ALBERT MAS AND XAVIER TOLSA

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1 L p -ESTIMATES FOR THE VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS ALBERT MAS AND XAVIER TOLSA Abstract. The L p 1 < p < ) and weak-l 1 estimates for the variation for Calderón- Zygmund operators with smooth odd kernel on uniformly rectifiable measures are proven. The L boundedness and the corona decomposition method are two key ingredients of the proof. 1. Introduction This article is devoted to obtain L p 1 < p < ) and weak-l 1 estimates for the variation for Calderón-Zygmund operators with smooth odd kernel with respect to uniformly rectifiable measures. As a matter of fact, we prove that if the L estimate holds then the L p and weak-l 1 estimates follow; the results in [17] deal with the L case. Regarding the Calderón-Zygmund operators, given 1 n < d integers, in this article we consider kernels K : R d \ {0} R such that K x) = Kx) for all x 0 K is odd) and 1) Kx) C x n, x i Kx) C x n+1 and xi xj Kx) C x n+ for all x = x 1,..., x d ) R d \{0} and all 1 i, j d, where and C > 0 is some constant. The growth estimate on the second derivatives required in 1) comes from the fact that it is also assumed in [17, Theorem 1.3 and Corollary 4.], which are used in this article see Theorem 3.). We should mention that this growth estimate is usually required in what concerns to L boundedness of singular integral operators and uniformly rectifiable measures, see for example [5, 6, 16, 17, 0]. However, in Theorem 1.4 below we consider more general kernels. Given a Radon measure µ in R d, f L 1 µ) and x R d, we set ) T ɛ µ fx) T ɛ fµ)x) := Kx y)fy) dµy), x y >ɛ and we denote T µ fx) = sup ɛ>0 T ɛ µ fx), T = {T ɛ } ɛ>0 and T µ = {T ɛ µ } ɛ>0. Given ρ > and f L 1 loc µ), the ρ-variation operator acting on T µ f = {T ɛ µ f} ɛ>0 is defined as ) 1/ρ 3) V ρ T µ )fx) := sup T ɛ µ m fx) T ɛ µ m+1 fx) ρ {ɛ m} m Z where the pointwise supremum is taken over all the non-increasing sequences of positive numbers {ɛ m } m Z. 010 Mathematics Subject Classification. Primary 4B0, 4B5. A.M. was supported by the Juan de la Cierva program JCI MEC, Gobierno de España), ERC grant of the European Research Council FP7/ ), MTM and MTM MICINN, Gobierno de España), and IT DEUI, Gobierno Vasco). X.T. was supported by the ERC grant of the European Research Council FP7/ ) and partially supported by MTM , MTM P MICINN, Spain), 014-SGR-75 Catalonia), and by Marie Curie ITN MAnET FP ). 1

2 ALBERT MAS AND XAVIER TOLSA Concerning the notion of uniform rectifiability, recall that a Radon measure µ in R d is called n-rectifiable if there exists a countable family of n-dimensional C 1 submanifolds {M i } i N in R d such that µe \ i N M i) = 0 and µ H n, where H n stands for the n- dimensional Hausdorff measure. Moreover, µ is said to be n-dimensional Ahlfors-David regular, or simply n-ad regular, if there exists some constant C > 0 such that C 1 r n µbx, r)) Cr n for all x suppµ and 0 < r diamsuppµ). Note that if diamsuppµ) < + then µr d ) < and so the condition µbx, r)) Cr n in the definition of AD regularity actually holds for all r > 0. Finally, one says that µ is uniformly n-rectifiable if it is n-ad regular and there exist θ, M > 0 so that, for each x suppµ and 0 < r diamsuppµ), there is a Lipschitz mapping g from the n-dimensional 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, uniform rectifiability implies rectifiability. A set E R d is called n-rectifiable or uniformly n-rectifiable) if H n E is n-rectifiable or uniformly n-rectifiable, respectively). We are ready now to state our main result. In the statement MR d ) stands for the Banach space of finite real Radon measures in R d equipped with the total variation norm. Theorem 1.1. Let µ be a uniformly n-rectifiable measure in R d. Let K be an odd kernel satisfying 1) and, for ρ >, consider the associated variation operator defined in 3). Then V ρ T µ : L p µ) L p µ) 1 < p < ) and V ρ T : MR d ) L 1, µ) are bounded operators. In particular, V ρ T µ : L 1 µ) L 1, µ) is bounded. The variation operator has been studied in different contexts during the last years, being probability, ergodic theory, and harmonic analysis three areas where variational inequalities turned out to be a powerful tool to prove new results or to enhace already known ones see for example [1, 8, 9, 10, 11, 13, 18], and the references therein). Inspired by the results on variational inequalities for Calderón-Zygmund operators in R n like [, 3], in [16] we began our study of such type of inequalities when one replaces the underlying space R n and its associated Lebesgue measure by some reasonable measure in R d, being the Hausdorff measure on a Lipschitz graph a first natural candidate. In this regard, Theorem 1.1 should be considered as a natural generalisation of variational inequalities for Calderón-Zygmund operators in R n from a geometric measure-theoretic point of view. A big motivation to prove Theorem 1.1 is its connection to the so called David-Semmes problem regarding the Riesz transform and rectifiability. Given a Radon measure µ in R d, one defines the n-dimensional Riesz transform of a function f L 1 µ) by R µ fx) = lim ɛ 0 R ɛ µ fx) whenever the limit exists), where R ɛ µ fx) = x y >ɛ x y x y n+1 fy) dµy), x Rd. Note that the kernel of the Riesz transform is the vector x 1,..., x d )/ x n+1 so, in this case, the kernel K in 1) is vectorial). We also use the notation R µ fx) := {R µ ɛ fx)} ɛ>0 and, as usual, we define the maximal operator R µ fx) = sup ɛ>0 R µ ɛ fx). G. David and S. Semmes asked more than twenty years ago the following question, which is still open see, for example, [19, Chapter 7]): Question 1.. Is it true that an n-dimensional AD regular measure µ is uniformly n- rectifiable if and only if R µ is bounded in L µ)?

3 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 3 By [5], the only if implication of this question above is already known to hold. Also in [5], G. David and S. Semmes gave a positive answer to the other implication 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 the if implication was proved in [14] using the notion of curvature of measures. Later on, the same implication was answered affirmatively for n = d 1 in the work [1] by combining quasiorthogonality arguments with some variational estimates which use the maximum principle derived from the fact that the Riesz kernel is a multiple) of the gradient of the fundamental solution of the Laplacian in R d when n = d 1. Question 1. is still open for the general case 1 < n < d 1. However, thanks to Theorem 1.1 and [17, Theorem.3] we get the following corollary, which characterizes uniform rectifiability in terms of variational inequalities for the Riesz transform and more general Calderón-Zygmund operators. Corollary 1.3. Let µ be an n-dimensional AD regular Radon measure in R d. Then, the following are equivalent: a) µ is uniformly n-rectifiable, b) for any odd kernel K as in 1) and any ρ >, V ρ T µ is bounded in L p µ) for all 1 < p <, and from L 1 µ) into L 1, µ), c) for some ρ > 0, V ρ R µ is bounded in L µ). Comparing Corollary 1.3 to Question 1., note that the corollary asserts that if we replace the L µ) boundedness of R µ by the stronger assumption that V ρ R µ is bounded in L µ), then µ must be uniformly rectifiable. On the other hand, the corollary claims that the variation for singular integral operators with any odd kernel satisfying 1), in particular for the n-dimensional Riesz transforms, is bounded in L p µ) for all 1 < p < and it is of weak-type 1, 1), which is a stronger conclusion than the one derived from an affirmative answer to Question 1.. The proof of c) = a) in Corollary 1.3 is not as hard as the converse implications. Essentally, a combination of the arguments in [0] with the fact that, in a sense, V ρ R µ controls R µ does the job see [17]). Theorem 1.1 is used to prove that a) = b) in Corollary 1.3, the corresponding result in [17] was only proved for p =. Theorem 1.1 allows us to get it in full generality, completing the whole picture on variation for singular integrals and uniform rectifiability. As far as we know, neither the L p estimates with 1 < p < nor the weak-l 1 estimate for V ρ T µ on uniform rectifiable measures µ were known, except for the case p = treated in [17] and the case where 1 < p < but suppµ is a Lipschitz graph with slope strictly smaller than 1, solved in [15]. Let us stress that from the latter result one can not easily deduce the L p estimates on uniformly rectifiable measures as in the standard situation in Calderón-Zygmund theory), basically because the good-λ method does not work properly for V ρ T. To avoid this obstacle, our method relies on the corona decomposition technique combined with some ideas from the Lipschitz case in [15] and from [] and [13] to deal with variational inequalities, as well as the L result from [17]. Finally we wish to remark that the same techniques used to prove Theorem 1.1 yield the following result, which applies to more general Calderón-Zymund operators. See Section 5 for the proof. Theorem 1.4. For 1 n < d, let µ be a uniformly n-rectifiable measure in R d. K : R d R d \ {x, y) : x = y} R be a kernel such that Kx, y) C x y n for all x y R d, Let

