to appear in Birkäuser Advanced courses in Mathematics C.R.M. Barcelona A COURSE ON SINGULAR INTEGRALS AND WEIGHTS

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1 to appear in Birkäuser Advanced courses in Mathematics C.R.M. Barcelona A COURSE ON SINGULAR INTEGRALS AND WEIGHTS CARLOS PÉREZ Abstract. This article is an expanded version of the material covered in a minicourse given at the Centre de Recerca Matematica in Barcelona during the week May 4-8, We provide details and different proofs of known results as well as new ones. We also survey on several recent results related to the core of this course, namely weighted optimal bounds for Calderón-Zygmund operators with weights. The basic topics covered by these lecture revolved around the Rubio de Francia iteration algorithm, the extrapolation theorem with optimal bounds, the Coifman-Fefferman estimate, the Besicovith covering lemma and the rearrangements of functions. The paper can be seen as a modern introduction to the theory of weights.. Introduction The Hardy-Littlewood maximal function is the operator defined by Mf(x) = sup f(y) dy x where the supremum is taken over all the cubes containing x, and where f is any locally integrable function. It is clear that M is not a linear operator but it is a sort of self-dual operator since the following inequality holds (.) sup λ w{x R n : Mf(x) > λ} c f Mw dx, λ>0 R n for any nonnegative functions f and w. It is crucial here that the constant c is independent of both functions f and w. Here we use the standard notation w(e) = w(x)dx E where E is any measurable set. The inequality (.) is interesting on its own because 99 Mathematics Subject Classification. Primary 42B20, 42B25. Secondary 46B70, 47B38. Key words and phrases. Maximal operators, weighted norm inequalities, multilinear singular integrals, Calderón-Zygmund theory, commutators. The author would like to thank Professors Joan Mateu and Joan Orobitg for coordinating two special minicourses at the Centre de Recerca Mathemática on Multilinear Harmonic Analysis and Weights presented by Professor Loukas Grafakos and by the author during the period May 4-8, These mincourses were part of a special research program for the academic year entitled Harmonic Analysis, Geometric Measure Theory, and uasiconformal Mappings coordinated by Professors Xavier Tolsa and Joan Verdera. The author would also like to thank the Centre de Recerca Matemàtica for the invitation to spend the semester there and to give this course. The author would like to acknowledge the support of the grant Spanish Ministry of Science and Innovation grant MTM and the grant from the Junta de Andalucía, proyecto de excelencia FM-4745.

2 2 CARLOS PÉREZ 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 holds (.2) (Mf) p w dx c p R n f p Mw dx R n f, w 0. This estimate follows from the classical interpolation theorem of Marcinkiewicz. Both results (.) and (.2) were proved by C. Fefferman and E.M. Stein in [29] 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 ( ) (.3) (Mf j ) q q ( ) c f j q q 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 (.). Nevertheless, the proof of (.) does not follow from the classical maximal theorem (corresponding to the case w ) but the proof is nearly identical and is based on a covering lemma of Vitali type as can be seen for instance in [3] or [27]. We show in Section 2 a simpler and direct proof based on the classical Besicovith covering lemma. The Muckenhoupt-Wheeden conjecture In these lectures we are mainly interested in 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 [33]. Conjecture. (The Muckenhoupt-Wheeden conjecture). There exists a constant c such that for any function f and any weight w (.4) sup λw{x R n : T f(x) > λ} c f Mwdx. λ>0 R n We remark that the author was studying this problem during the 90 s, only much later found out that was studied by B. Muckenhoupt and R. Wheeden during the 70 s. In particular, it seems that these authors conjectured that (.4) should hold for T = H, namely the simplest singular integral operator, the Hilbert transform: Hf(x) = pv R j f(y) x y dy, We could think that to obtain a vector-valued extension of the classical Calderón- Zygmund tgeorem for singular integral operators similar to (.3), namely ( ) T f j q q ( ) c f j q q L p (R n ) L p (R ). n j j

3 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 3 we would need to prove (.4). However, less refined estimates such as sup λw{x R n : T f(x) > λ} c r f M r wdx λ>0 R n would do the job where M r w = (M(w r )) /r, r >. This was shown in [6] by A. Córdoba and C. Fefferman being the key fact that M r w A if r > (Coifman- Rochberg estimate (5.)). This result shows that inequalities with weights having some additional properties yield interesting results as well. It seems that the best result obtained is due to the author and can be found in [62] 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.2 (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 (.5) sup λw{x R n : T f(x) > λ} c f M L(log L) ɛ(w)dx w 0. λ>0 ɛ R n We remark that the operator M L(log L) ɛ is pointwise smaller than M r, r >. Remark.3. We remark that very recently and in a joint work with T. Hytönen we have improved this theorem replacing T by T, the maximal singular integral operator. The main difficulty is that T is not linear and the approach used in these notes cannot be applied. See [39]. The author conjectured in [62] that inequality (.4) would be false. This was confirmed in [9] in the case of fractional integrals I α which are positive operators. However, there were some evidences suggesting that the conjecture could be true. Indeed, the conjecture was confirmed by Chanillo and Wheeden in [] for certain continuos Lttlewood-Paley square function S (see (.23)) was considered instead of H, namely (.6) sup t w{x R n : Sf(x) > t} c f Mwdx w 0. t>0 R n Also in [65] a similar result was obtained for the vector-valued maximal function (see (.3)): ( (.7) sup t w({x R n : t>0 i= (Mf i (x)) q ) /q > t}) c q R n f(x) q Mw(x)dx, where w 0, < q <. Since many evidences show that the vector-valued maximal function ( i= (Mf i(x)) q ) /q behaves somehow like a singular integral, both inequalities (.7) and (.6) suggested that it would be the same for the case of singular integrals. Muckenhoupt-Wheeden conjecture has been opened even for the Hilbert transform until the end of 200 when was disproved for the Hilbert transform by M. C. Reguera and C. Thiele [74]. Previously, Reguera had shown that (.4) is false for dyadic type

