VARIATION FOR THE RIESZ TRANSFORM AND UNIFORM RECTIFIABILITY

Transcription

1 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY ALBERT MAS AN XAVIER TOLSA Abstract. For 1 n < d integers and ρ >, we prove that an n-diensional Ahlfors- avid regular easure µ in R d is uniforly n-rectifiable if and only if the ρ-variation for the Riesz transfor with respect to µ is a bounded operator in L µ. This result can be considered as a partial solution to a well known open proble posed by G. avid and S. Sees which relates the L µ boundedness of the Riesz transfor to the unifor rectifiability of µ. 1. Introduction In this paper we characterize the notion of unifor rectifiability in the sense of avid and Sees [S] in ters of the L boundedness of the ρ-variation for the Riesz transfor, with ρ >. Given 1 n < d integers and a Radon easure µ in R d, one defines the n-diensional Riesz transfor of a function f L 1 µ by R µ fx = li ɛ 0 R ɛ µ fx whenever the liit exists, where R ɛ µ x y fx = x y n+1 fy dµy, x Rd. x y >ɛ We will use the notation R µ fx := {R ɛ µ fx} ɛ>0. When d = i.e., µ is a Radon easure in C, one defines the Cauchy transfor of f L 1 µ by C µ fx = li ɛ 0 C ɛ µ fx whenever the liit exists, where C ɛ µ fx = x y >ɛ fy dµy, x C. x y To avoid the proble of existence of the preceding liits, it is useful to consider the axial operators R µ fx = sup ɛ>0 R ɛ µ fx and C µ fx = sup ɛ>0 C ɛ µ fx. Notice that the Cauchy transfor coincides with the 1-diensional Riesz transfor in R odulo conjugation, since 1/x = x/ x for all x C \ {0}. The Cauchy and Riesz transfors are two very iportant exaples of singular integral operators with a Calderón-Zygund kernel. Given d, the kernels K : R d \ {0} R that we consider in this paper satisfy 1 Kx C x n, x ikx C x n+1 and x i x jkx C x n+, for all 1 i, j d and x = x 1,..., x d R d \{0}, where 1 n < d is soe integer and C > 0 is soe constant; and oreover K x = Kx for all x 0 i.e. K is odd. Notice that ate: Septeber, Matheatics Subject Classification. Priary 4B0, 4B5. Key words and phrases. ρ-variation and oscillation, Calderón-Zygund singular integrals, Riesz transfor, unifor rectifiability. Both authors are partially supported by grants 009SGR Generalitat de Catalunya and MTM Spain. Albert Mas is also supported by grant AP FPU progra, Spain. 1

2 A. MAS AN X. TOLSA the n-diensional Riesz transfor corresponds to the vector kernel x 1,..., x d / x n+1, and the Cauchy transfor to x 1, x / x so, we ay consider K to be any scalar coponent of these vector kernels. For f L 1 µ and x R d, we set T ɛ µ fx T ɛ fµx := Kx yfy dµy, and we denote T µ fx = {T µ ɛ fx} ɛ>0. x y >ɛ efinition 1.1 ρ-variation and oscillation. Let F := {F ɛ } ɛ>0 be a faily of functions defined on R d. Given ρ > 0, the ρ-variation of F at x R d is defined by 1/ρ V ρ Fx := sup F ɛ+1 x F ɛ x ρ, {ɛ } Z where the pointwise supreu is taken over all decreasing sequences {ɛ } Z 0,. Fix a decreasing sequence {r } Z 0,. The oscillation of F at x R d is defined by 1/ OFx := F ɛ x F δ x, sup {ɛ },{δ } Z where the pointwise supreu is taken over all sequences {ɛ } Z and {δ } Z such that r +1 ɛ δ r for all Z. The ρ-variation and oscillation for artingales and soe failies of operators have been studied in any recent papers on probability, ergodic theory, and haronic analysis see [Lp], [Bo], [JKRW], [CJRW1], [JSW], [LT], and [OSTTW], for exaple. In this paper we are interested in the ρ-variation and oscillation of the faily T µ f. That is, given a Radon easure µ in R d and f L 1 µ we will deal with V ρ T µ fx := V ρ T µ fx, O T µ fx := OT µ fx. We are specially interested in the case T µ = R µ. Notice, by the way, that T µ fx V ρ T µ fx for any copactly supported function f L 1 µ and all x R d. When µ coincides with the Lebesgue easure in the real line and Kx = 1/x is the kernel of the Hilbert transfor, Capbell, Jones, Reinhold and Wierdl [CJRW1] showed that V ρ T µ and O T µ are bounded in L p µ, for 1 < p <, and of weak type 1, 1. This result was extended to other singular integral operators in higher diensions in [CJRW]. The case of the Cauchy transfor and other odd Calderón-Zygund operators on Lipschitz graphs was studied recently in [MT]. Let us turn our attention to unifor rectifiability now. Recall that a Radon easure µ in R d is called n-rectifiable if there exists a countable faily of n-diensional C 1 subanifolds {M i } i N in R d such that µe \ i N M i = 0. Moreover, µ is said to be n-diensional Ahlfors-avid regular, or siply A regular, if there exists soe constant C > 0 such that C 1 r n µbx, r Cr n for all x suppµ and 0 < r diasuppµ. One also says that µ is uniforly n-rectifiable if there exist θ, M > 0 so that, for each x suppµ and r > 0, there is a Lipschitz apping g fro the n-diensional ball B n 0, r R n into R d such that Lipg M and µ Bx, r gb n 0, r θr n, where Lipg stands for the Lipschitz constant of g. In particular, unifor rectifiability iplies rectifiability. Given a set E R d, we denote by HE n the n-diensional Hausdorff easure restricted to E. Then E is called, respectively, n-rectifiable, A regular, or uniforly n-rectifiable if HE n is so. By the Lebesgue differentiation theore, any n-diensional A regular easure µ is of the for µ = fhsuppµ n with C 1 fx C for soe constant C > 0 and all x suppµ.

3 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 3 G. avid and S. Sees asked ore than twenty years ago the following question, which is still open see, for exaple, [Pa, Chapter 7]: Question 1.. Is it true that an n-diensional A regular easure µ is uniforly n- rectifiable if and only if R µ is bounded in L µ? Soe coents are in order. By the results in [S1], the only if iplication of the question above is already known to hold. Also in [S1], G. avid and S. Sees gave a positive answer to Question 1. if one replaces the L boundedness of R µ by the L boundedness of T µ for a wide class of odd kernels K. In the case n = 1 in particular, for the Cauchy transfor, the if iplication was proved by P. Mattila, M. Melnikov and J. Verdera in [MMV] using the notion of curvature of easures. Later on, G. avid and J. C. Léger [Lé] proved that the L boundedness C µ iplies that µ is rectifiable, even without the A regularity assuption with n = 1. When µ is the n-diensional Hausdorff easure on a set E R d such that µe <, the rectifiability of µ is also related with the existence µ-a.e. of the principal value of the Riesz transfor of µ, that is, the existence of R µ 1x = li ɛ 0 R ɛ µ 1x for µ-a.e. x E. In [MPr], P. Mattila and. Preiss proved that, under the additional assuption that li inf r 0 r n µbx, r > 0 for µ-a.e. x E, the rectifiability of E is equivalent to the existence of R µ 1x µ-a.e. x E. Later on, in [To3] X. Tolsa reoved the assuption and proved the result in full generality, i.e., he proved that a set E R d with µe < is rectifiable if and only if R µ 1x exists for µ-a.e. x E. Let us ention that, for the case n = 1 and d = that is, for the Cauchy transfor, the analogous results had been obtained previously by [Ma] under the assuption, and in [To1], in full generality, by using the notion of curvature of easures. In this paper we prove the following: Theore 1.3. Let 1 n < d and ρ >. An n-diensional A regular Radon easure µ in R d is uniforly n-rectifiable if and only if V ρ R µ is a bounded operator in L µ. Moreover, if µ is n-uniforly rectifiable, then for any kernel K satisfying 1, the operator V ρ T µ is bounded in L µ. Let us copare this result with the avid-sees Question 1.. Notice that the preceding theore asserts that if we replace the L µ boundedness of R µ by the stronger assuption that V ρ R µ is bounded in L µ, then µ ust be uniforly rectifiable. On the other hand, the theore clais that the variation for odd singular integral operators with any kernel satisfying 1, in particular for the n-diensional Riesz transfors, is bounded in L µ. A natural question then arises. Given an arbitrary easure µ on R d, without atos say, does the L µ boundedness of R µ iplies the L µ boundedness of V ρ R µ, for ρ >? By the results of [MMV] and Theore 1.3, this is true in the case n = 1 if µ is A regular 1-diensional. Clearly, a positive answer in the general case n 1 would solve the avid- Sees proble in the affirative. Nevertheless, such an approach to try to solve this proble looks quite difficult. In fact, we recall that, it is not even known if the L µ boundedness of R µ ensures the µ-a.e. existence of the principal values of R µ 1, which is a necessary condition for the L µ boundedness of V ρ R µ. Concerning the proof of Theore 1.3, in our previous paper [MT] we showed that, if µ stands for the n-diensional Hausdorff-easure on an n-diensional Lipschitz graph, then the ρ-variation for Riesz transfors and odd Calderón-Zygund operators with sooth truncations are bounded in L µ. This is a fundaental step to prove that V ρ R µ and, ore generally, V ρ T µ, are bounded in L µ if µ is uniforly n-rectifiable. Another basic