4 4 ALBERT MAS AND XAVIER TOLSA and Kx, y) Kx, y) + Ky, x) Ky, x ) C x x x y n+1 for all x, x, y R d with x x 1 x y. For ɛ > 0, denote T ɛ µ fx) T ɛ fµ)x) := Kx, y)fy) dµy). x y >ɛ Let T µ f = {T µ ɛ f} ɛ>0 and let V ρ T µ ) be defined as in 3). If V ρ T µ is bounded in L µ), then it is also bounded in L p µ) for 1 < p < and from L 1 µ) to L 1, µ). Also, V ρ T is bounded from MR d ) to L 1, µ). Acknowledgement We are very grateful to the anonymous referee for his/her careful reading of the paper and for his/her comments and suggestions that improved its readability.. Preliminaries and auxiliary results.1. Notation and terminology. As usual, in the paper the letter C or c ) stands for some constant which may change its value at different occurrences, and which quite often only depends on n and d. Given two families of constants At) and Bt), where t stands for all the explicit or implicit parameters involving At) and Bt), the notation At) Bt) At) Bt)) means that there is some 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). Throughout all the paper we assume that 1 n < d are integers and that µ is an n- dimensional AD-regular measure in R d. Given a bounded Borel set A R d and f L 1 loc µ), we write the mean of f on A with respect to µ as follows: m A f := 1 f dµ. µa) We consider the centered maximal Hardy-Littlewood operator: A Mfx) = sup m Bx,r) f. r>0 This is known to be bounded in L p µ), for 1 < p, and from MR d ) to L 1, µ). For 1 q <, we also set M q f := M f q ) 1/q. This is bounded in L p µ), for q < p, and from L q µ) to L q, µ). Given 0 a < b, consider the closed annulus Ax, a, b) := Bx, b) \ Bx, a). Given k Z, set I k := [ k 1, k ). One defines the short and long variation operators Vρ S T µ and Vρ L T µ, respectively, by ) 1/ρ Vρ S T µ )fx) := sup T ɛ µ m fx) T ɛ µ m+1 fx) ρ, {ɛ m} k Z ɛ m,ɛ m+1 I k ) 1/ρ Vρ L T µ )fx) := sup T ɛ µ m fx) T ɛ µ m+1 fx) ρ, {ɛ m} m Z: ɛ m I j, ɛ m+1 I k for some j<k

5 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 5 where, in both cases, the pointwise supremum is taken over all the non-increasing sequences of positive numbers {ɛ m } m Z. Given a finite Borel measure ν in R d, one defines V S ρ T )νx) and V L ρ T )νx) similarly. For convenience of notation, given 0 < ɛ δ we set 4) T δ,ɛ := T δ T ɛ and T ν δ,ɛ analogously. Let ϕ R : [0, + ) [0, + ) be a non-decreasing C function with χ [4, ) ϕ R χ [1/4, ) and set ϕ ɛ x) = ϕ R x /ɛ ). We define 5) T ϕɛ νx) := ϕ ɛ x y)kx y) dνy) for x R d with Kx y) replaced by Kx, y) if K is as in Theorem 1.4). Finally, write T ϕ := {T ϕɛ } ɛ>0. Compare the operator in 5) to T ɛ νx) = χ ɛ x y)kx y) dνy), where χ ɛ ) := χ 1, ) /ɛ), and the family T ϕ to T... Dyadic lattices. For the study of the uniformly rectifiable measures we will use the dyadic cubes built by G. David in [4, Appendix 1] see also [6, Chapter 3 of Part I]). These dyadic cubes are not true cubes, but they play this role with respect to a given n-dimensional AD regular Radon measure µ, in a sense. Let us explain which are the precise results and properties of this lattice of dyadic cubes. Given an n-dimensional AD regular Radon measure µ in R d for simplicity, here we may assume that diamsuppµ) = ), for each j Z there exists a family D µ j of Borel subsets of suppµ the dyadic cubes of the j-th generation) such that: a) each D µ j is a partition of suppµ, i.e. suppµ = Q D µ Q and Q Q = whenever j Q, Q D µ j and Q Q ; b) if Q D µ j and Q D µ k with k j, then either Q Q or Q Q = ; c) for all j Z and Q D µ j, we have j diamq) j and µq) jn ; d) there exists C > 0 such that, for all j Z, Q D µ j, and 0 < τ < 1, µ {x Q : distx, suppµ \ Q) τ j } ) 6) + µ {x suppµ \ Q : distx, Q) τ j } ) Cτ 1/C jn. This property is usually called the small boundaries condition. From 6), it follows that there is a point z Q Q the center of Q) such that distz Q, suppµ \ Q) j see [6, Lemma 3.5 of Part I]). We set D µ := j Z Dµ j. Given a cube Q Dµ j, we say that its side length is j, and we denote it by lq). Notice that diamq) lq). For λ > 1, we also write We denote λq = { x suppµ : distx, Q) λ 1) lq) }. 7) B Q := Bz Q, c 1 lq)), where c 1 1 is some big constant which will be chosen below, depending on other parameters. Let P Q) denote the cube in D µ j 1 which contains Q the parent of Q), and set ChQ) := {Q D µ j+1 : Q Q}, V Q) := {Q D µ j : distq, Q) C 1 lq)}