4 4 CARLOS PÉREZ singular operators T, more precisely for a special Haar multiplier operator. This was the main result of Reguera s PhD s thesis [7]. Haar multipliers can be seen as dyadic versions of singular integrals and are used as models to understand them. It is remarkable that Reguera disproved an stronger weighted L 2 result of special type, namely of the form (.8) T f 2 w dx c f 2 ( Mw w )2 w dx w 0. R R Indeed, it was shown in [22] that if (.4) holds for an arbitrary operator T then this weighted L 2 estimate holds for T. In other words, Reguera disproved an stronger inequality than the original one of Muckenhoupt-Wheeden at least for these special dyadic singular integrals. This result gave strong evidence for a negative answer of the Muckenhoupt-Wheeden conjecture. This scheme was used in the subsequent paper [74] by Reguera and Thiele where they gave a simplified construction of the weight given in [72] and finally showed that (.4) result is really false for the Hilbert transform H by showing again that (.8) is false for H. This result is really interesting and is related to what is called the A conjecture that we will discuss below and which are really the main motivation of these lecture notes. The A conjecture This is a variant of Conjecture. and the idea is to assume an a priori condition on the weight w. This condition can be read directly from Fefferman-Stein s inequality (.) 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 (.9) Mw c w a.e. It is standard to denote by [w] A the smallest of these constants c. Then if w A (.0) sup t w{x R n : Mf(x) > t} c n [w] A f wdx t>0 R n and it is natural to ask wether the corresponding inequality holds for singular integrals (say for the Hilbert transform): Conjecture.4 (The A conjecture). Let w A, then (.) sup λ w{x R n : T f(x) > λ} c [w] A f wdx. λ>0 R n However, this inequality seems to be false too (see [57]) for T = H, the Hilbert transform. In this paper we will survey on some recent progress in connection with this conjecture exhibiting an extra logarithmic growth in (.) which in view of [57] could be the best possible result. To prove this logarithmic growth 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

5 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 5 in Sections 6 and 7 and are essentially taken from [52] and [50]. Strong type estimates To study inequality (.), it is natural to ask first the dependence of T L p (w), p >, in terms of [w] A. We discuss briefly some results before the papers [50] and [52]. Denote by δ the best possible exponent in the inequality (.2) T L p (w) c n,p[w] δ A, in the case when p = 2 and T = H is the Hilbert transform, R. Fefferman and J. Pipher [28] established that δ =. The proof is based on a sharp A bounds for appropriate square functions on L 2 (w) from the works [0, ], in particular, the following celebrated inequality of Chang-Wilson-Wolff was used: (Sf) 2 w dx C f 2 M(w) dx R n R n where S is any of the classical Littlewood-Paley square function as for instance (.23) (compare with inequality (.6)). One can show that this approach yields δ = also for p > 2. However, for < p < 2 the same approach gives the estimate δ /2 + /p. Also, that approach works only for smooth singular integrals of convolution type and recall that Calderón-Zygmund operators are non-convolution operators with a very minimal regularity condition. In [50] and [52] a different approach was used to show that for any Calderón-Zygmund operator, the sharp exponent in (.2) is δ = for all < p <. The method in this paper has its root in the classical Calderón-Zygmund theory but with several extra refinements. We believe that the circle of ideas in these papers may lead to another proof of the A 2 conjecture. below. We state now our main theorems. 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.5. [The linear growth theorem] Let T be a Calderón-Zygmund operator and let < p <. Then (.3) T L p (w) c pp [w] A where c = c(n, T ). As an application of this result we obtain the following endpoint estimate. Theorem.6. [The logarithmic growth theorem] Let T be a Calderón-Zygmund operator. Then (.4) T L, (w) c[w] A ( + log[w] A ), where c = c(n, T ).

6 6 CARLOS PÉREZ Remark.7. The result in (.3) is best possible and, as already mentioned, [57] strongly suggests that (.4) could also be the best possible result. On the other hand and very recently, a new improvement of these two theorems have been found by the author and T. Hytönen [38] in terms of mixed A A constants. See Section 9 for some details, in particular Theorem 9.3 and also Theorem.3. Remark.8. As in remark (.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 [39]. The weak (p, p) conjecture and the Rubio de Francia s algorithm If we could improve (.4) by removing the log term, namely if the A conjecture were to hold then we had the following result. Conjecture.9. [The weak (p, p) conjecture] Let < p < and let T be a Calderón- Zygmund singular integral operator. There is a constant c = c(n, T ) such that for any weight w, (.5) T L p, (w) c p [w]. We recall that a weight w satisfies the Muckenhoupt condition if ( ) ( p [w] Ap sup w(x)dx w(x) dx) /(p ) <. [w] Ap is usually called the constant (or often called characteristic or norm) of the weight. The case p = is understood by replacing the right hand side by (inf w) which is equivalent to the definition given above, (.9). Observe the duality relationship: [w] p = [σ] where we use the standard notation σ = w p = w p. Also observe that [w] Ap. In section 4. we will prove this conjecture assuming that the A conjecture were true. The argument will be based on an application of the Rubio de Francia s algorithm or scheme. The same argument applied to inequality (.4) yields the following result. Corollary.. Let < p < and let T be a Calderón-Zygmund operator. Also let w, then (.6) T f L p, (w) c p [w] Ap ( + log[w] Ap ) f L p (w), where c = c(n, T ). Observe that for p close to one, the behavior of the constant is much better than in (.2). The advantage here is that this method works for any Calderón-Zygmund operator. p