4 4 A. MAS AN X. TOLSA tool in our arguents is the geoetric corona decoposition of uniforly rectifiable easures introduced by avid and Sees in [S1], which, roughly speaking, describes how suppµ can be approxiated at different scales by n-diensional Lipschitz graphs. The proof of the fact that the L µ boundedness of V ρ R µ iplies the unifor rectifiability of µ is not so laborious as the one of the converse iplication. As rearked above, if V ρ R µ is bounded in L µ, then the principal values of R µ 1 exist µ-a.e., which iplies the n-rectifiability of µ, by the results of [MPr] or [To3]. However, this is not enough to ensure the unifor n-rectifiability of µ. We will prove the unifor n-rectifiability by arguents partially inspired by soe of the techniques in [To4]. Finally, let us reark that Theore 1.3 follows fro a ore general result, naely Theore.3 below, which also deals with the variation for Riesz transfors and odd Calderón- Zygund operators with sooth truncations. As usual, in the paper the letter C stands for soe constant which ay change its value at different occurrences, and which quite often only depends on n and d. Given two failies of constants At and Bt, where t stands for all the explicit or iplicit paraeters involving At and Bt, the notation At Bt At Bt eans that there is soe fixed constant C such that At CBt At CBt for all t, with C as above. Also, At Bt is equivalent to At Bt At..1. The ain theore.. Preliinaries efinition.1 failies of truncations. Let χ R := χ [1, and let ϕ R : [0, + [0, + be a non decreasing C function with χ [4, ϕ R χ [1/4,. Suppose oreover that ϕ R is bounded below away fro zero in [1/3, 3], i.e., χ [1/3,3] C ϕ R for soe C > 0. Given x R d, and 0 < ɛ δ, we set χ ɛ x := χ R x /ɛ and χ δ ɛx := χ ɛ x χ δ x, ϕ ɛ x := ϕ R x /ɛ and ϕ δ ɛx := ϕ ɛ x ϕ δ x. Notice that, for any finite Radon easure µ, T ɛ µx = Kχ ɛ µx. Given x = x 1,..., x d R d, we denote x = x 1,..., x n, 0,..., 0 R d, and we set ϕ ɛ x := ϕ ɛ x and ϕ δ ɛx := ϕ δ ɛ x. Finally, for f L 1 µ we set T µ f T fµ := {T µ ɛ f} ɛ>0, T ϕ µ ɛ fx T ϕɛ fµx := Kϕ ɛ µx and T ϕ µ f T ϕ fµ := {T ϕ µ ɛ f} ɛ>0, T µ ϕ ɛ fx T ϕɛ fµx := K ϕ ɛ µx and T µ ϕ f T ϕfµ := {T µ ϕ ɛ f} ɛ>0. Reark.. In the definition, the choice of [4,, [1/4,, and [1/3, 3] is not specially relevant, it is just for definiteness. One can replace the preceding intervals by other suitable intervals, and all the proofs in the paper reain alost the sae. We will prove the following. Theore.3 Main Theore. Let 1 n < d be integers. Let µ be an n-diensional A regular Radon easure on R d. The following are equivalent: a µ is uniforly n-rectifiable. b For any K satisfying 1 and any ρ >, the operator V ρ T ϕ µ is bounded in L p µ for all 1 < p <, and fro L 1 µ into L 1, µ. c For any K satisfying 1 and any ρ >, the operator V ρ T µ is bounded in L µ. d For soe ρ > 0, the operator V ρ R µ is bounded in L µ. e For Kx = x/ x n+1 and soe ρ > 0, the operator V ρ T µ ϕ is bounded in L µ.

5 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 5 Clearly, Theore 1.3 is a direct consequence of the preceding result. Reark.4. Let {r } Z 0, be a fixed decreasing sequence defining O. Then, the iplications a b,..., e in the theore above still hold if one replaces V ρ by O. If there exists C > 0 such that C 1 r r r +1 Cr for all Z, then the iplications b,..., e a also hold so Theore.3 reains true replacing V ρ by O, but we do not know if they are still true without this additional assuption see Reark 6.9. Notice that, by Theore.3, besides V ρ R µ and O R µ, the operators V ρ T ϕ µ and for Kx = x/ x n+1 characterize copletely the n-a regular easures µ which are O T µ ϕ uniforly n-rectifiable. One of the ain ingredients for the proof of Theore.3 is the following result, which strengthens one of the endpoint estiates obtained in [MT]. Let MR d be the space of finite real Radon easures on R d, with the nor induced by the variation of easures. Theore.5. Let ρ > and let µ be the n-diensional Hausdorff easure restricted to an n-diensional Lipschitz graph. Then, V ρ T ϕ is a bounded operator fro MR d to L 1, µ, i.e., there exists C > 0 such that, for all λ > 0 and all ν MR d, µ { x R d : V ρ T ϕ νx > λ } C λ ν. In particular, V ρ T µ ϕ is of weak type 1, 1. The constant C only depends on n, d, K, ρ, ϕ R, and the axial slope of Γ. By an n-diensional Lipschitz graph Γ R d we ean any translation and rotation of a set of the type {x R d : x = y, Ay, y R n }, where A : R n R d n is soe Lipschitz function with Lipschitz constant LipA, which coincides with the axial slope of Γ. Reark.6. The theore above reains valid if one replaces V ρ by O. Moreover, the nor of O T µ ϕ is bounded independently of the sequence that defines O. The plan to prove Theore.3 is the following: in Section 3 we deal with Theore.5, which is used in the subsequent Section 4 to obtain the iplication a = b of Theore.3. In Section 5 we prove a = c in Theore 5.1, and in Section 6 we prove Theore 6.8, which gives d = a and e = a, and finishes the proof of Theore.3, taking into account that the iplications b = e and c = d are trivial. Theores.3 and.5 are stated in ters of V ρ, but they also hold for O, as rearked above. However, we will only give the proof of these results for V ρ, because the case of O follows by very siilar arguents and coputations... Calderón-Zygund decoposition for easures. Given a cube Q R d and a > 0, we denote by lq the side length of Q and by aq the cube concentric with Q with side length alq. The cubes that we consider in this paper have sides parallel to the coordinate axes in R d. A proof of the following result can be found in [To5, Chapter ] or [M, Lea 5.1.]. Lea.7 Calderón-Zygund decoposition. Assue that µ := H n Γ B, where Γ is an n-diensional Lipschitz graph and B R d is soe fixed ball. For any ν MR d with copact support and any λ > d+1 ν / µ, the following holds:

