A theory of weights for singular integral operators Carlos Pérez University of the Basque Country and Ikerbasque 2 School on Nonlinear Analysis, Function Spaces and Applications T re st, June 204 Introduction: motivation The A p theorem In the celebrated paper [2], B. Muckenhoupt characterized the class of 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, namely M L p (w) is finite if and only if the quantity ( ) ( p dx) [w] Ap := sup w(x) dx w(x) p (.) is finite. We will call it the A p constant of the weight although some authors call it the characteristic or the norm of the weight. The operator norm M L p (w) will depend somehow upon the A p constant of w, but the first result expressing this dependenc was proved by S. Buckley [2] as part of his Ph.D. thesis: Let < p < and let w A p, then the Hardy-Littlewood maximal function satisfies the following operator estimate: namely, M L p (w) c n p [w] p A p, (.2) sup M w A p [w] Lp (w) c n p. p A p The author would like to thank Professors Stanislav Hencl and Lubo s Pick for their invitation to deliver these lectures. 2 The author would like to acknowledge the support of the Spanish grant MTM202-30748
Furthermore the result is sharp in the sense that: for any ɛ > 0 sup M w A p [w] p ɛ Lp (w) = (.3) A p Buckley showed with an specific example in (.2) that the exponent is optimal but the explanation why this exponent is the correct one is due to the following fact: M L p (R n ) p p as shown in [9] as part of a general phenomena. The Fefferman-Stein inequality More or less at the same time the following inequality was proved by C. Fefferman and E. Stein [9] for the Hardy-Littlewood maximal function: Mf L, (w) c f Mw dx, (.4) R n The inequality (.4) is interesting on its own because it is an improvement of the classical weak-type (, ) property of the Hardy-Littlewood maximal operator M. However, the crucial new point of view is that it can be seen as a sort of duality for M since the following L p inequality is a consequence: (Mf) p w dx c p R n f p Mw dx R n f, w 0. (.5) This estimate follows from the classical interpolation theorem of Marcinkiewicz although a direct proof can be given avoiding interpolation. Both results (.4) and (.5) were proved by C. Fefferman and E.M. Stein in [9] 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 (.6) 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 (.4). Nevertheless, th The A condition The A p condition defined above makes sense when p (, ), however there are two extreme cases A and A which play a central role in the theory. The A condition can be defined immediately from Fefferman-Stein s inequality (.4) and in fact was already introduced by these authors in that paper: the weight w is an A weight or satisfies the A condition if there is a finite constant c such that j Mw c w a.e. (.7) 2
and [w] A will denote the smallest of these constants c. Then if w A Mf L, (w) c n [w] A R n f wdx. This follows from (.4) but also it follows from (.5) that M L p (w) c n p [w] /p A We point out that the A condition can be defined from (.) by letting p go down to. The A condition To define the A class of weights the key observation is that the A p condition is decreasing in p: [w] Aq [w] Ap p < q implying that A p A q. Hence it is natural to define A := p> A p. The A class of weights shares a lot of interesting properties, we remit to [6] for more information. The problem is that this definition does not lead to any appropriate constant. It is known that another possible way of defining a constant for this class is by means of the quantity ( ) [w] exp A := sup w exp log w, as can be found in for instance in []. This constant was introduced by Hruščev in [2]. This definition is natural because it 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 [3] the authors use a new A constant (which was originally introduced by Fujii in [0] and rediscovered later by Wilson in [3]) which is more suitable. Definition.. [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 3
that [w] A c n [w] exp A See [3] for details. In fact it is shown in the same paper with explicit examples that [w] A is much smaller than [w] exp A. To be more precise one can find a family of weights {w t } t>t on the real line such that such that [w t ] A 4 log t and [w t ] exp A t for log t t > 3. We also refer the reader to the forthcoming work of Duoandikoetxea, Martin-Reyes and Ombrosi [7] for a discussion regarding different definitions of A classes (see also [6]). Improving the A p theorem: the mixed A p A approach As was mentioned above, the exponent in Buckley s estimate (.2) cannot be improved, however that estimate can be really improved in a different way. 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. (.8) [σ] p A p = [w] p A p, we see immediately that ( [w]ap [σ] A ) /p [w] p A p, yielding (.2). This result and Theorem.2 below were first proved in [3] with a simplified proof in [5]. A key estimate was the following result. 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 n, such that [w] A c n K r. 4
