p (R n ) T L p (R n ) c p0,α T,ɛ M α T ɛ L p (R n ) < p < p 0 and using that we have as when p is close to, M p p p (, ) p ( p )(α T ɛ) p 0 (p ) α T ɛ T L p (R n ) c p0,α T,ɛ. Finally letting p and since α = α(t ) we assume (.2.), namely we conclude the proof of the theorem. lim sup (p ) α T ɛ T L p (R n ) = p

12 .3 Two weight problem: sharp Sawyer s theorem Another highlight of the theory of weights is the two weight characterization of the maximal theorem due to E. Sawyer [48] (see also [9]): Let < p <, and let u, σ two unrelated weights, then there is a finite constant C such that M(fσ) L p (u) C f L p (σ) (.3.) if and only if there is a finite constant K such that for any cube ( /p M(σ χ ) udx) p K σ() /p < let K. Moen proved in [35] a quantitative version of Sawyer s theorem as follows: if we ( M(σ χ ) p udx) /p then we have. [u, σ] Sp = sup σ() /p Theorem.3.. Let < p < and let u, σ and M as above. Then [u, σ] Sp M c n p [u, σ] Sp Very recently and in a joint work with E. Rela [45] an application of this result was found obtaining an improvement of the mixed A p A Theorem.. as well as some other two weight estimates with bumps conditions providing new quantitative estimates of results derived in [4]. We present now the two weight version of Theorem.. as a corollary of Theorem.3.. Corollary.3.. Under the same condition as above M c n p ([u, σ] Ap [σ] A ) /p where [u, σ] Ap is the two weight A p constant: ( ) ( p [u, σ] Ap := sup w(x) dx σ(x) dx). 2

13 Proof. By Theorem.3. it is enough to prove that [u, σ] Sp c n ([u, σ] Ap [σ] A ) /p. (.3.2) Now, let a > a universal parameter to be chosen. Fix a cube and let k 0 the integer such that a k 0 σ < ak 0+. Then M(σ χ ) p u dx = M(σ χ ) p u dx+ M(σ χ ) p u dx M(σ χ ) a k 0 a k <M(σ χ ) a k+ ( σ) p u() + a p a kp u({x : M(σ χ ) > a k }) k=k 0 we know use the Calderón-Zygmund cubes locally and at each level k. Indeed, since a k > σ k k 0 we can find a family of dyadic cubes with respect to { k,j } such that for each of these k {x : M(σ χ ) > a k } = k,j k j with a k < σ 2 n a k j Z k j Furthermore the family of cubes satisfies the so called sparseness property (see Chapter 4). Then using the A p condition we continue with [u, σ] Ap σ() + a ( ) p p σ dx u( k j ) k,j k j [u, σ] Ap σ() + a p k,j ( k j [u, σ] Ap σ() + c n [u, σ] Ap k=k 0 p σ dx) u dx σ( k j ) k j σ( k j ) Now by the sparseness property of the cubes we can continue last sum with c n σ dx Ej k c n M(χ σ) dx c n M(χ σ) dx c n [σ] A σ() j,k k j j,k E k j since the family Ej k is pairwise disjoint and all of them are contained in. Finally, dividing by σ() and taking the supremum over all the cubes we obtain the desired estimate (.3.2). j,k 3

14 Chapter 2 Some applications 2. The factorization theorem with control on the bounds Muckenhoupt already observed in [37] that it follows from the definition of the A class of weights that if w, w 2 A, then the weight is an A p weight. Furthermore we have w = w w p 2 [w] Ap [w ] A [w 2 ] p A (2..) He conjectured that any A p weight can be written in this way. This conjecture was proved by P. Jones s showing that if w A p then there are A weights w, w 2 such that w = w w p 2. It is also well known that the modern approach to this question uses completely different path and it is due to J. L. Rubio de Francia as can be found in [9] and also in [3]. Here we present a variation of these ideas which appeared in [2] obtaining results with quantitative control on the constants. To be more precise we have the following version of the factorization theorem. Theorem 2... Let < p < and let w A p, then there are A weights u, v A, such that w = u v p in such a way that [u] A ( max{ M /p L p (w), M /p L p (σ) }) p and hence & [v] A ( max{ M /p L p (w), M /p }) p L p (σ) (2..2) [w] Ap ( max{ M /p L p (w), M /p L p (σ) }) 2p cn,p [w] 2 A p 4