6 6 ALBERT MAS AND XAVIER TOLSA for some constant C 1 > 0 big enough ChQ) are the children of Q, and V Q) stands for the vicinity of Q). Notice that P Q) is a cube from D µ but ChQ) and V Q) are collections of cubes from D µ. It is not hard to show that the number of cubes in ChQ) and V Q) is bounded by some constant depending only on n and the AD regularity constant of µ, and on C 1 in the case of the vicinitiy. The following assumptions will be used in the sequel: c 1 in 7) is big enough so that and C 1 is big enough so that Finally, we write Q B Q B Q for all Q ChQ) B Q suppµ Q V Q) Q. I Q := I j = [lq)/, lq))..3. The corona decomposition. Given an n-dimensional AD regular Radon measure µ on R d consider the dyadic lattice D µ introduced in Subsection.. Following [6, Definitions 3.13 and 3.19 of Part I], one says that µ admits a corona decomposition if, for each η > 0 and θ > 0, one can find a triple B, G, Trs), where B and G are two subsets of D µ the bad cubes and the good cubes ) and Trs is a family of subsets S G that we will call trees), which satisfy the following conditions:: a) D µ = B G and B G =. b) B satisfies a Carleson packing condition, i.e., Q B: Q R µq) µr) for all R Dµ. c) G = S Trs S, i.e., any Q G belongs to only one S Trs. d) Each S Trs is coherent. This means that each S Trs has a unique maximal element Q S which contains all other elements of S as subsets, that Q S as soon as Q D µ satisfies Q Q Q S for some Q S, and that if Q S then either all of the children of Q lie in S or none of them do recall that if Q D µ j, the children of Q is defined as the collection of cubes Q D µ j+1 such that Q Q). e) The maximal cubes Q S, for S Trs, satisfy a Carleson packing condition. That is, S Trs: Q R µq S) µr) for all R D µ. f) For each S Trs, there exists an n-dimensional Lipschitz graph Γ S with constant smaller than η such that distx, Γ S ) θ diamq) whenever x Q and Q S one can replace x Q by x c Q for any constant c given in advance, by [6, Lemma 3.31 of Part I]). It is shown in [5] see also [6]) that if µ is uniformly rectifiable then it admits a corona decomposition for all parameters k > and η, θ > 0. Conversely, the existence of a corona decomposition for a single set of parameters k > and η, θ > 0 implies that µ is uniformly rectifiable. We set Top G = {Q S : S Trs} and Top = Top G B. If µ is uniformly rectifiable, then, by the properties b) and e) above, for all R D µ we have µq) µr). Q Top: Q R If R S for some S Trs, we denote by TreeR) the set of cubes Q S such that Q R the tree of R). Otherwise, that is, if R B, we set TreeR) := {R}. Finally, StpR) stands for the set of cubes Q B G \ TreeR)) such that Q R and P Q) TreeR)

7 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 7 the stopping cubes relative to R), so actually Q R. Notice that if R B, then we have StpR) = ChR)..4. Auxiliary results. The following lemma follows directly from [1, Lemma.14] see also [15, Lemma.] for the case of Lipschitz graphs). Lemma.1 Calderón-Zygmund decomposition). Let µ be a compactly supported uniformly n-rectifiable measure in R d. For every positive measure ν MR d ) with compact support and every λ > d+1 ν / µ, the following hold: a) There exists a finite or countable collection of cubes {Q j } j centered at supp ν which are almost disjoint, that is j χ Q j C with C depending only on d), and a function f L 1 µ) such that 8) 9) 10) νq j ) > d 1 λµq j ), νηq j ) d 1 λµηq j ) for η >, ν = fµ in R d \ Ω with f λ µ-a.e, where Ω = j Q j. b) For each j, let R j := 6Q j and denote w j := χ Qj k χ Q k ) 1. Then, there exists a family of functions {b j } j with suppb j R j and with constant sign satisfying 11) b j dµ = w j dν, 1) 13) b j L µ)µr j ) CνQ j ), and j b j C 0 λ, where C 0 is some absolute constant. Let us remark that the cubes in the preceding lemma are true cubes, i.e. they do not belong to D µ. Notice that from 9) it follows that 4.5Q j suppµ, which implies that 14) µηq j ) lηq j ) n for η > 5 such that lηq j ) diamsuppµ). Additionally, if we assume that 15) supp ν U diamsuppµ) suppµ), where U t A) stands for the t-neighborhood of A, then we infer that lq j ) Cdiamsuppµ), for all j and for some absolute constant C. Otherwise, for C big enough we would deduce that suppµ suppν Q j, and thus µq j ) = µ and νq j ) ν, so by 8) ν > d 1 λ µ, but this contradicts the choice of λ. In particular, under the assumption 15), we infer that 16) µr j ) lr j ) n lq j ) n. We will need the following version of the dyadic Carleson embedding theorem. Theorem. Dyadic Carleson embedding theorem). Let µ be a Radon measure on R d. Let D be some dyadic lattice from R d and let {a Q } Q D be a family of non-negative numbers. Suppose that for every cube R D we have 17) a Q c 3 µr). Q D:Q R

8 8 ALBERT MAS AND XAVIER TOLSA Then every family of non-negative numbers {γ Q } Q D satisfies 18) γ Q a Q c 3 sup γ Q dµx). Q x Q D Also, for p 1, ), if f L p µ), 19) m Q f p a Q c c 3 f p L p µ), Q D where m Q f = Q f dµ/µq) and c is an absolute constant. In the preceding theorem, the lattice D can be, for example, either the usual dyadic lattice of R d or, in the case when µ is AD-regular, the lattice of cubes associated with µ. For the proof of this classical result, see [1, Theorem 5.8], for example. We say that C D is a Carleson family of cubes if µq) c 3 µr) for all R D. Q C:Q R By 19), it follows that for such a family C and any f L p µ), m Q f p µq) c c 3 f p L p µ). Q C Lemma.3. Let ν MR d ) be a positive measure with compact support and λ > d+1 ν / µ. Consider cubes {Q j } j and {R j } j as in Lemma.1. Denote ν b := j w j ν b j µ), where the b j s satisfy 11), 1) and 13), and w j := χ Qj k χ Q k ) 1. Let C D µ be a family of cubes and {a S } S C be a family of non-negative numbers such that 0) a S c 3 µr). S C:S R For each S C consider the ball B S given by 7), so it is centered on S, S B S and rb S ) ls). Suppose that there exists some constant c > 0 such that for each S C, the ball cb S contains some cube R j. Then, for every p 1, ), ) νb B S ) p 1) ls) n a S λ p 1 ν and ) S C S C ) νbs ) p ls) n a S λ p 1 ν, with the implicit constants depending on p, c 3, and c. In particular, this lemma applies to the case when a S = 1 for all S C and C is a Carleson family satisfying the additional conditions stated in the lemma. Proof. First we will show 1). By 18) in Theorem., one gets ) νb B S ) p ) ν b B S ) p 3) ls) n a S c 3 sup S x ls) n dµx). S C

9 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 9 Write ν b = j w j ν and g = j b j, so that, for every S C, ν b B S ) ν b B S ) + g dµ. B S Note that the measure ν b and the functions b j, g are positive because ν is assumed to be a positive measure. By 3) then we have ) νb B S ) p ) ν b B S ) p p 4) ls) n a S sup S x ls) n dµx) + sup m BS g) dµx), S x S C where m BS g = B S g dµ/µb S ) and we have taken into account that µb S ) ls) n. To deal with the last integral on the right hand side of 4) we use the non-centered maximal Hardy-Littlewood operator defined by sup S x 1 Mfx) = sup B x µb) B f dµ, where the supremum is taken over all the balls which contain x and whose center lies on suppµ. Recalling that M is bounded in L p µ), and using that g L µ) c λ by 13)) and g L 1 µ) c ν by 1)), we obtain p 5) m BS g) dµx) c M g) p dµ c g p dµ cλ p 1 g dµ cλ p 1 ν. Now we turn our attention to the first integral on the right hand side of 4). We write ) ν b B S ) p sup S x ls) n dµx) =... + j Q j R d \... =: I 1 + I. j Q j To estimate I 1, we claim that ν b B S ) λ. ls) n This follows from the fact that cb S contains some cube R j, which in turn implies that, for some η 6 with η ls)/lq j ), B S is contained in some cube ηq j with lηq j ) ls), and then ν b B S ) ls) n νηq j) lηq j ) n, which together with 14) and 9) yields the claim above. Then, using also 8) and the fact the cubes {Q j } j have finite overlap, we deduce that I 1 λ p j µq j ) λ p j νq j ) λ λ p 1 ν. Finally we deal with the integral I. Consider x R d \ j Q j and S such that x S C which, in particular, tells us that S \ j Q j ). Notice that ν b B S ) νq i ). i:q i B S