We mention that some similar one-dimensional results have been independently obtained by O. Beznosova and A. Reznikov in [] by means of the Bellman function technique. The original proof of this result can be found in [3] but it was simplified an improved in [5] (it can be found as well in the Paseky lectures notes [26]). Also it should be mentioned that this reverse Hölder property was completely avoided in [27] to derive the mixed A p A result (.8). The following corollary which is usually called the openess property of the A p condition, is an important consequence of the reverse Hölder property. Corollary. (The precise openess 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. 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. A similar result was previously obtained in [7] for A weights and played an important role in that and subsequent papers. Later on an interesting extension to A p weights, where the exponent r w is r w M L p (σ), was further obtained in [6]. Factorization. Muckenhoupt already observed in [2] that it follows from the definition of the A class of weights that if w, w 2 A, then the weight w = w w p 2 is an A p weight. Furthermore, with our definitions, it follows easily that [w] Ap [w ] A [w 2 ] p A. (.9) He conjectured that any A p weight can be factored out in this fashion. This conjecture was proved by P. Jones 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 5
uses completely different path and it is due to J. L. Rubio de Francia as can be found in [] and also in [6]. In these notes we will use the easy part of the theorem and more precisely the estimate (.9). 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 [30] (see also []): Let < p <, and let u, σ two be unrelated weights, then there is a finite constant C such that M(fσ) L p (u) C f L p (σ) (.0) 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 [20] a quantitative version of Sawyer s theorem as follows: if we ( M(σ χ ) p udx) /p [u, σ] Sp = sup then we have the following theorem. σ() /p Theorem.3. Let < p < and let u, σ and M be as above. Then [u, σ] Sp M c n p [u, σ] Sp In a recent joint work with E. Rela [27] an application of this result has been 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 [24]. 2 Singular Integrals: The A theory Very recently, the so called Muckenhoupt-Wheeden conjecture has been disproved by Reguera-Thiele in [29]. This conjecture claimed that there exists a constant c such that for any function f and any weight w Hf L, (w) c f Mwdx. (2.) where H is the Hilbert transform. The failure of the conjecture was previously obtained by M.C. Reguera in [28] for a special model operator T instead of H. This conjecture 6 R
was motivated by Fefferman-Stein inequality (.4) for the Hardy-Littlewood maximal function. That this conjecture was believed to be false was already mentioned in [23] where the best positive result in this direction so far can be found, and where M is replaced by M L(log L) ɛ, i.e., a maximal type operator that is ɛ-logarithmically bigger than M: T f L, (w) c T,ɛ R n f M L(log L) ɛ(w)dx w 0. where T is the Calderón-Zygmund operator T. Until very recently the constant of the estimate did not play any essential role except, perhaps, for the fact that it blows up. If we check the computations in [23] we find that c ɛ e ɛ. It turns out that improving this constant would lead to understanding interesting questions in the area. Recently this estimate has been improved in [4] where the exponential blow up e ɛ has been reduced to a linear blow up. A second improvement consists of replacing T by the ɛ maximal singular integral operator T. The method in [23] cannot be used directly since the linearity of T played a crucial role in the argument. Theorem 2.. Let T be a Calderón-Zygmund operator with maximal singular integral operator T. Then for any 0 < ɛ, T f L, (w) c T f(x) M L(log L) ɛ(w)(x) dx, w 0. (2.2) ɛ R n It seems that the right conjecture is the following: T f L, (w) c T R n f(x) M L log log L (w)(x) dx w 0. (2.3) We remark that the operator M L(log L) ɛ is pointwise smaller than M r = M L r, r > where M r (w) = M(w r ) /r. In these lectures we will present a baby version of this result, Theorem 2.3, which can be essentially found in [8] and [7] with the improvements obtained in [3]. 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 4 and 5. Again we remit to [25] 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. Our goal is to prove the following results. Theorem 2.2. Let T be a Calderón Zygmund operator and let < p <. Then for any weight w and r >, T f L p (w) c T pp (r ) p f L p (M rw). (2.4) 7