15 Proof. We use Rubio de Francia s iteration scheme or algorithm to our situation. Define S (f) p := w /p M( f p ), w/p and S 2 (f) p := w /p M( f p w /p ), Observe that S i : L pp (R n ) L pp (R n ) with constants S L pp (R n ) M /p L p (w) & S 2 L pp (R n ) M /p L p (σ) where as usual σ = w p Hence, the operator S = S + S 2 is also bounded on L pp (R n ) with S L pp (R n ) 2 max{ M /p L p (w), M /p L p (σ) } ( c n,p[w] /p A p ) Now, define the Rubio de Francia algorithm R as R(h) := k=0 S k (h) 2 k ( S. L pp (R n ) )k Observe that R is also bounded on L pp (R n ). R(h) A (S) L pp (R n ). More precisely Now, if h L pp (R n ) is fixed, and hence R(h) A (S i ) i =, 2, with Hence and S(R(h)) 2 S L pp (R n ) R(h), S i (R(h)) 2 S L pp (R n ) R(h) i =, 2 M(R(h) p w /p ) (2 S L pp (R n ) )p R(h) p w /p M(R(h) p w /p ) (2 S L pp (R n ) )p R(h) p w /p. If we let v := R(h) p w /p & u := R(h) p w /p, then we have u, v A, with w = uv p and and hence [v] A (2 S L pp (R n ) )p & [u] A (2 S L pp (R n ) )p [v] A ( max{ M /p L p (w), M /p L p (σ) }) p & [u] A ( max{ M /p L p (w), M /p L p (σ) }) p 5

16 (observe that p max{[v] A, [u] p A } max{ M /p L p (w), M /p L p (σ) } c n[w] p M L p (w) L p, (w) M L p (σ) L p, (σ) ) A p which produces namely [v] A c n [w] p A p & [u] A c n [w] Ap max{[u] A, [v] A } [w] max{, p } A p. We finish the section by remarking that there is no good extrapolation theorem with mixed A p A bounds. A first result can be found in [25] but it seems not to be sharp. 2.2 Commutators of operators with BMO functions: quadratic estimates In this section we consider commutators of linear operators T with BMO functions defined by appropriate functions by [b, T ]f(x) = b(x)t (f)(x) T (b f)(x) When T is a singular integral operator, these operators were considered by Coifman, Rochberg and Weiss in [8]. If the kernel of the operator is K then formally we have [b, T ]f(x) = (b(x) b(y))k(x, y)f(y) dy, R n where K is a kernel satisfying the standard Calderón-Zygmund estimates. Although the original interest in the study of such operators was related to generalizations of the classical factorization theorem for Hardy spaces many other applications have been found. The main result from [8] states that [b, T ] is a bounded operator on L p (R n ), < p <, when b is a BMO function. In fact, the BMO condition of b is also a necessary condition for the L p -boundedness of the commutator when T is the Hilbert transform. We may think that these operators behave as Calderón-Zygmund operators, however there are some differences. For instance, simple examples show that in general [b, T ] fails to be of weak type (, ) when b BMO. This was observed in [42] where it 6

17 is also shown that there is an appropriate weak-l(log L) type estimate replacement. To stress this point of view it is also shown in [43] that the right operator controlling [b, T ] is M 2 = M M, instead of the Hardy-Littlewood maximal function M explaining the reason of why these operators are more singular.. We pursue in this way by showing that commutators have an extra bad behavior from the point of view of A p weights when comparing with Theorem 4... In particular we will prove weighted estimates for these commutators with quantitate control on the bounds using a very general method. 2.3 A preliminary result: a sharp connection between the John-Nirenberg theorem and the A 2 class In this section we outline the main results from [5]. We want to stress the relevance of the reverse Hölder property of the A 2 weights in conjunction with the following sharp version of the classical and well known John-Nirenberg theorem. For a locally integrable b : R n R we define b BMO = sup b(y) b dy < where the supremum is taken over all cubes R n with sides parallel to the axes, and b = b(y) dy The main relevance of BM O is due to the fact that has an exponential selfimproving property, namely the celebrated John-Nirenberg s Theorem. We need a very precise version of it, as follows: Theorem [Sharp John-Nirenberg inequality] There are dimensional constants 0 α n < < β n such that ( ) αn sup exp b(y) b dy β n (2.3.) b BMO In fact we can take α n = 2 n+2. For the proof of this we remit to the Lecture Notes by J.L. Journé [29] p The result there is not so explicit but it follows from the proof which is very interesting and different from the usual ones that can be found in many references. We derive from Theorem 2.3. the following relationship between BM O and the A 2 class of weights Lemma that will be used in the proof of the main theorem of 7