6 6 A. MAS AN X. TOLSA a There exists a finite or countable collection of alost disjoint cubes {Q j } j R d that is, j χ Q j C and a function f L 1 µ such that ν Q j > d 1 λµq j, ν ηq j d 1 λµηq j for η >, ν = fµ in R d \ j Q j with f λ µ-a.e. b For each j, let R j := 6Q j and denote w j := χ Qj k χ 1. Q k Then, there exists a faily of functions {b j } j with suppb j R j and with constant sign satisfying b j dµ = w j dν, 7 b j L µµr j C ν Q j, and 8 j b j C 0 λ where C 0 is soe absolute constant..3. yadic lattices. For the study of the uniforly rectifiable easures we will use the dyadic cubes built by G. avid in [a, Appendix 1] see also [S, Chapter 3 of Part I]. These dyadic cubes are not true cubes, but they play this role with respect to a given n-dienasional A regular Radon easure µ, in a sense. To distinguish the fro the usual cubes, we will call the µ-cubes. Let us explain which are the precise results and properties about the lattice of dyadic µ-cubes. Given an n-diensional A regular Radon easure µ in R d for siplicity, we ay assue diasuppµ =, for each j Z there exists a faily j of Borel subsets of suppµ the dyadic µ-cubes of the j-th generation such that: a each j is a partition of suppµ, i.e. suppµ = Q j Q and Q Q = whenever Q, Q j and Q Q ; b if Q j and Q k with k j, then either Q Q or Q Q = ; c for all j Z and Q j, we have j diaq j and µq jn ; d there exists C > 0 such that, for all j Z, Q j, and 0 < τ < 1, µ {x Q : distx, suppµ \ Q τ j } 9 + µ {x suppµ \ Q : distx, Q τ j } Cτ 1/C jn. This property is usually called the sall boundaries condition. Fro 9, it follows that there is a point z Q Q the center of Q such that distz Q, suppµ \ Q j see [S, Lea 3.5 of Part I]. We denote := j Z j. For Q j, we define the side length of Q as lq = j. Notice that lq diaq lq. Actually it ay happen that a µ-cube Q belongs to j k with j k. In this case, lq is not well defined. However, this proble can be solved in any ways. For exaple, the reader ay think that a µ-cube is not only a subset of suppµ, but a couple Q, j, where Q is a subset of suppµ and j Z is such that Q j. Given a > 1 and Q, we set aq := { x suppµ : distx, Q a 1lQ }. Observe that diaaq diaq + a 1lQ a 1lQ..4. Corona decoposition. Given an n-diensional A regular Radon easure µ on R d, let := {Q j : j Z} be the dyadic lattice associated to µ introduced in Subsection.3. Following [S, efinitions 3.13 and 3.19 of Part I], one says that µ adits a corona decoposition if, for each η > 0 and θ > 0, one can find a triple B, G, Trs, where B and G are two subsets of the bad µ-cubes and the good µ-cubes and Trs is a faily of subsets S G that we will call trees, which satisfy the following conditions::

7 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 7 a = B G and B G =. b B satisfies a Carleson packing condition, i.e., Q B: Q R µq µr for all R. c G = S Trs S, i.e., any Q G belongs to only one S Trs. d Each S Trs is coherent. This eans that each S Trs has a unique axial eleent Q S which contains all other eleents of S as subsets, that Q S as soon as Q satisfies Q Q Q S for soe Q S, and that if Q S then either all of the children of Q lie in S or none of the do if Q j, the children of Q is defined as the collection of µ-cubes Q j+1 such that Q Q. e The axial µ-cubes Q S, for S Trs, satisfy a Carleson packing condition. That is, S Trs: Q S R µq S µr for all R. f For each S Trs, there exists an n-diensional Lipschitz graph Γ S with constant saller than η such that distx, Γ S θ diaq whenever x Q and Q S one can replace x Q by x C cor Q for any constant C cor given in advance, by [S, Lea 3.31 of Part I]. It is shown in [S1] see also [S] that if µ is uniforly rectifiable then it adits a corona decoposition for all paraeters k > and η, θ > 0. Conversely, the existence of a corona decoposition for a single set of paraeters k > and η, θ > 0 iplies that µ is uniforly rectifiable..5. The α and β coefficients. Let µ be an n-diensional A regular Radon easure in R d and as in Subsection.3. Given 1 p < and a µ-cube Q, one sets see [S] { 1 disty, L p } 1/p β p,µ Q = inf L lq n dµy, lq Q where the infiu is taken over all n-planes L in R d. For p = one replaces the L p nor by the supreu nor. The β,µ coefficients were first introduced by P. Jones in his celebrated work on rectifiability [Jn], while the β p,µ s for 1 p < were introduced by G. avid and S. Sees in their pioneering work on unifor rectifiability see [S1] for exaple. Other coefficients that have been proved useful in the study of unifor rectifiability and boundedness of Calderón-Zygund operators are the α coefficients introduced in [To4]. Let F R d be the closure of an open set. Given two finite Radon easures σ, ν on R d, one sets dist F σ, ν := sup { f dσ f dν } : Lipf 1, suppf F. Finally, given a µ-cube Q, consider the closed ball B Q := Bz Q, 6 dlq, where z Q denotes the center of Q. Then one defines 10 α µ Q := 1 lq n+1 inf dist B Q µ, chl, n c 0,L where the infiu is taken over all constants c 0 and all n-planes L in R d. The following result characterizes the unifor rectifiability of µ in ters of the α and β coefficients see [S1] for a b and [To4] for a c. Theore.8. Let p [1, ] and let µ be an n-diensional A regular Radon easure in R d. The following are equivalent: a µ is uniforly n-rectifiable. b Q : Q R β p,µq lq n lr n for all µ-cubes R. c Q : Q R α µq lq n lr n for all µ-cubes R. For the case µ = H n Γ for soe Lipschitz graph Γ = {x Rd : x = y, Ay, y R n }, one can take = { Q R d n Γ : Q R n }, where R n denotes the standard dyadic

8 8 A. MAS AN X. TOLSA lattice of R n. For Q = Q R d n Γ, we set 11 α µ Q := 1 l Q n+1 inf c 0,L dist 6 Q R d n µ, ch n L, where the infiu is taken over all constants c 0 and all n-planes L in R d. Then, it is easy to show that α µ Q α µ Q for all Q. One can also define β p,µ Q in an analogous anner. By Theore.8, 1 β p,µ Q + α µ Q lq n ClR n Q : Q R for all R, with C independent of R. Moreover, one can also show that this last inequality also holds replacing Q and R by k 1 Q and k R for any k 1, k 1 given in advance, where kq := k Q R d n Γ for k > If Γ is an n-diensional Lipschitz graph, then V ρ T ϕ : MR d L 1, HΓ n is a bounded operator The following result is contained in [MT, Theore 1.1] see also [M, Main Theore 3.0.1]. Theore 3.1. Let ρ > and let µ be the n-diensional Hausdorff easure restricted to an n-diensional Lipschitz graph. Then, the operator V ρ T µ ϕ is bounded in L µ. The bound of the nor only depends on n, d, K, ρ, ϕ R, and the slope of the graph. By very siilar techniques to the ones used in the proof of the theore above, one can prove the following. Theore 3.. Let ρ > and let µ be the n-diensional Hausdorff easure restricted to an n-diensional Lipschitz graph. Then, the operator V ρ T ϕ µ is bounded in L µ. The bound of the nor only depends on n, d, K, ρ, ϕ R, and the slope of the graph. Sketch of the proof. The first step consists in obtaining the following basic estiate: Fix a cube P R n. Set Γ := {x R d : x = y, Ay, y R n }, where A : R n R d n is a Lipschitz function supported in P, and set P := P R d n Γ. Set µ := fhγ n, where fx = 1 for all x Γ \ P and C0 1 fx C 0 for all x P, for soe constant C 0 > 0. For each x Γ, define 13 W µx := Z and 14 Sµx := sup {ɛ } j Z Kϕ µx K ϕ µx. Z: ɛ, I j Kϕ ɛ ɛ+1 µx, where I j = [ j 1, j and the supreu is taken over all decreasing sequences of positive nubers {ɛ } Z. Then, we clai that 15 W µ L µ + Sµ L µ Q αµ C 1 Q + β,µ Q lq n, where C 1 > 0 only depends on C 0, n, d, K, ϕ R, and LipA, and where denotes the dyadic lattice associated to HΓ n defined below Theore.8.