10 10 ALBERT MAS AND XAVIER TOLSA From the conditions Q i B S and S \ j Q j, we infer that rb S ) 1 lq i). So we deduce that Q i c 4 B S, for some constant c 4 1. Hence, ν b B S ) νq i ) b i dµ, i:q i c 4 B S i:q i c 4 B S where we used 11) for the last estimate. Observe now that if Q i c 4 B S, then R i c 5 B S, for some absolute constant c 5 c 4. So recalling that g = j b j, we obtain ν b B S ) g dµ, c 5 B S Therefore, ν b B S ) ls) n 1 g dµ µb S ) M gx) c 5 B S for every x S. So arguing as in 5) we deduce that I M gx)) p dµx) λ p 1 ν. Together with the estimate we obtained for I 1, this yields ) ν b B S ) p 6) sup S x ls) n dµx) λ p 1 ν, and so using 5) we get 1). In order to show ), recall that ν = ν b + fµ with f as in 10). Thus, νb S ) = ν b B S ) + f dµ ν b B S ) + m BS f ls) n, B S and then ) νbs ) p 7) ls) n a S ) νb B S ) p ls) n a S + m BS f) p a S. S C S C S C We easily get ) from 7), combinig 18) and 19) in Theorem. with 6) and the fact that f p L p µ) λp 1 ν by 10). Let µ be a uniformly n-rectifiable measure in R d. Consider the splitting D µ = B T Trs T ) given by the corona decomposition of µ. For a fixed constant A 1, we denote by T the family of cubes Q T for which either Q = Q T with Q T as in d) in Section.3 or there exists some P D µ \ T such that 1 8) lp ) lq) lp ) and distp, Q) A lq). We call T the boundary of T. If T = TreeR), with R Top G, we also write TreeR) := T. We set Trs := T. T Trs Notice that T T. The following lemma has been proved in [6, 3.8) in page 60]. Lemma.4. Let µ be a uniformly n-rectifiable measure in R d. The family Trs is a Carleson family. We will also need the following auxiliary result.

11 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 11 Lemma.5 Annuli estimates). Assume that the constants η and θ in property f) of the corona decomposition see Section.3) are small enough. Let Q D µ, x Q and ɛ [lq)/, lq)]. Let k Z be such that k lq). Given R V Q) and C > 0, denote Λ k := { } P TreeR) StpR) : lp ) = k, P Ax, ɛ C k, ɛ + C k ). Then ) 9) µ P P k lr) n 1, Λk where the implicit constant in the last inequality above only depends on n, d, µ and C. In the lemma, if ɛ C k < 0 we set Ax, ɛ C k, ɛ + C k ) := Bx, ɛ + C k ). For the proof, see [17, Lemma 5.9]. In fact, in this reference the annuli estimates are proved only for R G. However, for R B, the inequality 9) is trivial. Further, in [17, Lemma 5.9] one states that the result holds only for some constant C depending on n, d, and the AD-regularity constant of µ, and with a slight difference in the definition of V Q). However, it is trivial to check that this extends to the more general version above. 3. V ρ T : MR d ) L 1, µ) is a bounded operator In this section we will prove the following result. Theorem 3.1. Let µ be a uniformly n-rectifiable measure in R d. Let K be an odd kernel satisfying 1) and consider the operator T associated to K defined in ). Then, for ρ >, i) V S ρ T : MR d ) L 1, µ) is bounded, ii) V L ρ T : MR d ) L 1, µ) is bounded. In particular, V ρ T is a bounded operator from MR d ) to L 1, µ) for all ρ >. Notice that by the triangle inequality we can easily split the variation operator into the short and long variations, that is, V ρ T µ )f V S ρ T µ )f + V L ρ T µ )f. Therefore, that V ρ T is a bounded operator from MR d ) to L 1, µ) for all ρ > follows from i) and ii) above, whose proofs are given below. We will use the next result, which is contained in [17, Theorem 1.3 and Corollary 4.]. Theorem 3.. Let µ be a uniformly n-rectifiable measure in R d. Let K be an odd kernel satisfying 1) and consider the operator T associated to K defined in ). Then, for ρ >, i) V ρ T µ : L µ) L µ) is bounded, ii) V ρ T ϕ : MR d ) L 1, µ) is bounded. Proof of Theorem 3.1ii). We will deal with the long variation V L ρ T by comparing it with the smoothened version V ρ T ϕ, using Theorem 3.ii), estimating the error terms by the short variation V S ρ T, and applying Theorem 3.1i). More precisely, the triangle inequality yields T ɛ νx) T δ νx) T ϕɛ νx) T ϕδ νx) + T ɛ νx) T ϕɛ νx) + T δ νx) T ϕδ νx)

12 1 ALBERT MAS AND XAVIER TOLSA for any 0 < δ ɛ. Therefore, V L ρ T )νx) ) ρ Vρ T ϕ )νx) ) ρ 30) + sup {ɛ m} m Z: ɛ m I j, ɛ m+1 I k for some j<k V ρ T ϕ )νx) ) ρ + T ɛm νx) T ϕɛm νx) ρ + T ɛm+1 νx) T ϕɛm+1 νx) ρ) sup {ɛ m}: ɛ m I m for all m Z T ɛm νx) T ϕɛm νx) ρ. m Z Let us estimate the second term on the right hand side of 30). Since χ [4, ) ϕ R χ [1/4, ) by definition, we have χ [1, ) t) ϕ R t) = 4 1/4 ϕ R s)χ [1, ) χ [s, ) )t) ds for all t 0. This means that χ [1, ) ϕ R is a convex combination of the functions χ [1, ) χ [s, ) for 1/4 s 4. Then, Fubini s theorem gives ) T ɛ νx) T ϕɛ νx) = χ [1, ) x y /ɛ ) ϕ R x y /ɛ ) Kx y) dνy) 4 = ϕ R s) χ [ɛ, ) χ 31) [ɛ s, ) ) x y )Kx y) dνy) ds = 1/4 4 1/4 ) ϕ R T s) ɛ νx) T ɛ s νx) ds. It is easy to see that ) 1/ρ 3) T ɛm νx) T ɛm s νx) ρ Vρ S T )νx) m Z for all s [1/4, 4] with uniform bounds, where {ɛ m } m Z is any sequence such that ɛ m I m for all m Z. Using 31), Minkowski s integral inequality and 3), we get ) 1/ρ T ɛm νx) T ϕɛm νx) ρ 33) sup {ɛ m}: ɛ m I m for all m Z m Z 4 sup {ɛ m}: ɛ m I m for all m Z 4 1/4 Finally, applying 33) to 30) yields 1/4 ϕ R s) m Z T ɛm νx) T ɛm s νx) ρ ) 1/ρ ds ϕ R s)vs ρ T )νx) ds V S ρ T )νx). V L ρ T )νx) V ρ T ϕ )νx) + V S ρ T )νx), and Theorem 3.1ii) follows by Theorems 3.ii) and 3.1i). Proof of Theorem 3.1i). We have to prove that there exists some constant C > 0 such that 34) µ { x R d : V S ρ T )νx) > λ }) C λ ν

