POSITIVE SPARSE DOMINATION OF VARIATIONAL CARLESON OPERATORS

Transcription

1 POSITIVE SPARSE DOMINATION OF VARIATIONAL CARLESON OPERATORS FRANCESCO DI PLINIO, YEN Q. DO, AND GENNADY N. URALTSEV Abstract. Due to its nonlocal nature, the r-variation norm Carleson operator C r does not yield to the sparse domination techniques of Lacey [15, 16], Lerner [19, 17] and Di Plinio and Lerner [6]. We overcome this difficulty and prove that the dual form toc r can be dominated by a positive sparse form involving L p averages. Our result strengthens the L p -estimates by Oberlin et. al. [20]. As a corollary, we obtain quantitative weighted norm inequalities improving on [8] by Do and Lacey. Our proof is based on the intrinsic domination technique of Culiuc, Di Plinio and Ou [4], and relies on the localized outer L p -embeddings of Di Plinio and Ou [7] and Uraltsev [21]. 1. Introduction and main results Set in motion by Lerner with his seminal alternative proof [19] of Hytönen s sharp A 2 - weight theorem [14], the technique of controlling the Calderón-Zygmund singular integrals by local averaging operators over sparse grids, referred to as positive sparse operators, has recently emerged as a leading trend in Euclidean Harmonic Analysis. We briefly review the advancements which are most relevant for the present article and postpone further references to the body of the introduction. The original domination in norm result of [19] for Calderón- Zygmund operators has since been upgraded to a pointwise positive sparse domination by Conde and Rey [2] and Lerner and Nazarov [18], and later by Lacey [15] by means of an inspiring stopping time argument forgoing local mean oscillation. Lacey s approach was further clarified in [17], resulting in the following principle which we attribute to Lacey and Lerner: if T is a sublinear operator of weak-type (p,p and in addition the maximal operator (1.1 f sup T (f 1 R\3Q 1 L (Q Q Q R interval embodying the nonlocality of T, is of weak-type (s, s, for some 1 p s <, then T is pointwise dominated by a positive sparse operator involving L s averages of f. The Lacey-Lerner principle extends to certain modulated singular integrals: most recently, oscillatory integrals with polynomial phase [16]. Of interest for us is the maximal partial Fourier transform N C f (x = sup f (ξ e ixξ dξ N also known as Carleson s operator on the real line. The crux of the matter is that (1.1 follows for T = C from its representation as a maximally modulated Hilbert transform, a fact already exploited in the classical weighted norm inequalities for C by Hunt and Young [13], and in 2010 Mathematics Subject Classification. Primary: 42B20. Secondary: 42B25. Key words and phrases. Positive sparse operators, variational Fourier series, weighted norm inequalities. Y. Do was partially supported by the National Science Foundation under the grant DMS F. Di Plinio was partially supported by the National Science Foundation under the grant NSF-DMS

2 2 FRANCESCO DI PLINIO, YEN Q. DO, AND GENNADY N. URALTSEV the more recent work [12]. Together with sharp forms of the Carleson-Hunt theorem near the endpoint p = 1 [5] this allows, as observed by the first author and Lerner in [6], the domination of C by sparse operators and thus leads to sharp weighted norm inequalities for C. In this note we consider the r-variation norm Carleson operator, which is defined for Schwartz functions on the real line as N C r f (x = sup sup N N ξ 0 < <ξ N j=1 ξj r f (ξ e ixξ dξ ξ j 1 The importance of C r is revealed by the transference principle, presented in [20, Appendix B], which shows how r-variational convergence of the Fourier series of f L p (T;w for a weight w on the torus T follows from L p (R;w-estimates for the sublinear operator C r. T Values of interest for r are 2 < r <. Indeed the main result of [20] is that in this range, C r maps into L p whenever p > r, while no L p -estimates hold for variation exponents r 2. Unlike the maximal case, C r does not have an explicit kernel form and thus yield to Hunt-Young type techniques. The same essential difficulty is encountered in the search for L q -bounds for the nonlocal maximal function (1.1 when T = C r. Therefore, the Lacey-Lerner approach does not seem to be applicable to C r. In the series [9, 8], the second author and Lacey circumvented this issue through a direct proof of A p -weighted inequalities for C r and its Walsh analogue, based on weighted phase plane analysis. The main result of the present article is that a sparse domination principle for C r holds in spite of the difficulties described above. More precisely, we sharply dominate the dual form to the r-variational Carleson operator C r by a single positive sparse form involving L p averages, leading to an effortless strenghtening of the weighted theory of [8]. Our argument abandons the Lacey-Lerner principle in favor of a stopping time construction, relying on the localized Carleson embeddings for suitably modified wave packet transforms of [7] by the first author and Yumeng Ou, and [21] by the third author. In particular, our technique requires no a-priori weak-type information on the operator T : for this reason, we refer to it as the intrinsic sparse domination argument. The intrinsic approach was first employed by Culiuc, Ou and the first author in [4] in the proof of a sparse domination principle for the family of modulation invariant multilinear multipliers whose paradigm is the bilinear Hilbert transforms. We believe that intrinsic sparse domination can prove useful in the study of other classes of multilinear operators lying way beyond the scope of Calderón-Zygmund theory, such as the iterated Fourier integrals of [10] and the subdyadic multipliers of [1]. To formulate our main theorem, we recall the notation ( 1 1 f I,p := f p p dx, 1 p < I where I R is any interval, which simplifies to f I when p = 1, and the notion of a sparse collection of intervals. We say that the countable collection of intervals I S is η-sparse for some 0 < η 1 if there exist measurable sets {E I : I S} such that E I I, E I η I, I, J S, I J = E I E J =. 1/r.