18 this section. Indeed, it is well known that if w A 2 then b = log w BMO. A partial converse also holds, if b BMO there is an s 0 > 0 such that w = e sb A p, s s 0. We need a more precise version of this. Lemma Let b BMO and let α n < < β n be the dimensional constants from (2.3.). Then s R, s α n b BMO = e s b A 2 and [e s b ] A2 β 2 n Proof. By Theorem 2.3., if s and then and αn b BMO exp( s b(y) b ) dy and if is fixed α n exp(s(b(y) b )) dy β n exp( s(b(y) b )) dy β n. exp( b(y) b ) dy β n b BMO If we multiply the inequalities, the b parts cancel out: ( ) ( ) exp(s(b(y) b )) dy exp(s(b b(y))) dy ( ) ( ) = exp(sb(y)) dy exp( sb(y)) dy βn 2 namely e s b A 2 with [e s b ] A2 β 2 n 2.4 Results within the A p context In this section we use the extrapolation theorem with sharp bounds (see [3] or [0] ) to get the optimal estimates for commutators. We emphasize that next results are stated for very general operators. 8

19 Theorem Let T be a linear operator such that T L 2 (w) c [w] A2 w A 2. (2.4.) Then there is a constant c independent of w and b such that [b, T ] L 2 (w) c [w] 2 A 2 b BMO. (2.4.2) These results can be found in [5] as well some generalizations. Observe the quadratic exponent which makes it different from the non commutator case. As an easy consequence of the extrapolation theorem with sharp bounds we have the following. Corollary Let T be a linear operator satisfying (2.4.). Let < p <, the there is a constant c n,p such that [b, T ] L p (w) c n,p [w] 2 max{, p } A p b BMO. (2.4.3) These results have been improved in [25] by replacing the A p constant by appropriate mixed A p A constants. As a sample if T is as above we have [b, T ] L 2 (w) c [w] /2 A 2 ( [w]a + [w ] A ) 3/2 b BMO. See Chapter 4 for similar results for Caldern-Zygmund operators. Proof. We use a beautiful idea that goes back to the original paper [8]. The method is very general and goes beyond the case of linear operators as shown in []. We conjugate the operator as follows T : if z is any complex number we define T z (f) = e zb T (e zb f). Then, a computation gives (for instance for nice functions), [b, T ](f) = d dz T z(f) z=0 = T z (f) dz, ɛ > 0 2πi z 2 by the Cauchy integral theorem. Now, by Minkowski s inequality [b, T ](f) L 2 (w) 2π ɛ 2 z =ɛ z =ɛ T z (f) L 2 (w) dz ɛ > 0. (2.4.4) The key point is to find the appropriate radius ɛ. To do this we look at the inner norm T z (f) L 2 (w) T z (f) L 2 (w) = T (e zb f) L 2 (we 2Rez b ), 9

20 and try to find appropriate bounds on z. To do this we use the main hypothesis, namely that T is bounded on L 2 (w) if w A 2 with T L 2 (w) c [w] A2. Hence we should estimate [we 2Rez b ] A2. But since w A 2 we can use the sharp reverse Holder inequality to obtain: where r = r w = + c n[w] A2 + c n[w] A to apply Lemma 2.3.2, Hence for these z, [we 2Rez b ] A2 4 [w] A2 [e 2Rez r b ] r A 2 < 2. Now, since b BMO we are in a position if 2Rez r α n b BMO then [e 2Rez r b ] A2 β 2 n. [we 2Rez b ] A2 4 [w] A2 β 2 r n 4 [w] A2 β n, since < r < 2. Using this estimate and for these z we have T z (f) L 2 (w) 4β n [w] A2 f L 2 (w). Finally, choosing the radius finish the proof of the theorem. ɛ = α n 2r b BMO, 20