13 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 13 for all ν MR d ) and all λ > 0. The proof of 34) combines the Calderón-Zygmund decomposition developed in Lemma.1, the corona decomposition of µ described in Subsection.3, and other standard techniques for proving variational inequalities. We will start following the lines of the proof of [15, Theorem 1.4], until the application of the corona decomposition. Since V S ρ T is sublinear, we can assume without loss of generality that ν is a positive measure. Let us first check that we can also assume both µ and ν to be compactly supported. Given ν MR d ) and M N, set ν M := χ B0, M )ν. If diamsuppµ) < + then µ is compactly supported. In case diamsuppµ) = + we are going to restrict µ to a set K N R d such that µ KN it is still uniformly rectifiable with constants independent of N). For this purpose, for each N N consider the family of cubes Pi N D µ N, i I N, thus lpi N ) = N for all i I N ) such that B0, N ) Pi N. We denote K N = i and µ N = µ KN. i I N P N It is immediate to check that µ P N is uniformly rectifiable for each i, N. Since K N is a finite i union of uniformly rectifiable sets because #I N is uniformly bounded), µ N is also uniformly rectifiable, with constants independent of N. Suppose that there exists some constant C > 0 such that µ N { x R d : V S ρ T )ν M x) > λ }) C λ ν M for all λ > 0, all ν MR d ) and all M, N N. This implies that 35) µ { x B0, N ) : V S ρ T )ν M x) > λ }) C λ ν M for all λ > 0, all ν MR d ) and all M, N N. It is not hard to show that V S ρ T )νx) V S ρ T )ν N x) C M N ) n ν R d \ B0, M ) ) for all x B0, N ) and all M > N > 1. In particular, if M then Vρ S T )ν M x) Vρ S T )νx) uniformly in B0, N ). Since 35) holds for ν M by assumption, we deduce that it also holds for ν. Now, by letting N and using monotone convergence, 35) with ν M replaced by ν yields 34), as desired. In conclusion, for proving the theorem, we only have to verify 34) when µ and ν have compact support. Moreover, since 34) obviously holds for λ d+1 ν / µ, we can also restrict ourselves to the case λ > d+1 ν / µ. We are going to verify that we can assume 15), which will allows us to use 16) in the sequel, when we pursue the Calderón-Zygmund decomposition of ν with respect to µ. Let M := diamsuppµ) < + and set ν c := χ R d \U M suppµ)ν. Then distsuppν c, suppµ) M. By Chebyshev s inequality, µ { x R d : Vρ S T )ν c x) > λ }) 1 Vρ S T )ν c x) dµx) λ 36) C x y n dν c y) dµx) C λ M n λ ν c µ. For any x suppµ, µ = µbx, M)) M n by the AD regularity assumption on µ. Thus 36) yields 37) µ { x R d : V S ρ T )ν c x) > λ }) C λ ν c C λ ν,

14 14 ALBERT MAS AND XAVIER TOLSA with C independent of M. Note that ν = ν c +ν ν c ) and suppν ν c ) U diamsuppµ) suppµ). Using that V S ρ T is sublinear and 37) we see that, in order to prove the theorem, it is enough to show that µ { x R d : V S ρ T )ν ν c )x) > λ }) C λ ν, that is, we can assume that ν satisfies 15). In conclusion, for proving 34), from now on we assume that both µ and ν are compactly supported and they satisfy 15). Let {Q j } j be the almost disjoint family of cubes of Lemma.1, 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 b j s satisfy 11), 1) and 13), and w j := χ Qj k χ Q k ) 1. Since 15) holds, in the sequel we can also assume that 16) holds. Since Vρ S T is sublinear, µ { x R d : Vρ S T )νx) > λ }) 38) µ { x R d : Vρ S T µ )gx) > λ/ }) + µ { x R d : Vρ S T )ν b x) > λ/ }). We obviously have Vρ S T µ V ρ T µ, so Theorem 3.i) yields that Vρ S T µ is bounded in L µ). Note that g Cλ by 10) and 13). Hence, using 1), µ { x R d : Vρ S T µ )gx) > λ/ }) 1 λ Vρ S T µ )g dµ 1 λ g dµ 39) 1 λ 1 λ g dµ 1 λ νr d \ Ω) + j νr d \ Ω) + j ) νq j ) ν λ. ) b j dµ R j Set Ω := j Q j. By 8), we have µ Ω) j µq j) λ 1 j νq j) λ 1 ν. We are going to prove that 40) µ { x R d \ Ω : V S ρ T )ν b x) > λ/ }) ν λ. Then 34) follows directly from 38), 39), 40) and the estimate µ Ω) λ 1 ν abovementioned, finishing the proof of Theorem 3.1i). To prove 40), given x R d \ Ω we first write 41) Vρ S T )ν b x) Vρ S T ) χ Rj x)ν j b j ) x) + Vρ S T ) j χ R d \R j x)ν j b ) x). Notice that χ Rj x) and χ R d \R j x) are evaluated at the fixed point x on the right hand side. The first term on the right hand side of 41) is easily handled using the L µ) boundedness of Vρ S T µ and standard estimates. More precisely, since Vρ S T is sublinear, ) Vρ S T ) χ Rj x)ν j b x) 4) j j χ Rj x)v S ρ T µ )b j x) + j χ Rj x)v S ρ T ν )w j x)

15 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 15 because ν j b = w jν b j µ. On one hand, using Theorem 3.i), that µr j ) µr j ) by 16)) and 1), we get 43) ) 1/ Vρ S T µ )b j dµ Vρ S T µ )b j dµ µr j ) 1/ R j R j b j L µ)µr j ) 1/ b j L µ)µr j ) νq j ). On the other hand, if x R j \ Q j then distx, Q j ) lq j ). Therefore, given k Z, 44) Bx, k ) Q j = distx, Q j ) k lq j ) k. Since the l ρ -norm is not bigger than the l 1 -norm for ρ 1, and since suppw j Q j and w j 1, from 44) and 4) we get Vρ S T ν )w j x) sup Tɛ ν m,ɛ m+1 w j x) {ɛ m} k Z ɛ m,ɛ m+1 I k νq j ) kn νq j )lq j ) n, k Z: Bx, k ) Q j and therefore, using again that µr j ) µr j ) lr j ) n lq j ) n by 16), we obtain 45) Vρ S T ν )w j dµ νq j )lq j ) n µr j ) νq j ). R j \Q j Finally, applying 43) and 45) to 4), we conclude that ) Vρ S T ) χ Rj x)ν j b x) dµx) 46) R d \ Ω j j R j V S ρ T µ )b j dµ + j R j \Q j V S ρ T ν )w j dµ j νq j ) ν. Thanks to 41), 46) and Chebyshev s inequality, to prove 40) it is enough to verify that { ) }) 47) µ x R d \ Ω : Vρ S T ) χ R d \R j x)ν j b x) > λ/4 ν λ. j Our task now is to prove 47). Given x suppµ, let {ɛ m } m Z be a non-increasing sequence of positive numbers which depends on x, i.e. ɛ m ɛ m x)) such that 48) Vρ S T ) j χ R d \R j x)ν j b ) x) k Z ɛ m,ɛ m+1 I k ρ) 1/ρ χ R d \R j x)t ɛm,ɛm+1 ν j b x). Typically, the problem of the existence of such a sequence can be avoided by defining an auxiliary operator Vρ,I S T along the same lines of VS ρ T and requiring the supremum to be taken over a finite set of indices I thus the supremum is a maximum in this case). One then proves the desired estimate for Vρ,I S T with bounds independent of I and deduces the general result by taking the supremum over all finite sets I and using monotone convergence. For the sake of shortness, we omit the details. j