3 SPARSE DOMINATION OF VARIATIONAL CARLESON OPERATORS 3 Theorem 1. Let 2 < r < and p > r. Given f,д C0 (R there exists a sparse collection S = S(f,д,p and an absolute constant K = K (p such that (1.2 C r f,д K (p I f I,p д I. I S Fixing q (r, ], choose p (r,q. Denoting by M p f (x = sup f I,p x I the p-th Hardy-Littlewood maximal function, and omitting the subscript p when p = 1, the estimate of Theorem 1 and the sparsity of S yields C r f,д E I f I,p д I M p f, Mд M p f q Mд q f q д q. I S In other words, a corollary of Theorem 1 is that C r extends to a bounded sublinear operator on L q (R whenever q > r. This was first proved in [20], where it is also shown that the restriction q > r is necessary for L q -boundedness, whence no sparse domination of the type occurring in Theorem 1 will hold for p < r. We can thus claim that Theorem 1 is sharp up to the endpoint p = r. In fact, sparse domination as in (2.5 also entails C r : L p (R L p, (R and it is not known whether such estimate holds for p = r. However, Theorem 1 yields much more precise information than mere L q -boundedness. In particular, we obtain precisely quantified weighted norm inequalities for C r as an immediate corollary. Recall the definition of the A t constant of a locally integrable nonnegative function w as 1 t 1 sup w [w] At := I w t 1 1 < t < I I R inf{a : Mw (x Aw (x a.e.x} t = 1 Theorem 2. Let 2 < r < and q > r be fixed. Then (i there exists K : [1, q r (0, nondecreasing such that C r L q (R;w L q (R;w K (t[w] max{ 1, t q(t 1 } A t ; (ii there exists a positive increasing function Q such that for t = q r (1.3 C r L q (R;w L q (R;w Q ( [w] At. We omit the standard deduction of Theorem 2 from Theorem 1, which follows along lines analogous to the proofs of [4, Theorem 3] and [18, Theorem 17.1]. Estimate (i of Theorem 2 yields in particular that w A t = C r L q (R;w L q (R;w < r > max { 2, } q q t an improvement over [8, Theorem 1.2], where L q (R;w boundedness is only shown for variation exponents r > max { 2t, qt } q t when w At. Fixing r instead, part (ii of Theorem 2 is sharp in the sense that t = q r is the largest exponent such that an estimate of the type of (1.3 is allowed to hold. Indeed, if (1.3 were true for any q = q 0 (r, and some t = q 0 s with s < r, a version of the Rubio de Francia extrapolation theorem (see for instance [3, Theorem 3.9]