21 Chapter 3 Singular Integrals: The A theory The Hardy-Littlewood maximal function is not a linear operator but it is a sort of self-dual operator since the following inequality holds Mf L, (w) c f Mw dx, (3.0.) R n for any nonnegative functions f and w and where the constant c is independent of both functions f and w. The inequality (3.0.) it is much more than an interesting improvement of the classical weak-type (, ) property of M. M is also self-dual from the L p point of view: (Mf) p w dx c p f p Mw dx f, w 0. (3.0.2) R n R n This estimate follows from the classical interpolation theorem of Marcinkiewicz. Both results (3.0.) and (3.0.2) were proved by C. Fefferman and E.M. Stein in [6] to derive the following vector-valued extension of the classical Hardy-Littlewood maximal theorem: for every < p, q <, there is a finite constant c = c p,q such that ( ) (Mf j ) q q ( ) c f j q q (3.0.3) L p (R n ) L p (R ). n j This is a very deep theorem and has been used a lot in modern harmonic analysis explaining the central role of inequality (3.0.). One of the main purposes of these lectures is to find corresponding estimates for Calderón-Zygmund operators T instead of M. Here we use the standard concept of Calderón-Zygmund operator as can be found in many places as for instance in [20] or []. B. Muckenhoupt and R. Wheeden during the 70 s he Muckenhoupt-Wheeden considered the following problem: j T f L, (w) c T R n f Mwdx. (3.0.4) 2

22 when T = H is the the Hilbert transform: Hf(x) = pv R f(y) x y dy. It is very unfortunate that this inequality is false as shown first in [46] in the case of a very special dyadic singular integral operator and then in [47] for the Hilbert transform. We remit to [44] for a discussion on this estimate and some other related for other operators. It seems that the best result can be found in [40] where M is replaced by M L(log L) ɛ is a ɛ-logarithmically bigger maximal type operator than M. If ɛ = 0 we recover M but the constant c ɛ blows up as ɛ 0. The precise result is the following. Theorem (The L(log L) ɛ theorem). There exists a constant c depending on T such that for any ɛ > 0, any function f and any weight w sup λw{x R n : T f(x) > λ} c f M L(log L) ɛ(w)dx w 0. (3.0.5) λ>0 ɛ R n We remark that the operator M L(log L) ɛ is pointwise smaller than M r = M L r, r >. Remark We also remark that in [40] the constant in front does not appear ɛ explicitly. However, the way that estimate (3.0.5) blows up is relevant these days since its connection with questions related to the precise quantitate two weight estimates for Calderón-Zygmund operators (see [26]). In fact, in this work this theorem has been improved by replacing T by T, the maximal singular integral operator. The main difficulty is that T is not a linear operator and the approach used in these notes cannot be applied. Observe that the A class of weights can be defined immediately from Fefferman- Stein s inequality (3.0.) and in fact this is what the authors did in that paper: if the weight w A then Mf L, (w) c n [w] A R n f wdx (3.0.6) and it is natural to ask wether the corresponding inequality holds for singular integrals (say for the Hilbert transform). Then for a while there was a conjecture called the A conjecture: if w A, then T f L, (w) c [w] A R n f wdx. (3.0.7) However, this inequality seems to be false too (see [38]) for T = H, the Hilbert transform. In this chapter we will present some recent progress in connection with this conjecture exhibiting an extra logarithmic growth in (3.0.7) which in view of [38] could be 22