16 16 ALBERT MAS AND XAVIER TOLSA Define the interior and boundary sum, respectively, by ρ) 1/ρ S i x) := χ R d \R j x)t ɛm,ɛm+1 ν j b x), S b x) := k Z ɛ m,ɛ m+1 I k k Z ɛ m,ɛ m+1 I k j: R j Ax,ɛ m+1,ɛ m) j: R j Ax,ɛ m+1,ɛ m) χ R d \R j x)t ɛm,ɛ m+1 ν j b x) ρ) 1/ρ. If R j Ax, ɛ m+1, ɛ m ) = then T ɛm,ɛm+1 ν j b x) = 0, thus ) Vρ S T ) χ R d \R j x)ν j b x) S i + S b ) by 48) and the triangle inequality, and so { ) }) µ x R d \ Ω : Vρ S T ) χ R d \R j x)ν j b x) > λ/4 49) j j µ { x R d \ Ω : S i x) > λ/16 }) + µ { x R d \ Ω : S b x) > λ/16 }). To estimate µ { x R d \ Ω : S i x) > λ/16 }) we use the fact that the l ρ -norm is not bigger than the l 1 -norm for ρ 1, and that suppν j b ) R j: S i x) χ R d \R j x)t ɛm,ɛm+1 ν j b x) m Z j: R j Ax,ɛ m+1,ɛ 50) m) χ R d \R j x) T ɛm,ɛm+1 ν j b x) χ R d \R j x) T ν j b x), j m Z: Ax,ɛ m+1,ɛ m) R j j Recall that ν j b R j) = 0 and ν j b νq j) by 1). Thus, if z j denotes the center of R j, we have T ν j b dµ Kx y) Kx z j ) d ν j b y) dµx) R d \R j R d \R j R j y z j 51) R d \R j R j x z j n+1 d νj b y) dµx) ν j b lr j ) R d \R j x z j n+1 dµx) νj b νq j). Finally, from Chebyshev s inequality, 50) and 51) we conclude that µ { x R d \ Ω : S i x) > λ/16 }) 16 T ν j λ b dµ 1 5) νq j ) ν R d \R j λ λ. By 49), 5) and Chebyshev s inequality once again we see that, in order to prove 47), it is enough to show that 53) dµ λ ν. Sb R d \ Ω The proof of this estimate is much more involved than the previous ones and requires the use of the corona decomposition of µ, that is, we need to introduce the splitting D µ = j j

17 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 17 B S Trs S). We denote Tj,mx) := χrd\rj x)tɛm,ɛm+1 νj b x). Recall that for P D k we write I P = [ k 1, k ). Since ρ >, the l ρ -norm is not bigger than the l -norm, and we get Sb dµ T j,m x) dµx) R d \ Ω P B P \ Ω ɛ m,ɛ m+1 I P j: R j Ax,ɛ m+1,ɛ m) 54) + T j,m x) dµx). Observe that S Trs P S P \ Ω ɛ m,ɛ m+1 I P j: R j Ax,ɛ m+1,ɛ m) 55) T j,m x) lp ) n χ R d \R j x) ν j b Ax, ɛ m+1, ɛ m )) for all ɛ m, ɛ m+1 I P. If in addition x P \ R j and R j Ax, ɛ m+1, ɛ m ), taking into account that ɛ m ɛ m+1 lp ) distx, R j ) lr j ), we deduce that 56) R j B P, assuming the constant c 1 in 7) big enough. Concerning the first term on the right hand side of 54), from 55) and using that ν j b νq j ), that the Q j s have bounded overlap and that Q j B P for all j such that R j B P, we get T j,m x) dµx) 57) P B P \ Ω ɛ m,ɛ m+1 I P P B P B P P j: R j Ax,ɛ m+1,ɛ m) ɛ m,ɛ m+1 I P j: R j B P ν ) lp ) n ν j b Ax, ɛ m+1, ɛ m )) dµx) j: R j B P j b ) dµx) ) νbp ) lp ) n λ ν, lp ) n lp ) n P B: R j B P where we also used Lemma.3 in the last inequality, because B is a Carleson family. From now on, all our efforts are devoted to estimate the second term on the right hand side of 54). Claim 3.3. Assume c 1 in 7) is big enough, and let also α > 0 be big enough depending on n, d, and on the AD regualrity constants of µ. Given Q Top G, P TreeQ) and R j B P, at least one of the following holds: i) There exists R TreeQ) such that R αb P, R j B R and lr j ) I R. ii) There exists R TreeQ) such that R αb P and R j B R. We postpone the proof of the preceding statement till the end of the proof of the theorem. Thanks to this claim, given Q Top G and P TreeQ) we can split {j : R j B P } J 1 J, where J 1 : = {j : R j B P, R TreeQ) such that R αb P, R j B R, lr j ) I R }, J : = {j : R j B P, R TreeQ) such that R αb P, R j B R }.

18 18 ALBERT MAS AND XAVIER TOLSA Recall that if x P \ R j, ɛ m, ɛ m+1 I P and R j Ax, ɛ m+1, ɛ m ) then R j B P see 56)). Thus, we can decompose the second term on the right hand side of 54) using J 1 and J as follows 58) S Trs P S P \ Ω ɛ m,ɛ m+1 I P Q Top G P TreeQ) + j: R j Ax,ɛ m+1,ɛ m) P \ Ω ɛ m,ɛ m+1 I P Q Top G P TreeQ) P \ Ω ɛ m,ɛ m+1 I P T j,m x) dµx) j J 1 : R j Ax,ɛ m+1,ɛ m) j J : R j Ax,ɛ m+1,ɛ m) T j,m x) T j,m x) dµx) dµx). Despite that the arguments to estimate both terms on the right hand side of 58) are similar, we will deal with them separately, due to its different nature with respect to the structure of the corona decomposition. Claim 3.4. Let Q, P, x, ɛ m and ɛ m+1 be as on the right hand side of 58). We have 59) j J 1 : R j Ax,ɛ m+1,ɛ m) λlp ) n k: k lp ) ν j b Ax, ɛ m+1, ɛ m )) ) k 1/ ν j lp ) b Ax, ɛ m+1, ɛ m )). j J 1 : lr j ) I k Given j J, denote by Rj) TreeQ) some cube such that Rj) αb P and R j B Rj), where α > 0 is as in Claim 3.3. We have 60) j J : R j Ax,ɛ m+1,ɛ m) λ 1/ lp ) n/ νb P ) 1/ ν j b Ax, ɛ m+1, ɛ m )) R TreeQ): R αb P j J : Rj)=R ) lr) 1/4 ν j lp ) b B R Ax, ɛ m+1, ɛ m )). Again we postpone the proof of the preceding claim till the end of the proof of the theorem.

19 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 19 For the case j J 1 in 58), using 55), 59) and 56) we get 61) Q Top G P TreeQ) λ λ λ j P \ Ω ɛ m,ɛ m+1 I P Q Top G P TreeQ) lp ) n P \ Ω ɛ m,ɛ m+1 I P k: k lp ) Q Top G P TreeQ) k: k lp ) νq j ) k: lr j ) I k j J 1 : R j Ax,ɛ m+1,ɛ m) T j,m x) dµx) ) k 1/ ν j lp ) b Ax, ɛ m+1, ɛ m )) dµx) j J 1 : lr j ) I k ) k 1/ ν j lp ) b j J 1 : lr j ) I k ) k 1/ λ νq j ) λ ν. lp ) P D µ : R j B P k lp ) j In the third inequality we used that j J 1 implies that R j B P. Concerning the case j J in 58), by 55) and 60) we see that Q Top G P TreeQ) λ 1/ λ 1/ λ 1/ P \ Ω ɛ m,ɛ m+1 I P Q Top G P TreeQ) j J : R j Ax,ɛ m+1,ɛ m) ) 1/ lp ) n νbp ) lp ) n P \ Ω ɛ m,ɛ m+1 I P R TreeQ): j J : R αb P Rj)=R Q Top G P TreeQ) νbp ) lp ) n ) 1/ Q Top G P TreeQ) R TreeQ): R αb P R j B R T j,m x) dµx) ) lr) 1/4 ν j lp ) b B R Ax, ɛ m+1, ɛ m )) dµx) R TreeQ): j J : R αb P Rj)=R lr) lp ) ) lr) 1/4 ν j lp ) b ) 1/4 νbp ) lp ) n ) 1/ ) νbr ) lr) n lr) n,