9 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 9 Let us prove the clai. If we define Sµ like Sµ but replacing ϕɛ ɛ +1 by ϕ ɛ ɛ +1, in the proof of Theore 3.1 in [MT] it is shown that Sµ L µ is bounded above by the right hand side of 15. The proof for Sµ L µ is alost the sae. Let us deal now with W µ. Fix := R d n Γ with l = and x. Let L be an n-plane that iniizes α µ C 1 in 11, where C 1 > 0 is soe constant big enough which will be fixed later, and let σ := c HL n be a iniizing easure for α µ C 1. Let L x be the n-plane parallel to L which contains x, and set σ x := c HL n x. Since x and l =, ϕ x ϕ x Kx is a function supported in C 1 R d n for soe constant C 1 big enough and with Lipschitz constant saller than C n+1. Moreover, by the antisyetry of the function ϕ x ϕ x Kx, and since σ x is a ultiple of the n-diensional Hausdorff easure on an n-plane which contains x, we have Kϕ σ x x K ϕ σx x = 0. Therefore, 16 Kϕ µx K ϕ µx = Kϕ ϕ µx = Kϕ ϕ µ σ x + Kϕ ϕ σ σ x x. Using the definition of α µ, we get 17 Kϕ ϕ µ σ x n+1 dist C1 R d nµ, σ α µ C 1. Since L x is a translation of L, by standard estiates it is not hard to show that 18 Kϕ ϕ σ σ x x distx, L = distx, L /l. Let dist H E, F denote the Hausdorff distance of two given sets E, F R d, and set B := 6 R d n. If L 1 and L denote a iniizing n-plane for β 1,µ and β,µ, respectively, one can show that dist H L B, L 1 B α µ l and that dist H L 1 B, L B β,µ l. This easily iplies that distx, L distx, L + β,µ l + α µ l for all x. Applying this to 18, and using also 17 and 16, we obtain W µ L µ = Z = Z Z Kϕ ϕ µx dµx : l= : l= αµ C 1 + β,µ l n, Kϕ ϕ µx dµx distx, L /l + β,µ + α µ C 1 dµx which proves 15. Let now µ be as in Theore 3.. Using 15 and Theore 3.1, one can show that there exists C > 0 such that, for any cube R n and any g L µ supported in where := R d n, Vρ T ϕ µ g dµ C g L µ µ. This yields the endpoint estiates V ρ T µ ϕ : H 1 µ L 1 µ and V ρ T µ ϕ : L µ BMOµ, where H 1 µ denotes the atoic Hardy space related to µ. Then, by interpolation, one finally

10 10 A. MAS AN X. TOLSA deduces that V ρ T ϕ µ is bounded in L µ. Since this part of the proof is analogous to the one in the proof of Theore 3.1 see [MT, Theore 1.1], we oit it Proof of Theore.5. The proof of Theore.5 uses the Calderón-Zygund decoposition of Lea.7 and rather standard arguents. Set µ := HΓ B n, where is soe fixed ball B R d. Let ν MR d be a finite Radon easure with copact support and λ > d+1 ν / µ. We will show that 19 µ { x R d : V ρ T ϕ νx > λ } C λ ν, where C > 0 depends on n, d, K, ρ and Γ, but not on B. Let us check that this iplies that V ρ T ϕ is bounded fro MR d into L 1, HΓ n. First, we show that 19 also holds for ν without copact support. Set ν N = χ B0,N ν and let N 0 be such that suppµ B0, N 0. Then it is not hard to show that, for x suppµ, V ρ T ϕ νx V ρ T ϕ ν N x C ν Rd \ B0, N N N 0, thus V ρ T ϕ ν N x V ρ T ϕ νx for all x suppµ, and since the estiate 19 holds by assuption for ν N, letting N, we deduce that it also holds for ν. Now, by increasing the size of the ball B and by onotone convergence, we deduce that HΓ{ n x R d : V ρ T ϕ νx > λ } Cλ 1 ν, as desired. To prove 19 for ν MR d with copact support, let {Q j } j be the alost disjoint faily of cubes of Lea.7, and set Ω := j Q j and R j := 6Q j. Then we can write ν = gµ + ν b, with gµ = χ R d \Ων + j b j µ and ν b = j ν j b := j w j ν b j µ, where the functions b j satisfy 6, 7, 8 and w j = χ Qj k χ 1. Q k By the subadditivity of V ρ T ϕ, we have µ { x R d : V ρ T ϕ νx > λ } 0 µ { x R d : V ρ T µ ϕ gx > λ/ } + µ { x R d : V ρ T ϕ ν b x > λ/ }. Since V ρ T Hn Γ ϕ is bounded in L HΓ n by Theore 3., it is easy to show that V ρ T ϕ µ is bounded in L µ, with a bound independent of B. Notice that g Cλ by 5 and 8. Then, using 7, µ { x R d : V ρ T ϕ µ gx > λ/ } 1 λ V ρ T ϕ µ g dµ 1 λ g dµ 1 1 λ 1 λ g dµ 1 λ ν R d \ Ω + j ν R d \ Ω + j ν Q j ν λ. b j dµ R j Let Ω := j Q j. By 3, we have µ Ω j µq j λ 1 j ν Q j λ 1 ν. We are going to show now that µ { x R d \ Ω : V ρ T ϕ ν b x > λ/ } C λ ν,

11 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 11 and then 19 is a direct consequence of 0, 1, and the estiate µ Ω λ 1 ν. Since V ρ T ϕ is sublinear, µ { x R d \ Ω : V ρ T ϕ ν b x > λ/ } 1 V ρ T ϕ ν j λ b dµ j R d \ Ω 3 1 V ρ T ϕ ν j λ b dµ + 1 V ρ T ϕ ν j R d \R j λ b dµ. R j \Q j j We are going to estiate the two ters on the right of 3 separately. Let us start with the first one. Given j and x suppµ \ R j, let {ɛ } Z be a decreasing sequence of positive nubers which depends on j and x, i.e. ɛ ɛ j, x such that j 4 V ρ T ϕ ν j b x Z Kϕ ɛ ν j b x ρ 1/ρ. If we set I k := [ k 1, k, we can decopose Z = S L, where L := { Z : ɛ I k, I i, for i > k}, S := k Z S k, S k := { Z : ɛ, I k }. Let z j denote the center of Q j and of R j. Then, since ν j b R j = 0 and suppν j b R j, Kϕ ɛ ν j b x = ϕ ɛ x ykx y dν j b y 5 ϕ ɛ x ykx y ϕ ɛ x z j Kx z j d ν j b y. If L, it is easy to see that ϕ ɛ Kt ϕ ɛ+1 Kt + ϕ ɛ Kt t n 1 for all t R d \ {0}. Moreover, since x R d \ R j and suppν j b R j, there are finitely any L which depends only on n and d such that Kϕ ɛ ν j b x 0, and this nuber only depends on n and d. On the other hand, if S k, it is not hard to show that ϕ ɛ Kt k ɛ t n 1. Actually, this follows fro the fact that ϕ ɛ Kt 0 only if t k and the estiates ϕɛ ɛ +1 t = t t ϕ R ϕ R ϕ R ɛ t L R t ɛ 6 = ϕ R t ɛ k ɛ ɛ and 7 t i ϕ ɛ ɛ+1 t t 1 t 1 ϕ R ϕ R ɛ ɛ t 1 ϕ R 1 t ɛ ɛ + ϕ R ɛ ϕ R + ϕ R t t ϕ 1 R ɛ ɛ k ɛ t 1, where 1 i d and t i denotes the i th coordinate of t R d recall that ɛ k for S k and we assued t k. Siilarly to the case L, there are finitely any