Now, using the sharp exponent in the reverse Hölder inequality for weights in the A class from Theorem.2 the following result follows easily. Corollary 2.. Let T be a Calderón Zygmund operator and let < p <. Then if w A, we obtain and if w A, and hence T f L p (w) c T pp [w] /p A f L p (Mw), (2.5) T f L p (w) c T pp [w] /p A [w] /p A f L p (w) (2.6) T L p (w) c T pp [w] A. (2.7) The exponent in inequality (2.7) is best possible similarly as in (.2). A very nice application of a very especial extrapolation theorem obtained by J. Duoandikoetxea in [5] can be deduced from (2.7). Corollary 2.2. Under the same assumption as before we have, if q < p, T L p (w) c T,p,q [w] Aq. As an application of (2.4) we obtain the following endpoint estimate. Theorem 2.3. Let T be a Calderón Zygmund operator. Then for any weight w and r >, T f L, (w) c T log (e + r ) f L (M rw). (2.8) Now, using again the reverse Hölder inequality for weights from Theorem.2 we get the following consequence. Corollary 2.3. [The logarithmic growth theorem] Let T be a Calderón Zygmund operator. Then. If w A, 2. If w A, T f L, (w) C log (e + [w] A ) f L (Mw). T f L, (w) c T [w] A log (e + [w] A ) f L (w). 8
We remark that these theorems can be further improved by replacing T by T, the maximal singular integral operator. Again, the method presented now cannot be applied because it is based on the fact that T is linear while T is not. See [4]. In view of [22] this could be the best possible result, namely [w] A log (e + [w] A ) cannot be replaced by [w] A log (e + [w] A ) α, with 0 α <. This came as a big surprise. 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. 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 every 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 2.2. Estimate (3.4) was proved by Coifman- Fefferman in the celebrated paper [3]. 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 ) 9
This is the main improvement in [8] of [7] where we had obtained logarithmic growth on p. It is an important step towards the proof of Theorem 2.2. The above mentioned good-λ inequality 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 variant 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 <. The original proof given in [8] of this estimate was based on an idea of R. Fefferman and J. Pipher from [8] 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 establishes a very interesting exponential improvement of the good-λ estimate of above mentioned Coifman-Fefferman estimate as can be found in [2]: {x R n : T (f) > 2λ, Mf < γλ} c e c 2/γ {T (f) > λ} λ, γ > 0 (3.5) where T is the maximal singular integral operator. New more flexible arguments can be found in [4] leading to the following. Lemma 3.2. Let w A. Then T f L (w) c [w] A Mf L (w). We now finish this section by proving the tricky Lemma 3.. The proof is based on the following lemma which is another variant of the Rubio de Francia algorithm. Lemma 3.3. Let < s < and let w be a weight. Then there exists a nonnegative sublinear operator R with 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. 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). 0
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 R n 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 (.9) [w] Ap [w ] A [w 2 ] p A. Second, if r >, then (Mf) r A by the Coifman-Rochberg theorem from [4] 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. Therefore, by Lemma 3.2 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)
4 Proof of Theorem 2.2 We will prove T f L p ((M rw) p ) cp (r ) /pr f L p (w p ) from which (2.4) follows since t /t 2, t. We consider the equivalent dual estimate: T t f L p ((M rw) p ) cp (r ) /pr f L p (w p ). Then use the tricky Lemma 3. since T t is also a Calderón-Zygmund operator: T t f M r w L p (M rw) p c Mf M r w L p (M rw). We could use now Theorem.3 but we use a more direct method. Indeed, 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) cp (r ) /pr f. w 5 Proof of Theorem 2.3 L p (w) The proof is based on initial ideas from [23]. 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 2
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) 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. λ R n Now, by Chebyshev s inequality and Theorem 2.2: if p > and r >, we have λ w{x ( Ω) c : T g(x) > λ/2} λ c T (p ) p (r ) p λ p R n g p M r (wχ ( Ω) c)dx c T (pp ) p (r ) p R n g M r (wχ ( Ω) c)dx. By known standard geometric arguments we have R n g M r (wχ ( Ω) c)dx c n R n f M r wdx. Now if we choose p = + we can continue with log(e+r ) c T log(r ) f M r wdx r >. R n This estimate combined with the previous one completes the proof. 3
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