7 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 7 Rubio de Francia s algorithm is a technique which has become ubiquitous in the modern theory of weights and beyond. It is also very flexible as will be shown in Sections 3 and 4 as it will be applied to five different scenarios of interest for these lecture notes. We refer the reader to the monograph [9] for a full account of the technique. The A 2 conjecture In [54] B. Muckenhoupt 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 condition of Muckenhoupt. Of course the operator norm M L p (w) will depend on the condition of w but it seems that the first precise result was proved by S. Buckley [8] as part of his Ph.D. thesis. Theorem.0. Let w, then the Hardy-Littlewwod maximal function satisfies the following operator estimate: namely, M L p (w) c n p [w] p (.7) sup M c [w] L p (w) n p. p Furthermore the result is sharp in the sense that: for any θ > 0 (.8) sup w [w] w p θ M L p (w) = In fact, we cannot replace the function ψ(t) = t p by a smaller function ψ : [, ) (0, ) in the sense that inf t> ψ(t) t p = 0 (or lim t ψ(t) t β = 0, or sup t> t β ψ(t) = or lim t tβ ψ(t) (.9) sup w ψ([w] Ap ) M L p (w) = = ) since then The original proof of Buckley is delicate because is based on a sharp version of the so called Reverse Hölder Inequality for weights. However, very recently, A. Lerner [46] has found a very nice and simple proof of this result that we will given in section 2. It is based on the Besicovich lemma which can be avoided when M is dyadic maximal function. To see the power of this new proof we remark that the constant c p in Buckley s proof cannot be that precise. In fact in these lecture notes we avoid the use of The Besicovich by considering first the dyadic case and finally shifting the dyadic cubes.

8 8 CARLOS PÉREZ (.7) should be compared with the weak-type bound (.20) M L p (w) L p, (w) c n [w] /p whose proof is much simpler and will be shown in Lemma 2.3. Buckley s theorem attracted renewed attention after the work of Astala, Iwaniec and Saksman [3] on the theory of quasirregular mapppings. They proved sharp regularity results for solutions to the Beltrami equation, assuming that the operator norm of the Beurling-Ahlfors transform grows linearly in terms of the constant for p 2. This linear growth was proved by S. Petermichl and A. Volberg in [70]. This result opened up the possibility of considering some other operators such as the classical Hilbert Transform. Finally S. Petermichl [68, 69] has proved the corresponding results for the Hilbert transform and the Riesz Transforms. To more precise, in [70] [68, 69] it has been shown that if T is either the Ahlfors-Beurling, Hilbert or Riesz Transforms and < p <, then (.2) T L p (w) c p,n[w] max{, p }. Furthermore the exponent max{, } is best possible by examples similar to the one p related to Theorem.0. It should be compared this result with the linear growth theorem.5. Indeed, recall that A, and that [w] Ap [w] A. Therefore, (.2) clearly gives that δ = in (.2) when p 2. However, (.2) cannot be used in Theorem.5 to get the sharp exponent δ in the range < p < 2, becoming the exponent worst when p gets close to. In view of these results and others (for instance in the case of paraproducts [?] due to O. Beznosova) It was then believed that the conjecture that should be true is the following. Conjecture. (the A 2 conjecture). Let < p < and let T be a Calderón- Zygmund singular integral operator. Then, there is a constant c = c(n, T ) such that for any weight w, { } (.22) T c p L p (w) p [w] max, p. This conjecture has been proved by Tuomas Hytönen in [36]. We will briefly mention in next paragraphs some of the previous steps done toward this conjecture although the main topic of these lectures is more related to the A conjecture already mentioned. The maximum in the exponent reflects the duality of T, namely that T is also a Calderón-Zygmund operator. In fact it can be shown that if T is selfadjoint (or essentially like Calderón-Zygmund operators) and if (.22) is proved for p > 2 then the case < p < 2 follows by duality (see Corollary 3.). What it is more interesting is that by the sharp Rubio de Francia extrapolation theorem obtained in [26] (or see

9 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 9 Corollary 3. again) it is enough to prove (.22) only for p = 2. This is the reason why the result has been called the A 2 conjecture. Observe that in this case the growth of the constant is simply linear. In these lecture notes we will prove an special case of the extrapolation theorem which is enough for our purposes, namely Theorem 3.3 and Corollary 3.. In each of the previously known cases, the proof of (.22) (again, just in the case p = 2 by the sharp Rubio de Francia extrapolation theorem) is based on a technique developed by S. Petermichl [68] reducing the problem to proving the analogous inequality for a corresponding Haar shift operator. The norm inequalities for these dyadic operators were then proved using Bellman function techniques. Much more recently, Lacey, Petermichl and Reguera-Rodriguez [42] gave a proof of the A 2 conjecture for a large family of Haar shift operators that includes all the dyadic operators needed for the above results. Their proof avoids the use of Bellman functions, and instead uses a deep, two-weight testing type T b theorem for Haar shift operators due to Nazarov, Treil and Volberg [56]. A bit later and motivated by [42], a completely different proof was found by the author together with D. Cruz-Uribe and J.M. Martell in [2] which avoids both Bellman functions and two-weight norm inequalities such as the T b theorem. Instead, it is used a very interesting and flexible decomposition formula for general functions f due to Andrei Lerner [48] whose main ideas go back to the work of Fuji [30]. The main new ingredient is the use of the local mean oscillation instead of the usual oscillation which was the one considered by Fuji. We remit the reader to [2] for details of how to apply Lerner s formula to generalized Haar shift operators. An important advantage of this approach (again by means of Lerner s formula) is that it also yields the optimal sharp one weight norm inequalities for other operators such as dyadic square functions and paraproducts for the vector-valued maximal function of C. Fefferman-Stein as well as some very sharp two weight bump type conditions. All these results can be found in [20]. As a sample we mention the following result for dyadic square functions. Let denote the collection of dyadic cubes in R n. Given, let be its dyadic parent: the unique dyadic cube containing whose side-length is twice that of. The dyadic square function is the operator ( /2 (.23) S d f(x) = (f f ) 2 χ (x)), where f = f. For the properties of the dyadic square function we refer the reader to Wilson [78]. Theorem.2. Given p, < p <, then for any w, S d f L p (w) C n,p [w] max{ 2, { } Further, the exponent max, is the best possible. 2 p p } f L p (w).