12 1 A. MAS AN X. TOLSA k Z such that suppϕ k x R k 1 j, and the nuber only depends on n and d notice that suppϕ ɛ x suppϕ k x for all S k 1 k. Fro these estiates and rearks, and 4, 5, we obtain V ρ T ϕ ν j b x Kϕ ɛ ɛ+1 ν j b x + Kϕ ɛ ν j b x k Z S k L k ɛ x z j n 1 lr j ν j b S k k Z: suppϕ k k 1 x R j + L: suppϕ ɛ x R j x z j n 1 lr j ν j b x z j n 1 lr j ν j b for all j and x suppµ \ R j. Therefore, using that µ has n-diensional growth, that ν j b ν Q j, and that the Q j s are seidisjoint, V ρ T ϕ ν j b dµ lr j ν j b x z j n 1 dµ 8 ν j b ν. R d \R j j R d \R j j j Let us now estiate the second ter on the right hand side of 3. As above, given j and x R j \ Q j, let {ɛ } Z be a decreasing sequence of positive nubers such that 1/ρ V ρ T ϕ w j νx Kϕ ɛ w j νx ρ, Z where w j = χ Qj k χ Q k 1. Since ρ >, Vρ T ϕ is sublinear, and since ν j b = w jν b j µ, for x R j \ Q j we have V ρ T ϕ ν j b x V ρ T ϕ w j νx + V ρ T b j µx Kϕ ɛ w j νx + V ρ T ϕ µ b j x Z ν Q j x z j n + V ρ T µ ϕ b j x. Since V ρ T ϕ µ is bounded in L µ, using the estiate above and Cauchy-Schwarz we get V ρ T ϕ ν j b dµ ν Q j R j \Q j j R j \Q j x z j n dµx + V ρ T ϕ µ b j dµ j R j \Q j j j ν Q j µr j lq j n + j V ρ T µ ϕ b j L µµr j 1/ j ν Q j + j b j L µµr j j ν Q j ν. Together with 8 and 3, this proves, and Theore.5 follows. 4. If µ is a uniforly n-rectifiable easure, then V ρ T µ ϕ : L p µ L p µ is a bounded operator for 1 < p < The purpose of this section consists in proving the following theore and the subsequent corollary.

13 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 13 Theore 4.1. Let µ be an n-diensional A regular Radon easure in R d and let ρ >. Assue that there exist constants C 0 and C 1 such that, for each ball B centered on suppµ, there is a set F = F B such that: a µf B C 0 µb, b V ρ T ϕ is bounded fro MR d to L 1, HF n with constant bounded by C 1. Then V ρ T ϕ is bounded fro MR d to L 1, µ, and V ρ T ϕ µ is a bounded operator in L p µ for all 1 < p <. Corollary 4.. If µ is an n-diensional A regular uniforly n-rectifiable easure, then V ρ T ϕ µ is a bounded operator in L p µ for all 1 < p < and ρ >. Moreover, the operator V ρ T ϕ is bounded fro MR d to L 1, µ, so V ρ T ϕ µ is also of weak type 1, 1. Proof. Recall fro [S, efinition 1.6] that a Radon easure ν in R d has BP LG big pieces of Lipschitz graphs if ν is n-diensional A regular and if there exist constants C 1 > 0 and θ > 0 such that, for any x suppν and 0 < r < diasuppν, there is a rotation and translation of an n-diensional Lipschitz graph Γ with constant less than C 1 such that νγ Bx, r θr n. Thus, if ν has BP LG, the assuption a of Theore 4.1 is satisfied for ν by taking F = Γ, while Theore.5 iplies that the assuption b holds with a unifor constant. Therefore, fro Theore 4.1 we deduce that, if ν has BP LG and ρ >, then V ρ T ϕ is bounded fro MR d to L 1, ν. Siilarly, a easure ν has BP LG big pieces of big pieces of Lipschitz graphs if there exist constants C g, θ, and 0 < α 1 so that, if B is any ball centered on suppν, then there is an n-diensional A regular set F R d with constant bounded by C g such that νf B ανb and such that HF n has BP LG with unifor constants. So V ρ T ϕ is a bounded operator fro MR d to L 1, HF n, by the coents above. Hence, we can apply once again Theore 4.1 to ν now b is satisfied for the big pieces F of ν, and we deduce that, for any easure ν which has BP LG, V ρ T ϕ is bounded fro MR d to L 1, ν. Siilar arguents yield that V ρ Tϕ ν is a bounded operator in L p ν for all 1 < p <. Finally, fro [S, page ] and the reark given in [S, page 16], we know that if µ is n-diensional A regular, then being uniforly n-rectifiable is equivalent to having BP LG. Therefore, the corollary is proved by applying the coents above to ν = µ. Since the arguents for proving Theore 4.1 are ore or less standard in Calderón- Zygund theory, for the sake of shortness we will only sketch its proof see [To5, Chapter ] or [S, Proposition 1.8 of Part I] for a siilar arguent. Sketch of the proof of Theore 4.1. The proof follows by the so-called good λ inequality ethod. Fix ρ > and let M µ denote the Hardy-Littlewood axial operator M µ ν Bx, r νx := sup r>0 µbx, r, for ν MRd and x suppµ. The good λ inequality: one shows that there exists soe absolute constant η > 0 such that for all ɛ > 0 there exists δ := δɛ > 0 such that µ { x R d : V ρ T ϕ νx > 1 + ɛλ, M µ νx δλ } 9 1 ηµ { x R d : V ρ T ϕ νx > λ } for all λ > 0 and ν MR d. It is easy to check that this iplies that V ρ T ϕ is bounded fro MR d to L 1, µ, and that V ρ T ϕ µ is bounded in L p µ for all 1 < p <, by standard arguents recall that M µ is bounded in these spaces.