23 the best possible result. To prove this result we have to study first the corresponding weighted L p (w) estimates with < p < and w A being the result this time fully sharp. The final part of the proofs of both theorem can be found in Sections 3.2 and 3.3 and are essentially taken from [33] and [32] and the recent improvement in [25]. Again we remit to [44] for a more complete discussion about these estimates and their variants. We state now the main theorems of this chapter. From now on T will always denote any Calderón-Zygmund operator and we assume that the reader is familiar with the classical unweighted theory. Theorem Let T be a Calderón-Zygmund operator and let < p <. Then T L p (w) c T pp [w] /p A [w] /p A, (3.0.8) and hence T L p (w) c T pp [w] A. (3.0.9) The result (3.0.8) is best possible as well as (3.0.9) in a sense similar to (..). As an application of this result we obtain the following endpoint estimate. Theorem [The logarithmic growth theorem] Let T be a Calderón-Zygmund operator. Then T L, (w) c T [w] A log(e + [w] A ). (3.0.0) As mentioned, in view of [38], (3.0.0) seems to be the best possible result. Remark As in remark 3.0.3, these two theorems can be further improved by replacing T by T the maximal singular integral operator. Again, the method presented in these notes cannot be applied because is based on the fact that T is linear while T is not. See [26]. 3. Estimates involving A weights In harmonic analysis, there are a number of important inequalities of the form T f(x) p w(x) dx C R n Sf(x) p w(x) dx, R n (3..) where T and S are operators. Typically, T is an operator with some degree of singularity (e.g., a singular integral operator), S is an operator which is, in principle, easier to handle (e.g., a maximal operator), and w is in some class of weights. 23

24 The standard technique for proving such results is the so-called good-λ inequality of Burkholder and Gundy. These inequalities compare the relative measure of the level sets of S and T : for every λ > 0 and ɛ > 0 small, w({y R n : T f(y) > 2 λ, Sf(y) λɛ}) C ɛ w({y R n : Sf(y) > λ}). (3..2) Here, the weight w is usually assumed to be in the class A = p> A p. Given inequality (3..2), it is easy to prove the strong-type inequality (3..) for any p, 0 < p <, as well as the corresponding weak-type inequality In these notes the special case of T f L p, (w) C Sf L p, (w). (3..3) T f L p (w) c Mf L p (w) (3..4) where T is a Calderón-Zygmund operator and M is the maximal function, will play a central role in the proof of Theorem Estimate (3..4) was proved by Coifman- Fefferman in the celebrated paper [6]. In our context the weight w will also satisfy the A condition but the problem is that the behavior of the constant is too rough. We need a more precise result for very specific weights. Lemma 3... Let w be any weight and let p, r <. Then, there is a constant c = c(n, T ) such that: T f L p (M rw) p ) cp Mf L p (M rw) p ) This is the main improvement in [33] of [32] where we had obtained logarithmic growth on p. It is an important step towards the proof of the the linear growth Theorem The above mentioned good λ of Coifman-Fefferman is not sharp because instead of c p gives C(p) 2 p because [(M r w) p )] Ap (r ) p The proof of this lemma is tricky and it combines another variation the of Rubio de Francia algorithm together with a sharp L version of (3..4): T f L (w) c[w] Aq Mf L (w) w A q, q < (3..5) The original proof given in [33] of this estimate was based on an idea of R. Fefferman- Pipher from [5] which combines a sharp version of the good-λ inequality of S. Buckley together with a sharp reverse Hölder property of the weights. The result of Buckley 24

25 establishes a very interesting exponential improvement of the good-λ estimate of above mentioned Coifman-Fefferman estimate as can be found in [4]: {x R n : T (f) > 2λ, Mf < γλ} c e c 2/γ {T (f) > λ} λ, γ > 0 (3..6) where T is the maximal singular integral operator. This approach is interesting on its own but we will present in these lecture notes a more efficient approach based on the following estimate: Let 0 < p <, 0 < δ < and let w A q, q <, then f L p (w) c p[w] Aq M # δ (f) L p (w) (3..7) for any function f such that {x : f(x) > t} <. Here, M # δ f(x) = M # ( f δ )(x) /δ and M # is the usual sharp maximal function of Fefferman-Stein: M # (f)(x) = sup f(y) f dy. x This result can be found in [39] and the proof is based on properties of rearrangement of functions and corresponding local sharp maximal operator. To prove (3..5) we combine (3..7) with the following pointwise estimate [2]: Lemma Let T be any Calderón-Zygmund operator and let 0 < δ <, then there is a constant c T,δ such that M # δ (T (f ))(x) c Mf(x) and in fact we have. Corollary 3... Let 0 < p < and let w A q. Then T f L p (w) c [w] Aq Mf L p (w) for any f such that {x : T f(x) > t} <. We now finish this section by proving the tricky Lemma 3... The proof is based on the following lemma which is another variation of the Rubio de Francia algorithm. Lemma Let < s < and let w be a weight. Then there exists a nonnegative sublinear operator R satisfying the following properties: (a) h R(h) (b) R(h) L s (w) 2 h L s (w) (c) R(h)w /s A with [R(h)w /s ] A cs 25