10 0 CARLOS PÉREZ The exponent in Theorem.2 was first conjectured by Lerner [45] for the continuous square function; he also showed it was the best possible. An interesting fact concerning the proof of this theorem is that its proof is based again on the sharp Rubio de Francia extrapolation theorem but the novelty is that the extrapolation hypothesis is for the case p = 3, instead of p = 2 as in the case of singular integrals. Again, details can be found in [20]. The results mentioned above for generalized Haar shifts could be used to prove the A 2 conjecture when the kernel of the Calderón-Zygmund operator was sufficiently smooth (for instance C 2 would be enough by applying for instance the approximating result by A. Vagharshakyan [75]). However, this was not enough to prove the full A 2 conjecture since it is assumed that the kernel satisfies merely a Hölder-Lipschitz condition. Finally the conjecture was proved, as already mentioned, by T. Hytönen in August 200 [36]. The proof was based on an important reduction obtained by the author with S. Treil and A. Volberg in [67]. Very roughy this reduction says that a weighted L 2 weak type estimate is essentially equivalent to prove the corresponding strong type. A bit later a direct proof, avoiding this reduction, was found in [40]. One of the key points is to use a probabilistic representation formula due to Hytönen. Then the generalized shift operators act as building blocks of this representation. Therefore, an important and hard part of the proof of the A 2 conjecture was to obtain bounds for appropriate Haar shifts operators with complexity (m, n) that depend at most polynomially on the complexity (the problem in [20] is that the method is very flexible but the complexity s dependence is of exponential type). The estimate obtained in [40] is of polynomial degree k = 3 that was further improved to linear by [73]. Last result was based on the Bellman method using some ideas from another argument given in [58] with a slight worst estimate but with the advantage that can be transferred to the context of doubling metric spaces. Improving the A 2 conjecture, now theorem On the other hand, and in a direction that we think is more interesting, the A 2 conjecture, which is now a theorem, has been improved by the author and T. Hytönen [38] in terms of mixed A 2 A constants (see remark.7) and in [37] in the general p. We state this new result whose proof is based on a new sharp reverse Hölder property for A weights (Theorem 9.) that can also be found in [38]. The new idea is to derive results fractioning the A 2 constant in two fractions, one involves the A 2 constant as such and the other involves the A. Theorem.3. Let T be a Calderón-Zygmund operator. Then there is a constant depending on T such that T L 2 (w) c [w] /2 A 2 max{[w] A, [w ] A } /2. See Section 9 for some details of these mixed A constants mainly in the case p =.

11 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS The fractional A 2 conjecture We finish this introductory section by mentioning briefly some results about fractional integrals that confirm both Conjectures.9 and.. Of course these operators are different from singular integrals but still is a good indication. Recall that for 0 < α < n, the fractional integral operator or Riesz potential I α is defined, except perhaps for a constant, by I α f(x) = R n f(y) dy. x y n α In [55], B. Muckenhoupt and R. Wheeden characterized the weighted strong-type inequality for fractional operators in terms of the so-called,q condition. For < p < n α and q defined by = α, they showed that, q p n ( ) /q ( ) /p (.24) (wi α f) q dx c (wf) p dx f 0 R n R n if and only if w,q : [w] Ap,q sup ( ) ( w q dx w p In [43] it has been shown the following estimates: Suppose that p, q, α are as above. Then (The weak estimate) (The strong estimate) ) q/p dx <. I α f L q, (w q ) c [w] α n,q w f L p (R n ) f 0. wi α f L q, (R n ) c [w] ( α n ) max{, p q },q w f L p (R n ) f 0. Furthermore both exponents are sharp. Note that if we formally put α = 0 in these results we may think that the fractional integral becomes a singular integral operator and we recover the two conjectures already mentioned. 2. Three applications of the Besicovitch covering lemma to the maximal function In this section we take the maximal function as a model example to understand more difficult operators. Before considering the strong case, Theorem.0, we will show first a corresponding result for the weak type case. To prove both theorems we will use the classical Besicovitch covering lemma.