20 0 ALBERT MAS AND XAVIER TOLSA where we also used in the last inequality above that ν j b νq j) and that the Q j s have bounded overlap. Since a 1/ b a 3/ + b 3/ for all a, b 0, we obtain T j,m x) dµx) Q Top G P TreeQ) λ 1/ λ 1/ P \ Ω ɛ m,ɛ m+1 I P Q Top G P TreeQ) Q Top G P TreeQ) R j cb P j J : R j Ax,ɛ m+1,ɛ m) R TreeQ): R αb P R j B R νbp ) lp ) n ) νbp ) 3/ lp ) n + ) 3/ a P + λ 1/ Q Top G νbr ) lr) n R TreeQ) R j B R ) 3/ ) ) lr) 1/4 lr) n lp ) ) νbr ) 3/ lr) n lr) n, where we have set a P := R TreeQ): R αb P lr)/lp )) 1/4 lr) n whenever P TreeQ) for some Q Top G otherwise, we set a P = 0). Since Trs is a Carleson family, we see that the a P s satisfy a Carleson packing condition because, for a given T D µ, a P ) lr) 1/4 lr) n lp ) P T P T Q Top G : P TreeQ) R TreeQ): R αb P ) lr) 1/4 lr) n lp ) Therefore, 6) P T R Trs: R αb P αb T R Trs: R αb T lr) n Q Top G P TreeQ) λ 1/ P D µ P T : R αb P P \ Ω ɛ m,ɛ m+1 I P νbp ) lp ) n ) lr) 1/4 lp ) j J : R j Ax,ɛ m+1,ɛ m) R Trs: R αb T lr) n lt ) n. T j,m x) ) 3/ ap + lp ) n χ TrsP ) ) λ ν, dµx) because the coefficients a P + lp ) n χ Trs P ) satisfy a Carleson packing condition and thus we can use Lemma.3. Finally, 53) follows from 54), 57), 58), 61) and 6), so Theorem 3.1i) is proved except for the claims. Proof of Claim 3.3. Let Q Top G, P TreeQ) and R j B P. For the purpose of the claim, we can assume that lq) lr j ), otherwise we can take R = Q which fulfills ii). Without loss of generality, we can also assume that lp ) lr j ) recall that R j B P, so lp ) lr j )). Otherwise, we replace P by a suitable ancestor from TreeQ) with side length comparable to lr j ), which must exists thanks to the previous assumption lq) lr j ). Let R TreeQ) be a cube with minimal side length such that R j B R and lr) lr j ), that is, lr) ls) for all S TreeQ) with R j B S and ls) lr j ). In particular, notice that P may coincide with R, and in any case lr) lp ). If lr j ) I R, that is lr) lr j ) lr)/, then R fulfills i) if α is big enough, and we are done. On the contrary, assume that lr)/ > lr j ). Since R j B R and R j suppµ, there exists R D µ such that lr ) = lr), distr, R) lr) and R R j. Therefore, there exists

21 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 1 a son R of R such that R R j, so R j B R if c 1 is big enough. By the minimality of R, we must have R / TreeQ), thus R TreeQ) if A 1 in 8) is big enough, and then ii) is fulfilled for some α big enough. Proof of Claim 3.4. Let us first prove 59). If j J 1 then R j B P and, in particular, lr j ) lp ). Thus, by Cauchy-Schwarz inequality, 63) j J 1 : R j Ax,ɛ m+1,ɛ m) = k: k lp ) k: k lp ) ν j b Ax, ɛ m+1, ɛ m )) ) k 1/4 ) lp ) 1/4 lp ) k ν j b Ax, ɛ m+1, ɛ m )) j J 1 : lr j ) I k R j Ax,ɛ m+1,ɛ m) ) lp ) 1/ k ν j b Ax, ɛ m+1, ɛ m )). j J 1 : lr j ) I k R j Ax,ɛ m+1,ɛ m) Using that ν j b Ax, ɛ m+1, ɛ m )) νq j ) and that the Q j s have bounded overlap, from the definition of J 1 we see that ν j b Ax, ɛ m+1, ɛ m )) νb R ). 64) j J 1 : lr j ) I k R j Ax,ɛ m+1,ɛ m) R TreeQ): lr) I k, B R Ax,ɛ m+1,ɛ m), R αb P, R j B R If 6Q j = R j B R then ν6q j ) νb R ) λµb R ) λµr) by 9). From 64) we infer 65) j J 1 : lr j ) I k R j Ax,ɛ m+1,ɛ m) ν j b Ax, ɛ m+1, ɛ m )) λ R TreeQ): lr) I k, B R Ax,ɛ m+1,ɛ m), R αb P, R j B R µr). We want to show that the right hand side of 65) can be estimated by λ k lp ) n 1. To this end, we can suppose that lr) lp ), otherwise the estimate becomes trivial because we are already assuming k lp ) and lr) I k so in this last case there is only a finite and uniformly bounded number of terms in the sum above). Suppose now that lr) lp ). Since R αb P then R P V P ) P if the constant C 1 in the definition of V P ) is big enough. Thus, R P for some P V P ). Note that P TreeQ) because R TreeQ), and so we finally get R TreeP ). Then, from 65) and the estimates on annuli from Lemma.5 we obtain ν j b Ax, ɛ m+1, ɛ m )) λ µr) 66) j J 1 : lr j ) I k R j Ax,ɛ m+1,ɛ m) P V P ) R TreeP ): lr) I k, B R Ax,ɛ m+1,ɛ m) λ k lp ) n 1, as desired. Finally, 59) follows from 63) and 66). Let us turn our attention to 60) now. Recall that, given j J, Rj) TreeQ) denotes some cube such that Rj) αb P and R j B Rj). Similarly to 63), by Hölder s inequality

22 ALBERT MAS AND XAVIER TOLSA we get 67) j J : R j Ax,ɛ m+1,ɛ m) ν j b Ax, ɛ m+1, ɛ m )) R TreeQ): R αb P j J : Rj)=R B R Ax,ɛ m+1,ɛ m) k: k lp ) lp ) k ) 1/4 3/ ν j b B R Ax, ɛ m+1, ɛ m )) R TreeQ): j J : Rj)=R R αb P, lr)= k B R Ax,ɛ m+1,ɛ m) 3/ ν j b B R Ax, ɛ m+1, ɛ m )) 3/. For the cubes R = Rj) in the last sum above, note that R j B R see the definition of J ). So, as we did before 65), νb R ) λµb R ) λµr) by 9). Using that ν j b νq j), that the Q j s have bounded overlap and that νb R ) λµb R ), we deduce that 68) R TreeQ): R αb P, lr)= k j J : Rj)=R B R Ax,ɛ m+1,ɛ m) R TreeQ): j J : Rj)=R R αb P, lr)= k B R Ax,ɛ m+1,ɛ m) λ ν j b B R Ax, ɛ m+1, ɛ m )) R TreeQ): R αb P, lr)= k B R Ax, ɛ m+1,ɛ m) νq j ) µr). R TreeQ): R αb P, lr)= k B R Ax,ɛ m+1,ɛ m) νb R ) As we did in the case of J 1, now we want to show that the last term above can be estimated by λ k lp ) n 1. We argue similarly to what we did before 66). If R is as in the right hand side of the last inequality in 68), since R αb P we have lr) lp ), and thus we can assume lr) lp ) otherwise the estimate that we want to show becomes trivial). Since R αb P then R P V P ) P if the constant C 1 in the definition of V P ) is big enough. Thus, R P for some P V P ) and R TreeP ) recall that R TreeQ) implies R TreeQ)). Then, from 68) and the estimates on annuli from Lemma.5 we obtain 69) ν j b B R Ax, ɛ m+1, ɛ m )) R TreeQ): R αb P, lr)= k j J : Rj)=R B R Ax,ɛ m+1,ɛ m) λ P V P ) R TreeP ): lr) I k, B R Ax,ɛ m+1,ɛ m) µr) λ k lp ) n 1, as desired.