26 Proof. We consider the operator Since M L s s, we have S(f) = M(f w/s ) w /s S(f) L s (w) cs f L s (w). Now, define the Rubio de Francia operator R by S k (h) R(h) = 2 k ( S L s (w)). k It is very simple to check that R satisfies the required properties. k=0 Proof of Lemma 3... We are now ready to give the proof of the tricky Lemma, namely to prove T f M r w cp Mf L p (M rw) M r w L p (M rw) By duality we have, T f M r w = T f h dx T f h dx L p (M rw) R n R n for some h L p (M =. By Lemma 3..3 with rw) s = p and v = M r w there exists an operator R such that (A) h R(h) (B) R(h) L p (M 2 h rw) L p (M rw) (C) [R(h)(M r w) /p ] A cp. We want to make use of property (C) combined with the following two facts: First, if w, w 2 A, and w = w w p 2 A p, then by (2..) [w] Ap [w ] A [w 2 ] p A Second, if r > then (Mf) r A by the Coifman-Rochberg theorem from [7] but we need a more precise estimate which follows from the proof: Hence combining we obtain [(Mf) r ]A c n r. [R(h)] A3 = [R(h)(M r w) /p ( (Mr w) /2p ) 2]A3 [R(h)(M r w) /p ] A [(M r w) /2p ] 2 A cp. 26

27 Therefore, by Corollary 3.. and by properties (A) and (B), T f h dx T f R(h) dx R n R n c[r(h)] A3 M(f)R(h) dx R n cp Mf M r w h L p (M. rw) L p (M rw) 3.2 The main lemma and the proof of the linear growth theorem In this section we combine all the previous information to finish the proof of the linear growth Theorem We need a a lemma which immediately gives the proof. Lemma Let T be any Calderón-Zygmund singular integral operator and let w be any weight. Also let < p < and < r < 2. Then, there is a c = c n such that: ( ) /pr T f L p (w) cp f L r p (Mrw) In applications we will use the following consequence T f L p (w) cp (r ) /p f L p (M rw) since t /t 2, t. We are now ready to finish the proof of the linear growth theorem Proof of Theorem Indeed, since w A A we apply the sharp Reverse Hölder s exponent r = r w = + 2 n+ [w] A obtaining and then if further w A T f L p (w) c T pp [w] /p A f L p (M rw) T L p (w) c T pp [w] /p A [w] /p A. 27

28 Proof of the lemma. We consider to the equivalent dual estimate: ( ) /pr f L T f L p (M rw) p ) cp r p (w p ) Then use the tricky Lemma 3.. since T is also a Calderón-Zygmund operator T f M r w L p (M rw)) p c Mf M r w L p (M rw)) Next we note that by Hölder s inequality with exponent pr, ( ) /pr ( ) /(pr) fw /p w /p w r (fw /p ) (pr) and hence, (Mf) p (M r w) p M ((fw /p ) (pr) ) p /(pr) From this, and by the classical unweighted maximal theorem with the sharp constant, Mf ( p ) /(pr) f M r w c L p (M rw) p (pr) w L p (w) ( rp ) /pr f = c r w L p (w) ( ) /pr f cp. r w L p (w) 3.3 Proof of the logarithmic growth theorem. Proof of Theorem The proof is based on ideas from [40]. Applying the Calderón- Zygmund decomposition to f at level λ, we get a family of pairwise disjoint cubes { j } such that λ < f 2 n λ j j Let Ω = j j and Ω = j 2 j. The good part is defined by g = j f j χ j (x) + f(x)χ Ω c(x) 28