14 14 A. MAS AN X. TOLSA The proof of 9 is quite standard. The interested reader ay look at [M, Theore 5..1] for the detailed proof, or at [To5, Chapter ] for siilar arguents. The only point that we should ention is that, in order to pursue the good λ inequality ethod, one needs the following estiate: let ν MR d, consider a ball B R d and take x, z B. Then, 30 V ρ T ϕ χ R d \Bνx V ρ T ϕ χ R d \Bνz M µ νx. We finish the sketch of the proof of Theore 4.1 by showing 30. Since x, z B and V ρ T ϕ is sublinear and positive, by the ean value theore, 31 V ρ T ϕ χ R d \Bνx V ρ T ϕ χ R d \Bνz 1/ρ sup Kϕ ɛ χ R d \Bνx Kϕ ɛ χ R d \Bνz ρ ɛ Z sup ɛ Z ρ 1/ρ ϕ ɛ Ku x,z y y x z d ν y, B x,z where B x, z := R d \ B suppϕ ɛ x suppϕ ɛ z and u x,z y is soe point lying on the segent joining x and z. For each x and z, let ɛ ɛ x, z be a sequence that realizes the supreu in the right hand side of 31. Given ɛ > 0, let jɛ denote the integer such that ɛ [ jɛ 1, jɛ. For j Z set I j := [ j 1, j. As usual, we decopose Z = S L, where S := j Z S j, S j := { Z : ɛ, I j }, L := { Z : ɛ I i, I j for i < j}. Notice that if j+ < rb, where rb denotes the radius of B, then B x.z = for all S j. Therefore, we can assue that j log 4/rB. If S j, then B x, z Bx, j+3, and for t suppϕ ɛ K we have that ϕ ɛ Kt jn+ ɛ see 6 and 7. If L, we easily have ϕ ɛ Kt t n 1. Therefore, using 31, that ρ >, that the sets B x, z have bounded overlap for L, and that x z

15 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 15 rb, we get V ρ T ϕ χ R d \Bνx V ρ T ϕ χ R d \Bνz x z jn+ ɛ j log 4/rB S j + x z x y n 1 d ν y L B x,z rb jn+1 j log 4/rB j log 4/rB + rb k 1 M µ νx + k 1 rb j µbx, j+3 Bx, j+3 k+ rb x y k 1 rb Bx, j+3 Bx, j+3 d ν y + rb R d \B d ν y d ν y x y n+1 k µbx, k+ rb i Bx, k+ rb d ν y d ν y x y n+1 d ν y M µ νx. Reark 4.3. Notice that, to prove 30, it is a key fact that we are considering sooth truncations given by ϕ R in the definition of T ϕ. These coputations are no longer valid if one replaces T ϕ by T. 5. If µ is a uniforly n-rectifiable easure, then V ρ T µ : L µ L µ is a bounded operator This section is devoted to the proof of the following result. Theore 5.1. Let ρ > and let µ be an n-diensional A regular Radon easure on R d. If µ is uniforly n-rectifiable, then V ρ T µ is a bounded operator in L µ Short and long variation. Given j Z, set I j := [ j 1, j. Then, using the triangle inequality, we can split the variation operator into the so-called short variation and long variation operators, i.e., V ρ T µ fx Vρ S T µ fx + Vρ L T µ fx, where 3 Vρ S T µ fx := sup {ɛ } j Z V L ρ T µ fx := sup {ɛ } ɛ, I j Kχ ɛ ɛ+1 fµx ρ 1/ρ, Z: ɛ I j, I k for soe j<k Kχ ɛ fµx ρ 1/ρ, and, in both cases, the pointwise supreu is taken over all the sequences of positive nubers {ɛ } Z decreasing to zero. To prove Theore 5.1 we will show that both the short and long variation operators are bounded in L µ. 5.. L µ boundedness of Vρ L T µ. The L µ-nor of the long variation operator Vρ L T µ can be handled by coparing it with its soothed version V ρ T ϕ µ, using Corollary 4., and estiating the error ters by the short variation operator.

16 16 A. MAS AN X. TOLSA Lea 5.. We have V L ρ T µ f L µ V S ρ T µ f L µ + f L µ. Proof. We decopose V L ρ T µ fx ρ = sup 33 sup {ɛ } Z: ɛ I j, I k for soe j<k sup {ɛ } {ɛ } Z: ɛ I j, I k for soe j<k Z: ɛ I j, I k for soe j<k Kχ ɛ fµx ρ Kχ ɛ ϕ ɛ fµx ρ + Kϕ ɛ fµx ρ Kχ ɛ ϕ ɛ fµx ρ + V ρ T µ ϕ fx ρ. For siplicity, we denote by Vρ L T µ χ ϕ fx ρ the first ter on the right hand side of 33. Notice that, given ɛ, δ > 0, we have χ δ ɛ ϕ δ ɛ = χ ɛ ϕ ɛ χ δ ϕ δ. Recall that, in the definition of ϕ R in efinition.1, we have taken χ [4, ϕ R χ [1/4,. Hence, given t 0, χ R t ϕ R t = χ [1, t 4 1/4 ϕ R sχ [s, t ds = 4 1/4 ϕ R s χ [1, t χ [s, t ds that is, χ R ϕ R is a convex cobination of χ [1, χ [s, for 1/4 s 4, and thus, by Fubini s theore, Kχɛ ϕ ɛ fµ χr x = x y /ɛ ϕ R x y /ɛ Kx yfy dµy = = 4 1/4 4 1/4 ϕ R s χ [1, x y /ɛ χ [s, x y /ɛ Kx yfy dµy ds ϕ R s 4 χ ɛ s ɛ x ykx yfy dµy ds = ϕ R s Kχ ɛ s ɛ fµx ds. 1/4 Therefore, by the triangle inequality and Minkowski s integral inequality, we get Vρ L T µ χ ϕ f 1/ρ L L µ sup Kχ ɛ ϕ ɛ fµx ρ {ɛ I : Z} Z µ 4 ϕ R s 1/ρ L sup Kχ ɛ s ɛ fµx ρ ds. 1/4 {ɛ I : Z} Z µ s One can easily verify that sup {ɛ I: Z} Z Kχɛ ɛ fµx ρ 1/ρ V S ρ T µ fx for all s [1/4, 4] with unifor bounds. Hence 34 V L ρ T µ χ ϕ f L µ 4 1/4 Finally, using 33, 34, and Corollary 4., ϕ R s VS ρ T µ f L µ ds V S ρ T µ f L µ. V L ρ T µ f L µ V L ρ T µ χ ϕ f L µ + V ρ T µ ϕ f L µ V S ρ T µ f L µ + f L µ. Thus, to prove Theore 5.1, it only reains to show the L µ boundedness of V S ρ T µ.

17 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY L µ boundedness of V S ρ T µ. We will see that V S T µ is bounded in L µ, basically due to the big aount of cancellation given by the kernel defining T µ and the good geoetric properties of µ. Since V S ρ T µ V S T µ for ρ, we will be done. One could try the sae technique for V L ρ T µ, however V L T µ is not bounded in L µ in general, even for the case of the Hilbert transfor or in the setting of artingales see [JKRW] for a precise exaple, and this is why we should antain ρ > when we deal with V ρ T µ. Let us ention that to pass fro ρ > to ρ = in the study of the short variation operator is a rather standard arguent see [CJRW1] for exaple. Given f L µ and x suppµ, let {ɛ } Z be a decreasing sequence of positive nubers depending on x such that V S T µ fx j Z ɛ, I j Kχ ɛ ɛ+1 fµx. Given j see Section.3 for the definition of j and x, we set S x := { Z : ɛ, I j }. Since ρ, we have Vρ S T µ f L µ VS T µ f L µ Kχ ɛ ɛ+1 fµx dµx j Z ɛ, I j = Kχ ɛ fµx dµx. S x Let η and θ be two positive nubers that will be fixed below see the proofs of Clais 5.5 and 5.6. Consider a corona decoposition of µ with paraeters η and θ as in Subsection.4. Then, we can decopose = B S Trs S, so that Vρ S T µ f L µ Kχ ɛ fµx dµx B S x 35 + Kχ ɛ fµx dµx. S Trs S S x Since the µ-cubes in B satisfy a Carleson packing condition, we can use Carleson s ebedding theore to estiate the su on the right hand side of 35 over the µ-cubes in B. The Carleson s ebedding theore is a well known result in the area of haronic analysis see [To5, Chapter 5] for exaple, but the ost usual continuous version of this result can be found in [u, Theore 9.5] for exaple. Thus, if we set µ f := µ 1 f dµ for, we have Kχ ɛ fµx dµx 36 B S x B B S x 1 l n Kx y fy dµy dµx x y ɛ f dµ dµ µ 5 f µ f L µ. 5 Now we are going to estiate now the second ter on the right hand side of 35, that is the su over the µ-cubes in S, for all S Trs. To this end, we need to introduce soe notation. B