23 70) VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 3 Combining 69) with 67) we get ν j b Ax, ɛ m+1, ɛ m )) j J : R j Ax,ɛ m+1,ɛ m) λ 1/ lp ) n/ λ 1/ lp ) n/ k: k lp ) 3/ ) k 1/4 lp ) R TreeQ): R αb P R TreeQ): R αb P, lr)= k B R Ax,ɛ m+1,ɛ m) j J : Rj)=R lr) lp ) j J : Rj)=R Finally, 60) is a consequence of 70) and the trivial estimate ν j b Ax, ɛ m+1, ɛ m )) νb P ), j J : R j Ax,ɛ m+1,ɛ m) ν j b B R Ax, ɛ m+1, ɛ m )) ) 1/4 ν j b B R Ax, ɛ m+1, ɛ m )). which holds if c 1 in 7) is big enough because ν j b νq j) and the Q j s have bounded overlap. 4. V ρ T µ : L p µ) L p µ) is a bounded operator for 1 < p < Under the assumptions of Theorem 1.1, the boundedness of V ρ T µ in L p µ) for 1 < p < follows by interpolation, taking into account that it is bounded in L µ) and from L 1 µ) to L 1, µ), by Theorem 3. and Theorem 3.1. So it only remains to prove the boundedness in L p µ) for < p <. This task is carried out in the next theorem. Theorem 4.1. Let µ be a uniformly n-rectifiable measure in R d. Let K be an odd kernel satisfying 1) and consider the operator T associated to K defined in ). Then V ρ T µ is a bounded operator in L p µ) for all ρ > and all < p <. Proof. We are going to prove that if µ is a uniformly n-rectifiable measure then M D V µ ρ T µ is a bounded operator in L p µ) for all < p <, where M D denotes the dyadic sharp µ maximal function, that is, M D µ fx) = sup m D f m D f. D D µ : x D The theorem will then follow from the fact that the maximal operator defined by M D µfx) = sup D D µ : x D m D f can be controlled in L p µ) norm by M D µ. That is, M D µf L p µ) M D µ f L p µ) see [7, Lemma 6.9], for example). Fix f L p µ) and x 0 suppµ. Then, 71) M D V µ ρ T µ )fx 0 ) = sup m D V ρ T µ )f m D V ρ T µ )f). D D µ : x 0 D Given D D µ such that x 0 D, we decompose f = f 1 +f with f 1 := fχ 3D and f := f f 1. Since V ρ T µ is sublinear and positive, V ρ T µ )f V ρ T µ )f V ρ T µ )f 1 and so V ρ T µ )f c V ρ T µ )f 1 + V ρ T µ )f c for all c R. If we take c = V ρ T µ )f z D ),

24 4 ALBERT MAS AND XAVIER TOLSA where z D denotes the center of D we may assume that c < ), then 7) m D V ρ T µ )f m D V ρ T µ )f) m D V ρ T µ )f V ρ T µ )f z D ) m D V ρ T µ )f 1 + m D V ρ T µ )f V ρ T µ )f z D ) =: I 1 + I. A good estimate for I 1 can be easily derived using Cauchy-Schwarz s inequality, Theorem 3.i) and that µ is n-ad regular. More precisely, 1 1/ 1 1/ 73) I 1 V ρ T µ )f 1 dµ) f dµ) M fx 0 ). µd) µd) D The estimate of I is much more involved. Given x D, by the triangle inequality we have 74) V ρ T µ )f x) V ρ T µ )f z D ) sup T ɛ µ m,ɛ m+1 f x) T µ ) ɛ m,ɛ m+1 f z D ) ρ 1/ρ, {ɛ m} m Z m Z where the supremum is taken over all non-increasing sequences {ɛ m } m Z of positive numbers ɛ m. In order to estimate the right hand side of 74), take one of such sequences {ɛ m } m Z and note that, by the triangle inequality again, T ɛ µ m,ɛ m+1 f x) T ɛ µ m,ɛ m+1 f z D ) χ ɛm+1,ɛ m] x y ) Kx y) Kz D y) f y) dµy) 75) + χ ɛm+1,ɛ m] x y ) χ ɛm+1,ɛ m] z D y ) Kz D y) f y) dµy) =: a m + b m. Since x and z D belong to D and f vanishes in 3D, we can assume that ɛ m+1 > ld) in the definition of a m and b m for all m Z. Let us first look at the sum relative to the a m s for m Z. Using that ρ > 1, the regularity of the kernel K, that f vanishes in 3D, and that µ is n-ad regular, for each x D we have ) 1/ρ a ρ m Kx y) Kz D y) f y) dµy) ɛ m+1 < x y ɛ m 76) m Z m Z ld) m Z ld) R d \3D ɛ m+1 < x y ɛ m 3D f y) dµy) y z D n+1 fy) y z D n+1 dµy) Mfx 0) M fx 0 ), where we also used Cauchy-Schwarz s inequality in the last estimate above. The sum relative to the b m s for m Z requires a more delicate analysis. Z = J 1 J, where J 1 := {m Z : ɛ m ɛ m+1 > ld)}, J := {m Z : ɛ m ɛ m+1 ld)}. We split

25 VARIATION FOR SINGULAR INTEGRALS ON UNIFORMLY RECTIFIABLE SETS 5 To shorten notation, we also set A 1 mz D ) := Az D, ɛ m ld), ɛ m + ld)) and A mx) := Ax, ɛ m+1, ɛ m ). Since we are assuming ɛ m+1 > ld) for all m Z, both A 1 mz D ) and A 1 m+1 z D) are well defined for all m J 1. Moreover, since x z D ld) for all x D, we easily get 77) χ ɛm+1,ɛ m] x ) χ ɛm+1,ɛ m] z D ) χ A 1 m z D ) + χ A 1 m+1 z D ) for all m J 1, χɛm+1,ɛ m] x ) χ ɛm+1,ɛ m] z D ) χa m z D ) + χ A m x) for all m J. We are going to split the sum associated with the b m s in terms of J 1 and J, using in each case the corresponding estimate from 77). Concerning the sum over J 1, since ρ >, 77) yields 78) m J 1 b ρ m ) 1/ρ + m J 1 m J 1 =: S 1 + S. ) ) 1/ Kz D y) f y) dµy) A 1 mz D ) ) ) 1/ Kz D y) f y) dµy) A 1 m+1 z D) The arguments for estimating S 1 and S are almost the same, so we will only give the details for S 1. Since f vanishes in 3D, 79) S 1 = k Z m J 1 : ɛ m I k Kz D y) f y) dµy) A 1 mz D ) ) Q D µ : Q D m J 1 : ɛ m I Q f µ) A 1 mz D ) ) lq) n. Our task now is to bound f µ) A 1 mz D ) ). This is done by splitting the annulus A 1 mz D ), whose width equals ld), into disjoint cubes P D µ such that lp ) = ld) and grouping them properly in terms of the corona decomposition, in order to be able to apply Carleson s embedding theorem later. More precisely, for Q D and ɛ m I Q, we have A 1 mz D ) suppµ) R V Q) R R V Q) P TreeR): lp )=ld) P ) R V Q) P StpR): lp ) ld) Recall also that the number of cubes in V Q) is bounded independently of Q. Therefore, 80) f µ) A 1 mz D ) ) R V Q) + R V Q) P TreeR): lp )=ld) P StpR): lp ) ld) f µ) A 1 mz D ) P ) f µ) A 1 mz D ) P ). P ).