29 and the bad part b as b = j b j where b j (x) = (f(x) f j )χ j (x) Then, f = g + b. However, it turns out that b is excellent and g is really ugly. It is so good the b part that we obtain the maximal function on the right hand side: w{x ( Ω) c : T b(x) > λ} c f Mwdx λ R n by a well known argument using the cancellation of the b j and that we omit. Also the term w( Ω) is the level set of the maximal function and the Fefferman-Stein applies (again we obtain the maximal function on the right hand side). Combining we have w{x R n : T f(x) > λ} w( Ω) + w{x ( Ω) c : T b(x) > λ/2} + w{x ( Ω) c : T g(x) > λ/2}. and the first two terms are already controlled: w( Ω) + w{x ( Ω) c : T b(x) > λ/2} c f Mwdx c[w] A f wdx λ R λ n R n Now, by Chebyschev and the Lemma, for any p > and r > we have λ w{x ( Ω) c : T g(x) > λ/2} λ c(p ) p (r ) p λ p R n g p M r (wχ ( Ω) c)dx c(p ) p (r ) p R n g M r (wχ ( Ω) c)dx. By known standard geometric arguments we have g M r (wχ ( Ω) c)dx c f M r wdx. R n R n Now if we choose p = + we evan continue with log(r ) c T log(r ) f M r wdx r >. R n 29

30 Finally if we particularize r with the sharp reverse Holder s exponent r = + c n[w] A, we obtain λw{x ( Ω) c : T g(x) > λ/2} c T [w] A ( + log[w] A ) f wdx. R n This estimate combined with the previous one completes the proof. 30

31 Chapter 4 Singular Integrals: The A p theory In this chapter we plan to prove the A 2 theorem: if T is a Calderón-Zygmund operator, then T L p (w) c p,n [w] max{, p } A p. (4.0.) Now, by the sharp Rubio de Francia extrapolation theorem it is enough to prove this result only for the special case p = 2: T L 2 (w) c T [w] A2. (4.0.2) Observe that in this case the growth of the constant is simply linear. This is the reason why the A p result was called the A 2 conjecture. After several previous results assuming some extra regularity condition on the kernel stronger than the usual Hölder-Lipschitz condition, the theorem was finally proved by T. Hytönen in [23]. The proof that we will present here is based on A. Lerner s approach from [3]. The author found a highly interesting method of passing from the continuous case to the dyadic discrete case by means of Theorem 4..2 below. We remit the reader to [3] for a more complete account of the history of (4.0.). We plan to combine Lerner s reduction together with some nice ideas taken from Moen s work [36] which avoids the use of the extrapolation theorem and related to the main idea from [9]. An interesting point is that Lerner s approach seems not to be sharp enough to lead to the following recent improvement of the A 2 theorem that can be found in [25]. Following the same circle of ideas presented in the previous chapters, the A 2 constant in (4.0.2) is replaced by a mixed A 2 A constant. We will state the theorem without proof. Theorem Let T be a Calderón-Zygmund operator. Then there is a constant depending c T depending such that T L 2 (w) c T [w] /2 A 2 max{[w] A, [w ] A } /2. 3

32 Later on this estimate was improved in [24] for general p an even more recently the Hölder-Lipschitz condition of the kernel Calderón-Zygmund operator has been replaced by the Dini-type condition in [27] and for other type of singular integral type operators. The result is the following. Theorem Let T be a Calderón-Zygmund operator with kernel satisfying a Dinitype condition. Then there is a constant depending on T such that T L p (w) c T [w] /p A p max{[w] /p A, [w p ] /p A }. It would be very nice to derive this result from the previous by means of an appropriate extrapolation theorem. In fact, an extrapolation theorem in this direction can be found in [25] but the output it is not sharp when considering the case of singular integrals with exponents p The dyadic A p theory of Singular Integrals In this section we will prove the following theorem. Theorem 4... Let < p < and let T be a Calderón-Zygmund singular integral operator. Then, there is a constant c = c T such that for any A p weight w, { } T c p L p (w) p [w] max, p A p. (4..) To do this we will introduce some sort of dyadic singular integrals. To do this we say that a dyadic grid, denoted D, is a collection of cubes in R n with the following properties: ) each D satisfies = 2 nk for some k Z; 2) if, P D then P =, P, or ; 3) for each k Z, the family D k = { D : = 2 nk } forms a partition of R n. We say that a family of dyadic cubes S D is sparse if for each S, Given a sparse family, S, if we define S 2. E() := \ S then the family {E()} S is pairwise disjoint, E(), and 2 E(). Sparse families have long been used in Calderón-Zygmund theory. 32,