18 18 A. MAS AN X. TOLSA efinition 5.3. Given R j for soe j Z, let P R denote the µ-cube in j 1 which contains R the parent of R, and set ChR := {Q j+1 : Q R}, V R := {Q j : Q By, lr for soe y R} ChR are the children of R, and V R stands for the vicinity of R. If R S for soe S Trs, we denote by TrR the set of µ-cubes Q S such that Q R the tree of R. Otherwise, i.e., if R B, we set TrR :=. Finally, if TrR, let StpR denote the set of µ-cubes Q B G \ TrR such that Q R and P Q TrR the stopping µ-cubes relative to R, so actually Q R. On the other hand, if R B, we set StpR := {R}. Notice that P R is a µ-cube but ChR and V R are collections of µ-cubes. It is not hard to show that the nuber of µ-cubes in ChR and V R is bounded by soe constant depending only on n and the A regularity constant of µ. Fix S Trs, S, and x. To deal with the second ter on the right hand side of 35, we have to estiate the su S x Kχɛ fµx. By the definition of S x, we have 37 S x Kχ ɛ fµx = S x Kχ ɛ χ fµx, Since this union of µ-cubes is disjoint, we can decopose the where := R V R. function χ f using a Haar basis adapted to in the following anner: 38 χ f = µ R fχ R + Q f + Q f R V Q TrR where we have set Q f := χ U µ U f µ Q f, and Q f := U ChQ U ChQ Q StpR, χ U f µ Q f = χ Qf µ Q f. Using 38, we split the left hand side of 37 as follows: Kχ ɛ fµx Kχ ɛ µ R fχ Rµx 39 S x S x + S x + S x R V R V Q TrR R V Q StpR Kχ ɛ Q fµx Kχ ɛ Q fµx. In the following subsections, we will estiate each part separately. We could think that the leading ter in the right hand side of 39 is the second one, which corresponds to the µ-cubes Q TrR with R V. To control it, we will use that in these µ-cubes the easure µ is very close to a sufficiently flat Lipschitz graph, so good estiates can be achieved using approxiation arguents. To control the third ter in the right hand side of 39, we will basically use that the nuber of cubes Q S with S Trs or which belong to B is not too big see the packing conditions b and e in Subsection.4, so we will be able to apply the Carleson s ebedding theore. The first ter in 39 requires a uch ore detailed study, and we will need to use intensively the ultiscale analysis given by the α µ coefficients apart fro the Carleson s ebedding theore and the above-entioned ideas.

19 VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY Estiate of S x R V S Trs S S x R V Q TrR Q TrR Kχɛ Q fµx fro 39. Lea 5.4. Under the notation above, we have Kχ ɛ Q fµx dµx f L µ. Proof. Let C 0 > 0 be a sall constant to be fixed below. Given S x let A x := Ax,, ɛ = {y R d : y x ɛ }, and given R V let J 1,R := {Q TrR : Q A x, lq > C 0 ɛ }, J,R := {Q TrR : Q A x, lq C 0 ɛ }. Roughly speaking, J 1,R contains the µ-cubes which are big with respect to the thickness of A x, and J,R contains the sall ones. For the study of J 1,R, we will basically use that it does not contain too any µ-cubes. For J,R, using that Q f dµ = 0, we will be reduced to those µ-cubes that intersect the boundary of A x, which are not too any once again. For Q J 1,R, we write Kχ ɛ Q fµx l n χ Ax Q f L 1 µ. The following clai will be proved in Subsection 5.3. below. Clai 5.5. The following estiate holds: Q J 1,R lqn 1/ l n 1/. Using that V has finitely any eleents depending only on n and the A regularity constant of µ, Cauchy-Schwarz inequality, Clai 5.5, and the previous estiate, we obtain Kχ ɛ Q fµx 40 S x R V Q J 1,R R V S x R V S x Q J 1,R Q J 1,R R V S x Q TrR R V Q TrR lq l l n χ Ax Q f L 1 µ lq n 1/ Q J 1,R χ Ax Q f L 1 µ l n+1/ lq n 1/ 1/ Q f L 1 µ l n lq n. χ Ax Q f L 1 µ l n lq n 1/ We deal now with the µ-cubes Q J,R. Let z Q denote the center of Q. Since Q f dµ = 0, we can decopose 41 Kχ ɛ Q fµx = χaxykx y χ Axz Q Kx z Q Q fy dµy = χ Axy Kx y Kx z Q Q fy dµy + χ Axy χ Axz Q Kx z Q Q fy dµy =: T 1,µ Q fx + T,µ Q fx.

20 0 A. MAS AN X. TOLSA For the first ter on the right hand side of the last equality, we have the standard estiate by assuing C 0 sall enough, so any Q J,R is far fro x T 1,µ y z Q Q fx A x x y n+1 Qfy dµy lq l n+1 χ A x Q f L 1 µ. Fro this estiate and Cauchy-Schwarz inequality, we obtain S x R V Q J,R T 1,µ Q fx R V S x R V R V Q TrR Q TrR Q J,R lq l n+1 lq l n+1 χ A x Q f L 1 µ S x lq n+1 l n+1 Q TrR χ Ax Q f L 1 µ Q f L 1 µ lq n 1 l n+1. Since lr = l for all R V, we have Q TrR 1. Thus, using that t t for all t 1, we conclude lq n+1 l lq n+1 Q : Q R lr 4 S x R V Q J,R T 1,µ Q fx R V Q TrR lq 1/ Q f L 1 µ l lq n l n. We deal now with the second ter on the right hand side of 41. Given Q J,R, since supp Q f Q, if Q A x or Q A x c then we obviously have χ Axy χ Axz Q = 0 for all y supp Q f. Therefore, to estiate the su of T,µ Q fx over all Q J,R, we can replace J,R by J 3,R := {Q TrR : Q A x, Q A x c, lq C 0 ɛ }. For S x and Q J 3,R, we will use the estiate T,µ Q fx l n Q f L 1 µ. Clai 5.6. The following holds: Q J 3,R lqn 1/ l n 1 ɛ 1/.

21 S x VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 1 Hence, using that V has finitely any ters, Cauchy-Schwarz inequality, assuing Clai 5.6 see Subsection 5.3., and by the previous estiate, we deduce T,µ Q fx Q f L 1 µ l n R V S x Q J,R R V S x R V S x R V Q TrR R V Q J,R Q J 3,R ɛ l lq 1/ n lq n 1/ l n 1/ R V S x 1/ Q J 3,R l n+1/ Qf L 1 µ Q J 3,R Q J 3,R lq 1/ n l n+1/ Qf L 1 µ lq 1/ n l n+1/ Qf L 1 µ S x: A x Q, lq C 0 ɛ R V Q TrR ɛ ɛ 1/ +1. l The su over on the right hand side of the last inequality can be easily bounded by soe constant depending on C 0, thus we finally obtain T,µ Q fx lq 1/ Q f L 1 µ 43 l lq n l n. Finally, cobining 40, 41, 4, and 43, we conclude Kχ ɛ Q fµx 44 S x R V Q TrR R V Q TrR lq l 1/ Q f L 1 µ lq n l n, Since Q f L 1 µ Q f L µlq n/ by Hölder s inequality, since V has finitely any ters, and since lr = l for all R V, we get Kχ ɛ Q fµx dµx S Trs S S x R V Q TrR S Trs S R V Q TrR S Trs Q S R : R Q V R S Trs Q S Q f L µ Q lq 1/ Q f L l µ lq 1/ Q f L lr µ Q f L µ f L µ. To coplete the proof of Lea 5.4, it only reains to show Clais 5.5 and Proof of Clais 5.5 and 5.6. First of all, we need an auxiliary result whose easy proof is left for the reader. Lea 5.7. Let Γ := {x R d : x = y, Ay, y R n } be the graph of a Lipschitz function A : R n R d n such that LipA is sall enough. Then, H n Γ Ad z, a, b b ab n 1 for all 0 < a b and z Γ.