12 2 CARLOS PÉREZ Lemma 2. (The Besicovitch covering lemma). Let K de a bounded set in R n and suppose that for every x K there is an (open) cube (x) with center at x. Then we can find a sequence (possible finite) of points {x j } in K such that K j (x j ) and where c n is a finite dimensional constant. χ c (xj ) n j The proof of this result can be found in several places such as the classical lecture notes by M. De Guzman [25], also in [33]. We distinguish two cases, p =, and < p <. The first case will follow after proving the C. Fefferman-Stein basic initial estimate (.). Lemma 2.2. There is a dimensional constant c such that for any f, w (2.) sup λ w({x R n : Mf(x) > λ}) c f(x) Mw(x)dx λ>0 R n and hence M L, (w) c n [w] A. Proof. The proof is just an application of the Besicovitch covering lemma. Indeed, assuming that w is bounded as we may, the first observation is that (2.) is equivalent to sup λ w({x R n : M(f w )(x) > λ}) C f(x) w(x)dx λ > 0. λ>0 Mw R n The second is that we trivially have the pointwise inequality M(f w Mw )(x) c n Mwf(x), c where Mw c is the weighted centered maximal function (2.2) Mwf(x) c = sup f(y) w(y)dy. r>0 w( r (x)) Therefore (2.) follows from r(x) sup λ w({x R n : Mwf(x) c > λ}) C λ>0 R n f(x) w(x)dx which is a consequence of the Besicovitch covering lemma where C si a dimensional constant.

13 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 3 Lemma 2.3. Let w, < p <.There is a constant c = c n such that for any weight w, (2.3) M L p, (w) c n [w] p, This result is known but the point we want to make is to compare the exponent of the constant with the other exponents appearing in these lectures notes. For instance, compare it with both exponents in Conjecture.9 and Theorem.5. Another interesting observation here is that if we consider the dual estimate, the constant is essentially the same, namely: (2.4) M L p, (σ) c n [σ] p = c n [w] p. where recall σ = w p. Proof of the Lemma. Since w for each cube and nonnegative function f ( ) p f(y) dy w() [w] Ap f(y) p w(y)dy and hence Mf(x) M c f(x) [w] p M c w(f p )(x) p. We finish by applying again the Besicovtich covering lemma: Mf L p, (w) c n[w] p M c w(f p ) p L p, (w) c n[w] p M c w(f p ) p ( ) c n [w] /p p f p w dx R n L, (w) We remark that there are other proofs of these lemmas without appealing the Besicovitch lemma, just by a Vitali type covery lemma. We leave the proof to the interested reader. We now prove Buckley s Theorem.0 based on Lerner s proof [46]. This proof has as a bonus an improvement of the original constant c n,p. Proof of Theorem.0. To prove (.7) we set for any cube () = w() ( σ() ) p

14 4 CARLOS PÉREZ where as usual σ = w p. Now, we will consider first case of the dyadic maximal function. { f = () p ( ) } p p f w() σ() { } [w] p M d w() σ(fσ ) p p dx. where M d σ is the weighted dyadic maximal function. Hence We conclude by using that M d f(x) [w] p { M d w ( M d σ (fσ ) p w ) (x) } p M d µ L p (µ) p with bounds independent of µ which follows from the improved version of the Marcinkewicz interpolation theorem in the following form: T L p (µ) p T /p L, (µ) T /p L (µ) < p < where T is any sublinear operator bounded on L (µ) and of weak type (, ) with norms T L (µ) and T L, (µ) respectively. (see for instance [32] p. 42 exercise.3.3). This gives the estimate M d L p (w) [w] p p p /p p [w] p e p finishing the proof in the case of the dyadic maximal function. Observe that the constant here is the dyadic constant. The general situation follows easily by shifting the dyadic network applying Minkowski s inequality to the following wellknown Fefferman-Stein shifting lemma that can be found (cf. [3] p. 43): For each integer k M 2k f(x) 23n+ ( ) τ t M d τ t f(x) dt x R n 2 k+2(0) 2 k+2 (0) where τ t g(x) = g(x t), r (0) is the cube centered at the origin with side length r, and M δ, δ > 0, is the operator defined as M but with cubes having side length smaller than δ. For the sharpness we consider n = and 0 < ɛ <. Let It is easy to check that Let also w(x) = x ( ɛ)(p ). [w] p ɛ. f(y) = y +ɛ(p ) χ (0,) (y)

15 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 5 and observe that: f p L p (w) ɛ To estimate now Mf L p (w) we pick 0 < x <, hence and hence Mf(x) x x 0 f(y) dy = c p ɛ f(x) Mf L p (w) c p ɛ f L p (w) From which the sharpness (.8) follows easily. 3. Two applications of the Rubio de Francia algorithm: Optimal factorization and extrapolation. 3.. The sharp factorization theorem. Muckenhoupt already observed in [54] that it follows from the definition of the A class of weights that if w, w 2 A, then the weight is an weight. Furthermore we have w = w w p 2 (3.) [w] Ap [w ] A [w 2 ] p A He conjectured that any weight can be written in this way. This conjecture was proved by P. Jones s, namely if w 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 [33] where we remit the reader for more information about the theory of weights. Here we present a variation which appears in [36]. Here we give a proof of this result using sharp constants. To be more precise we have the following result. Lemma 3.. Let < p < and let w, then there are A weights u, v A, such that in such a way that w = u v p (3.2) [u] A c n [w] Ap & [v] A c n [w] p and hence [w] Ap [u] A [v] p A c n [w] 2

16 6 CARLOS PÉREZ Proof. We use Rubio de Francia s iteration scheme or algorithm to our situation. Define and S (f) p w /p M( f p ), w/p S 2 (f) p w /p M( f p w /p ), Observe that S i : L pp (R n ) L pp (R n ) with constant S i L pp (R n ) c n[w] /p i =, 2 by Buckley s theorem. Now, the operator S = S + S 2 is bounded on L pp (R n ) with S L pp (R n ) c n[w] /p. Define the Rubio de Francia algorithm R by 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 ). Now, if h L pp (R n ) is fixed, R(h) A (S). More precisely In particular R(h) A (S i ) i =, 2, with S(R(h)) 2 S L pp (R n ) c n[w] /p. S i (R(h)) c n [w] /p R(h) i =, 2 Hence and Finally, letting M(R(h) p w /p ) c n [w] p /p R(h) p w /p M(R(h) p w /p ) c n [w] Ap R(h) p w /p. u R(h) p w /p & v R(h) p w /p we have u, v A w = uv p with [u] A c n [w] Ap & [v] A c n [w] p