33 The sparseness property has already appeared in the proof of Corollary.3.. In fact the family of cubes that appear in the proof, they are Calderón-Zygmund cubes asocciated to a fix cube, satisfies this property and plays a crucial role in the proof. If S D is a sparse family we define the sparse Calderón-Zygmund operator associated to S as T S f := f dx χ. S As already mentioned the key idea is to transplant the continuous case to the discrete version by means of the following theorem due to A. Lerner [3]. Theorem Suppose that X is a Banach function spaces on R n Calderón-Zygmund operator, then there exists a constant c T T X c T sup T S X. S D and T is a We will not prove this theorem, we will simply mention that a key tool is the decomposition formula for functions found previously by Lerner in [?] using the local mean average (see also the Paseky lectures notes of 20). The main idea of this decomposition goes back to the work of Fujii [8] where the standard average is used instead. There is here a very interesting open problem related to this theorem, namely to go beyond the class of Banach function spaces as for instance the Marcinkiewicz space L,. The duality property of Banach function spaces plays a central role in the proof. In view of Theorem 4..2 the proof of Theorem 4.. is reduced to prove the weighted estimates for the sparse operators T S. The following properties of the family {E()} S will be used: disjointness, E(), and 2 E(). Theorem Suppose D is a dyadic grid, S D is a sparse family, < p <, and w A p. Then the following estimate holds where T S L p (w) c p [w] max{, p } A p c p = pp 2 max{p,p }. The main idea behind the proof of this theorem can be found in [9] but just for the case p = 2. Proof. Since T S is a positive operator we may assume f 0. We first consider the case p 2, and let σ = w p. A simple argument shows that T S L p (w) = T S ( σ) L p (σ),l p (w). 33

34 If g 0 belongs to L p (w), then by duality it suffices to estimate T S (fσ)gw dx = fσ dx gw dx. R n S Now, by definition of the A p constant, we have fσ dx gw dx where S = w()σ() p p fσ dx gw dx p w()σ() p S p [w] Ap fσ dx gw dx w()σ() p S = [w] Ap fσ dx gw dx p σ() 2 p σ() S w() 2 p [w] Ap A σ (f, )A w (g, ) E() p σ() 2 p S A σ (f, ) = fσ dx and A w (g, ) = gw dx, σ() w() and in the last inequality we have used 2 E(). Observe that we cannot use the [w] Ap constant again. However, since p 2 and E() we have σ() 2 p σ(e()) 2 p, (note: E() /2 > 0 so σ(e()) > 0) which in turn yields T S (fσ)gw dx R n 2 p [w] Ap A σ (f, )A w (g, ) E() p σ(e()) 2 p. (4..2) S By Hölder s inequality we have E() w(e()) p σ(e()) p, so E() p σ(e()) 2 p σ(e()) p w(e()) p. (4..3) 34

35 Utilizing inequality (4..3) in the sum in (4..2), followed by a discrete Hölder inequality, followed by standard dyadic type maximal function bounds. Then, we arrive at the desired estimate: A σ (f, )A w (g, ) E() p σ(e()) 2 p S S ( S A σ (f, )A w (g, )σ(e()) p w(e()) p ) A σ (f, ) p p σ(e()) M D σ f L p (σ) M D w g L p (w) pp f L p (σ) g L p (w). ( S ) A w (g, ) p p w(e()) The case < p < 2 follows from duality, since (T S ) = T S, we have T S L p (w) = T S L p (σ) pp 2 p [σ] Ap = pp 2 p [w] p A p. 35

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