22 A. MAS AN X. TOLSA Reark 5.8. Actually, to obtain the conclusion of the lea, one only needs LipA < 1 see [M, Lea 4.1.9]. Let us ention that this assuption is sharp in the sense that if LipA 1 then the lea fails. However, we do not need this stronger version for our purposes. Clais 5.5 and 5.6 follow fro the next lea, which will be proved using Lea 5.7. Lea 5.9. Let C 0 > 0 be soe constant depending only on n, d, and the A regularity constant of µ, and consider x j for soe j Z. Let ɛ [ j 1, j. Given k j and R V, set Λ k := {Q TrR k : Q Ax, ɛ C 0 k, ɛ + C 0 k }. Then, µ Q Λ k Q k l n 1 k jn 1. Proof. First of all, we can assue k j otherwise, the clai follows easily using the A regularity of µ, thus we ay assue that distx, Q 3 4 ɛ. For siplicity, set S TrR. By the property f of the corona decoposition of µ, there exists a rotation and translation of an n-diensional Lipschitz graph Γ S with LipΓ S η such that disty, Γ S θ diaq whenever y C cor Q and Q S, for soe given constant C cor. Since x and R V, we have x C cor Q assuing C cor big enough, and so distx, Γ S θ diaq. Hence, if η and θ are sall enough, one can easily odify Γ S inside Bx, 1 4 ɛ to obtain a Lipschitz graph Γ x S such that x Γx S, and oreover 45 LipΓ x S η for soe η sall enough, and Γ x S \ Bx, ɛ/4 = Γ S \ Bx, ɛ/4. Using that distx, Q 3 4 ɛ for all Q Λ k, that distz Q, Γ S θ diaq for the centre z Q of Q, and the last part of 45, we deduce that distz Q, Γ x S θ diaq for all Q Λ k. So Bz Q, θ diaq Γ x S, which in turn yields Hn Γ x S Bz Q, θ diaq θ diaq n. Therefore, since {Bz Q, θ diaq} Q Λk is a faily with finite overlap bounded by soe constant depending only on n, θ, and the A regularity constant of µ, we have µ Q lq n θ n H n Γ x S Bz Q, θ diaq Q Λ k Q Λ k Q Λ k θ n HΓ n x Bz S Q, θ diaq Q Λ k θ n H n Γ x S Ax, ɛ C0 k, ɛ + C 0 k θ n k jn 1, where we used Lea 5.7 and that ɛ j in the last inequality. The lea is proved. Proof of Clai 5.5. Recall that J 1,R := {Q TrR : Q A x, lq C 0 ɛ }, where R V and j. We have to check that Q J 1,R lqn 1/ l n 1/. We will split the su into different scales and we will apply Lea 5.9 at each scale. Given i Z such that i C 0 ɛ, the nuber of µ-cubes Q i such that Q R and Q A x is bounded by ClR n 1 in 1 jn 1+in 1, since for all these µ-cubes, Q Ax, C i, ɛ + C i Ax, ɛ C i+1, ɛ + C i+1 for soe constant C > 0 big enough, and then by Lea 5.9, µ Q J 1,R i Q i l n 1.

23 Therefore, VARIATION FOR THE RIESZ TRANSFORM AN UNIFORM RECTIFIABILITY 3 Q J 1,R lq n 1/ = i Z: i j i/ lq n Q J 1,R i j/ l n 1 = l n 1/. i Z: i j i/ i l n 1 Proof of Clai 5.6. Recall that J 3,R := {Q TrR : Q A x, Q A x c, lq C 0 ɛ }, where R V and j. We have to check that lq n 1/ l n 1 ɛ 1/. Q J 3,R As before, we will split the su into the different scales and we will apply Lea 5.9 at each scale. Given i Z such that i C 0 ɛ, since for any Q J 3,R i we have Q Ax, C i, + C i Ax, ɛ C i, ɛ + C i for soe constant C > 0 big enough, by Lea 5.9 applied to both annuli we have µ Q J 3,R i Q i l n 1. Therefore, Q J 3,R lq n 1/ = i Z: i log C 0 ɛ i Z: i log C 0 ɛ i/ lq n Q J 3,R i i/ l n 1 ɛ 1/ l n Estiate of S x R V Q StpR Kχɛ Q fµx fro 39. Lea Under the notation above, we have Kχ ɛ Q fµx dµx f L µ. S Trs S S x R V Q StpR Proof. Recall the definitions of V, TrR and StpR in efinition 5.3. Given R V, consider a µ-cube Q StpR. If TrR, then Q B G \ TrR, Q R and P Q TrR in particular, Q R. Take S Trs such that R S. By property f of the corona decoposition see Subsection.4, we have disty, Γ S θdiap Q for all y C cor P Q. Hence, disty, Γ S CθdiaQ for all y C cor Q. On the other hand, if TrR = we have set StpR = {R}. In this case, we have R B. Take S such that S. Since R V, we have R C cor if C cor is chosen big enough, and thus disty, Γ S CθdiaR for all y C R, where C is as above and C depends on C cor. Taking into account the coents above, one can prove the following clais using siilar arguents to the ones in the proof of Clais 5.5 and 5.6. Clai Let x, R V, and S x. If we set J 1,R := {Q StpR : Q A x, lq C 0 ɛ }, then Q J 1,R lqn 1/ l n 1/. Clai 5.1. Let x, R V, and S x. If we set J 3,R := {Q StpR : Q A x, Q A x c, lq C 0 ɛ }, then Q J 3,R lqn 1/ l n 1 ɛ 1/.

24 4 A. MAS AN X. TOLSA The only properties of Q f that we used to obtain 44 were that Q f is supported in Q and that Q f dµ = 0. The function Q f is also supported in Q and has vanishing integral. Thus, if we replace TrR by StpR, Clais 5.5 and 5.6 by Clais 5.11 and 5.1, and Q f by Q f, the sae arguents that gave us 44 yield the following estiate: Kχ ɛ Q fµx lq 1/ n 46 l 1/+n Q f L 1 µ. S x R V Q StpR R V Q StpR Below we will use that Q f L 1 µ lq n = Q f µ Q f dµ lq n µ Q f µq. Notice that, by the definition of StpR and since the corona decoposition is coherent property d, any Q StpR is actually a axial µ-cube Q S of soe S Trs or Q B and in this case TrR is epty. Hence, if we integrate 46 in, we su over all S Trs, and we change the order of suation, we get Kχ ɛ Q fµx dµx S Trs S S x R V Q StpR S Trs S R V Q StpR + = R V S Trs: Q S R R V Q B: Q R S Trs R : R Q S V R + Q B R : R Q V R lq 1/ Q f L 1 µ l lq n lqs l 1/ µ Q S f µqs lq 1/ µ Q l f µq lqs 1/ µ Q lr S f µqs lq 1/ µ Q lr f µq. Finally, using that V R has finitely any eleents, and that the µ-cubes Q S with S Trs and the µ-cubes Q B satisfy a Carleson packing condition so we can apply Carleson s ebedding theore, we deduce Kχ ɛ Q fµx dµx S Trs S S Trs S Trs S x R V Q StpR µ Q S f µqs µ Q S f µqs + Q B + Q B µ Q f µq f L µ. lq S 1/ lr 1/ R : R Q S µ Q f µq R : R Q lq 1/ lr 1/ Estiate of S x R V Kχɛ µ R fχ Rµx fro 39. Recall the definitions of V and ChR in efinition 5.3. We will need the following auxiliary lea, which we prove for copleteness, despite we think it is already known.