17 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS The sharp extrapolation theorem. One of the main results in modern Harmonic Analysis is the extrapolation theorem of J.L. Rubio de Francia for weights. This result is very useful because reduces matters to study one special exponent typically p = 2. We refer the reader to the monograph [9] for a new proof and for a full account of the theory. On the other hand, in [26] it has been shown a version of the extrapolation theorem with sharp constants which turns out to be very useful. The proof follows the classical method as exposed in [3] and it is based on the Rubio de Francia s algorithm. Here we will give this proof. Theorem 3.2. Let T be any operator such that for some exponent α > 0 (3.3) T L 2 (w) c [w] α A 2 w A 2 then (3.4) T L p (w) c p [w] α p > 2, w In [26] a corresponding result for < p < 2 can be found, but since all the applications we have in these lecture notes deal with linear operators whose adjoints behave like the operator itself we have the following corollary. Corollary 3.. Let T be a linear operator satisfying (3.3). Suppose also that the adjoint operator T (with respect to the Lebesgue measure) also satisfies (3.3). Then T L p (w) c p [w] α max{, p } The proof is simply a duality argument. < p < 2. Indeed, by standard theory T L p (w) = T L p (σ) p >, w All we have to do is to check the case where as usual σ = w p. Hence, if < p < 2, p > 2 and we can apply the theorem to T because it verifies (3.3) obtaining T L p (σ) c [σ]α = c [w] α p. Proof. The proof that follows is taken from Garcia-Cuerva and Rubio de Francia s book [3]. We only need to be absolutely precise with the exponents. Let so that (3.5) T (f) 2 L p (w) = sup h t = p 2 p R n T (f)(x) 2 h(x) w(x) dx, where the supremum runs over all 0 h L p /t (w) with h L p /t (w) =.

18 8 CARLOS PÉREZ We run the Rubio de Francia algorithm now as follows. Define the operator ( ) M(h /t t w) S w (h) = h 0. w It is easy to see that by Muckenhoupt s theorem S w is bounded on L p /t (w) if w. Furthermore if we use Buckley s theorem.0 we have S w L p /t (w) c p [w] t Define now D(h) = Then we have (A) h D(h) (B) D(h) L p /t (w) 2 h L p /t (w) k=0 2 k S k w(h) S w k L p /t (w) (C) S w (D(h)) 2 S w L p /t (w) D(h) and hence: (3.6) [D(h).w] A2 c [w] Ap. (A) and (B) and the first part of (C) are immediate. It only remains to prove (3.6). First we claim: for any h 0 ( ) ( ) [hw, S w (h)w] A2 = sup hw(x)dx (S w (h)w) dx 2 [w] t A p. Indeed, since for x it verifies that we have that ( I = ( = ( ( [w] t, M(h /t w)(x) ) ( hw(x)dx ) ( hw(x)dx ) t ( h /t w(x)dx h /t w(x)dx, ) (S w (h)w) dx M(h /t w) t w /(p ) ) ) t ( h /t w(x)dx ) t w(x)dx ) w(x) /(p ) dx

19 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 9 where we used Hölder s inequality and then that ( t)( p) =. Hence ( ) ( ) [D(h) w] A2 = sup D(h)w(x)dx (D(h)w) dx ( ) ( ) 2 S w L p /t (w) sup D(h)w(x)dx (S w (D(h)) w) dx = 2 S w L p /t (w) [D(h)w, S w(d(h))w] A2 2 S w L p /t (w) [w] t c [w] Ap. We are ready now to conclude the proof of the theorem using finally the extrapolation hypothesis (3.3). Indeed, fixing one of the h s from (3.5) we continue with T (f)(x) 2 h(x) w(x) dx T (f)(x) 2 D(h)(x) w(x) dx R n R n [D(h) w] 2α A 2 R n f(x) 2 D(h)(x) w(x) dx c [w] 2α R n f(x) 2 D(h)(x) w(x) dx This proves (3.4). c [w] 2α f 2 L p (w) D(h) L (p/2) (w) = c [w] 2α f 2 L p (w) D(h) L p /t (w) c [w] 2α f 2 L p (w) h L p /t (w) = c [w] 2α f 2 L p (w). 4. Three more applications of the Rubio de Francia algorithm. We have seen in the previous section two applications of the the Rubio de Francia algorithm. These application were already known except for the use of the sharp exponents. In this section we show some other applications, perhaps more technical, but they play a crucial role in the works [50] and [52]. It should be mentioned that other applications can be found in [7], [8], [24] and [53]. Also we remit to the monograph [9] for more information. 4.. Building A weights from duality. The following lemma, a variation of the Rubio de Francia iteration scheme, is the key link for proving Conjecture.9 assuming that the A conjecture holds. Lemma 4.. Let < p < and let w. Then there exists a nonnegative sublinear operator D bounded on L p (w) such that for any nonnegative h L p (w):

20 20 CARLOS PÉREZ (a) (b) (c) h D(h) D(h) L p (w) 2 h L p (w) D(h) w A with [D(h) w] A c p [w] Ap where the constant c is a dimensional constant. Proof. To define the algorithm we consider the operator S w (f) = M(fw) w and observe that for any < p <, by Muckenhoupt s theorem S w : L p (w) L p (w) w However we need the sharp version in both the constant and the constant (.7): S w L p (w) cp [w p ] p = cp [w] Ap Define now for any nonnegative h L p (w) S D(h) = w(h) k 2 k S w k L p (w) Hence properties (a) and (b) are immediate and for (c) simply observe that k=0 S w (D(h)) 2 S w L p (w) D(h) 2c p [w] D(h) or what is the same D(h) w A with [D(h) w] A 2 c p [w] Ap As an application of this lemma we prove Conjecture.9 assuming that the A conjecture.4 holds. Proof of Conjecture.9. Let w and let f C (R n ) with compact support. For each t > 0, let Ω t = {x R n : T f(x) > t}. This set is bounded, so w(ω t ) <. By duality, there exists a non-negative function h L p (w) such that h L p (w) = and w(ω t ) /p = χ Ωt L p (w) = h wdx. Ω t We consider now the operator D associated to this weight from Lemma 4.. Hence the operator D satisfies (a) h D(h) (b) Dh L p (w) 2 h L p (w) = 2 (c) [D(h) w] A c p [w] Ap

21 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 2 hence assuming that the weak Muckenhoupt conjecture holds, then w(ω t ) /p D(h) w dx = (D(h) w)(ω t ) Ω t f c [D(h) w] A D(h) w dx R t n c ( ) /p ( ) t p[w] f p w dx D(h) p p w dx R n R n This completes the proof. cp t [w] ( R n f p w dx) /p Improving inequalities with A weights. In harmonic analysis, there are a number of important inequalities of the form (4.) T f(x) p w(x) dx C Sf(x) p w(x) dx, R n R n 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 ɛ > 0 small, (4.2) w({y R n : T f(y) > 2 λ, Sf(y) λɛ}) C ɛ w({y R n : Sf(y) > λ}). Here, the weight w is usually assumed to be in the Muckenhoupt class A = p>. Given inequality (4.2), it is easy to prove the strong-type inequality (4.) for any p, 0 < p <, as well as the corresponding weak-type inequality (4.3) T f L p, (w) C Sf L p, (w). In these notes the special case of (4.4) T f L p (w) c Mf L p (w) where T is a Calderón-Zygmund operator and M is the maximal function, will play a central role. This theorem was proved by Coifman-Fefferman in the celebrated paper[4]. Here, the weight w also satisfies 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 4.2 (the tricky lemma). 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 )

22 22 CARLOS PÉREZ This is the main improvement in [52] of [50] where we had obtained logarithmic growth on p. It is an important step towards the proof of the the linear growth Theorem.5. 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 There is another proof by R. Bagby and D. Kurtz using rearrangements given in the middle of the 80 s which gives better estimates on p but not in terms of the weight constant. 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 (4.4): (4.5) T f L (w) c[w] Aq Mf L (w) w A q, q < The original proof given in [52] of this estimate was based on an idea of R. Fefferman- Pipher from [28] which combines a sharp version of the good-λ inequality of S. Buckley together with a sharp reverse Hölder property of the weights (Lemma 8.). 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 [8]: (4.6) {x R n : T (f) > 2λ, Mf < γλ} c e c 2/γ {T (f) > λ} λ, γ > 0 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 (4.7) f L p (w) c p[w] Aq M # δ (f) L p (w) 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 f = f(y) dy. We present this theory in section. To prove (4.5) we combine (4.7) with the following pointwise estimate [2]: Lemma 4.3. Let T be any Calderón-Zygmund singular integral operator and let 0 < δ < then there is a constant c such that and in fact we have. M # δ (T (f ))(x) c Mf(x)

23 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 23 Corollary 4.. 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} <. Hence the heart of the matter is estimate (4.7). It will proved in Section (see Corollary.) by a completely different path using instead properties of rearrangement of functions and corresponding local sharp maximal operator. We now finish this section by proving the tricky Lemma 4.2. The proof is based on the following lemma which is another variation of the Rubio de Francia algorithm. Lemma 4.4. 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 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 4.2. 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 4.4 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.

24 24 CARLOS PÉREZ 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, then by (3.) [w] Ap [w ] A [w 2 ] p A Second, if r > then (Mf) r A by Coifman-Rochberg theorem, furthermore we need to be more precise (5.) 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 Corollary 4. 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) 5. The sharp reverse Hölder property of the A weights We already encountered with weights of the form M r w, < r <. As we are going to see they play an important role in the theory. Of course, it is well known that these weights satisfy the A condition by the theorem of Coifman-Rochberg [5]. We will be using the following quantitative version of it: Let µ be a positive Borel and let < r < and furthermore (Mµ) r A (5.) [(Mµ) r ]A c n r. In fact they proved that any A weight can be essentially written in this way Recall that these weights satisfy an special important property, namely that if w A, then there is a constnat r > such that ( ) /r w r c w

25 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 25 However there is a bad dependence on the constant c = c(r, [w] A ). To prove our results we need a more precise estimate. Lemma 5.. Let w A, and let r w = + 2 n+ [w] A. Then for each ( ) /rw w rw 2 w i.e. M rw w(x) 2 [w] A w(x). Recall that M r w = M(w r ) /r Proof. We start with the layer cake formula ϕ(f) dν = ϕ (t) ν({x X : f(x) > t}) dt X 0 We fix a cube and denote by M d to the dyadic maximal operator restricted to the cube. Also we denote w = w. Hence, w +δ dx M (w)(x) d δ wdx = δ t δ w({x : M (w)(x) d > t}) dt 0 δ w t δ w({x : M (w)(x) d > t}) dt 0 + δ t δ w({x : M (w)(x) d > t}) dt w (w ) δ+ + δ t δ w({x : M (w)(x) d > t}) dt w We know use a sort of the reverse weak type (, ) estimate for the maximal function: if t > w w { x : Mw(x) d > t } 2 n t {x : Mw(x) d > t} that can be found, for instance, in [32]. Hence, (Mw) d δ wdx (w ) δ+ + 2n δ δ + (w ) δ+ + 2n δ[w] A δ + Setting here δ = 2 n+ [w] A, we obtain (M d w) δ wdx 2(w ) δ+ (M d w) δ+ dx (M d w) δ wdx

26 26 CARLOS PÉREZ 6. 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.5. We need a a lemma which immediately gives the proof. Lemma 6.. 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: T f L p (w) cp ( r ) /pr f L p (M rw) In applications we will often 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.5. Proof of Theorem.5. Indeed, apply the lemma to w A with sharp Reverse Holder s exponent r = r w = + 2 n+ [w] A obtaining T L p (w) c p [w] A 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 4.2 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) fw /p w /p w r (fw /p ) (pr) pr, and hence, (Mf) p (M r w) p M ((fw /p ) (pr) ) p /(pr)

27 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 27 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) 7. Proof of the logarithmic growth theorem. Proof of Theorem.6. The proof is based on ideas from [62]. 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) 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 have the full Muckenhoupt-Wheeden conjecture: 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 have the full Muckenhoupt conjecture). 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}.

28 28 CARLOS PÉREZ 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 > we have w{x ( Ω) c : T g(x) > λ/2} ( ) p c(p ) p r g p M r λ p r (wχ ( Ω) c)dx R ( n ) p c(p ) p r g M r (wχ r λ ( Ω) c)dx. R n By more or less standard arguments we have g M r (wχ ( Ω) c)dx c f M r wdx. R n R n Combining this estimate with the previous one, and then taking the sharp reverse Holder s exponent r = + 2 n+ [w] A, by the reverse Hölder s inequality lemma we get w{x ( Ω) c : T g(x) > λ/2} c(p [w] A ) p f wdx. λ R n Setting here p = + log( + [w] A ) gives w{x ( Ω) c : T g(x) > λ/2} c[w] A ( + log[w] A ) f wdx. λ R n This estimate combined with the previous one completes the proof. 8. Properties of the weights We have seen how important is the sharp reverse Hölder exponent for A weights, Lemma 5., for the proof of Theorem.5 and hence for that of Theorem.6. A natural question is then to find a similar result for the class of weights. The question is interesting on its own but it turns out that it is very useful as well as can be seen in the proof of the quadratic estimates for commutators given in Section 0. Recall that if w there are constants r > and c such that for any cube (8.) ( w r dx ) r c In the standard proofs both constants c, r depend upon the constant of the weight. We prove here a more precise version of (8.). w

29 A COURSE ON SINGULAR INTEGRALS AND WEIGHTS 29 Lemma 8.. Let w, < p < and let r w = + 2 2p+n+ [w] Ap. Then for any ( (8.2) w rw dx ) 2 rw w Remark 8.2. We remark that these result has been considerably improved in [38], it is stated below in Theorem 9. but we skip the proof which is different from the one we present here after next corollary. The classical proofs of this property for any weights produce non linear growth constants. We also remark that this result was stated and used in [8] with no proof. The author mentioned instead the work by Coifman-Fefferman [4] where no explicit statement can be found. Since this lemma plays an important role (specially the case p = 2) we supply below a proof. As a corollary we deduce the following useful result. Corollary 8.. Let < p < and let w. Denote p p w = p + Then w /pw and furthermore + 2 2p +n+ p [w] Ap [w] Ap/pw or what is equivalent w ɛ where ɛ = 2 p [w] Ap p +2 2p +n+ [w] p Ap p [w] p Ap The proof of the corollary is as follows, since w, σ lemma and hence by the with and hence ( ) ( w(x)dx namely r σ = + = + 2 2p +n+ [σ] Ap ( σ rσ dx ) 2 rσ σ 2 2p +n+ [w] p ) p ( ) ( ) p rσ σ(x) rσ dx 2 p 2 w(x)dx w p dx since [w] Ap/pw p r σ 2 p [w] Ap = p p w

30 30 CARLOS PÉREZ This theorem plays an important role in deriving a sharp version of the Coifman- Fefferman estimate as stated in Corollary.2 which plays a central role in (.5). Proof. Let w = w. w(x) δ w(x)dx = δ t δ w({x : w(x) > t}) dt t = δ + δ 0 w 0 = I + II. t δ w({x : w(x) > t}) dt t t δ w({x : w(x) > t}) dt w t Observe that I (w ) δ+. For II we make first the following observation: for any we let E = {x : w(x) w 2 p } [w] Ap Then we claim (8.3) E 2. Indeed, by Hölder s inequality we have for any f 0 ( p f(y) dy) w() [w] Ap f(y) p w(y)dy and hence if E, ( ) p E w(e) [w] Ap w() and in particular, ( ) p E w(e ) [w] Ap w() [w] w w() E = E 2 p [w] Ap 2 p, from which the claim follows. The second claim is the following for every λ > w (8.4) w({x : w(x) > λ}) 2 n+ λ {x : w(x) > λ 2 p [w] Ap w }. Consider the CZ decomposition of w at level λ, we find a family of disjoint cubes { i } contained in satisfying λ < w i 2 n λ for each i. Indeed, since except for a null set we have {x : w(x) > λ} {x : M d w(x) > λ} = i i,