4 4 FRANCESCO DI PLINIO, YEN Q. DO, AND GENNADY N. URALTSEV would yield that C r maps L q into itself for all q (s,, contradicting the already mentioned counterexample from [20]. We turn to further comments on the proof and on the structure of the paper. In the upcoming Section 2 we reduce the bilinear form estimate (2.5 to an analogous statement for a bilinear form involving integrals over the upper-three space of symmetry parameters for the Carleson operator of a wave packet transforms of f and a variational-truncated wave packet transform of д. The natural framework for L p -boundedness of such forms, the L p -theory of outer measures, has been developed by the second author and Thiele in [11]. In Section 3, we recall the basics of this theory as well as the localized Carleson embeddings of [7] and [21]. These will come to fruition in Section 4, where we give the proof of Theorem 1. Acknowledgments. This work was initiated during G. Uraltsev s visit to the Brown University Mathematics Department, whose hospitality is gratefully acknowledged. The authors would like to thank Amalia Culiuc, Michael Lacey, Ben Krause and Yumeng Ou for useful conversations about sparse domination principles. 2. Reduction to wave packet transforms In this section we reduce the inequality (2.5 to an analogous statement involving wave packet trasforms. Throughout this section, the variation exponent r (2, is fixed, and we take f,д C0 (R. Since C r is positive there is no loss in generality in the arguments below if we assume д 0. The first step is a suitable linearization of the variation norm in C r. Let S(f, x, ξ = ξ f (η e ixη dη. By the the uniform continuity of the map (x, ξ S(f, x, ξ and standard considerations there holds N ( C r f (x = sup sup sup a j S(f, x, ξj S(f, x, ξ j 1. N Ξ R,#Ξ N {a j } l r 1 Therefore, (2.5 will be a consequence of the estimate (2.1 Λ ξ, a (f,д := R j=1 j=1 N a j (x ( S(f, x, ξ j S(f, x, ξ j 1 д(x dx K (p I f I,p д I with right hand side independent of N N, Ξ R, #Ξ N, and of the measurable Ξ N +1 - valued function ξ (x = {ξ j (x} with ξ 0 (x < < ξ N (x, and C N +1 -valued a(x = {a j (x} with a(x l r = 1. The next step is to dominate uniformly Λ ξ, a (f,д by an outer form involving wave packet transforms of f and д; in the terminology of [11], embedding maps into the upper 3-space (u, t, η X = R (0, R. The parameters ξ, a will enter the definition of the embedding map for д. We introduce the wave packets ( x ψ t,η (x := t 1 e iηz ψ, η R, t (0, t I S

5 SPARSE DOMINATION OF VARIATIONAL CARLESON OPERATORS 5 where ψ is a real valued, even Schwartz function with frequency support of width b containing the origin. The wave packet transform of f is thus defined by (2.2 F (f (u, t, η = f ψ t,η (u, (u, t, η X. For our fixed choice of ξ, a we introduce the modified wave packet transform of д by (2.3 A(д(u, t, η := sup Ψ R д(x N j=1 a j (xψ ξ j (x,ξ j+1 (x u,η,t (x dx, (u, t, η X, with supremum being taken over all choices of truncated wave packets Ψ c,c + η,y,t. We send to [21] for a precise definition. Then, it is a consequence of [21, Eq. (1.11] that (2.4 Λ ξ, a (f,д B ξ, a (f,д := F (f (u, t, ηa(д(u, t, η dudtdη X with uniform implied constant. Summarizing inequality (2.1 and the last display, the estimate (2.5 of Theorem 1 will follow from the next proposition. Proposition 2.1. Let p > r be fixed. For all f,д C0 (R there exists a sparse collection S = S(f,д,p and an absolute constant K = K (p such that (2.5 sup sup sup B ξ, a (f,д K (p I f I,p д I N #Ξ N ξ, a I S where ξ, a range over Ξ N +1, C N +1 -valued functions as above. 3. Localized outer-l p embeddings We now turn to the description of the analytic tools which are relied upon in the proof of estimate (2.5. We work in the framework of outer measure spaces [11], see also [4, 7]. In particular, we define a distinguished collection of subsets of the upper 3-space X = R (0, R which we refer to as tents above the time-frequency loci (I, ξ where I is an interval of center c(i and length I, and ξ R: T(I, ξ := T l (I, ξ T o (I, ξ, T o (I, ξ := { (u, t, η : tη tξ Θ i, t < I, u c(i < I t } T l (I, ξ := { (u, t, η : tη tξ Θ \ Θ o, t < I, u c(i < I t } where Θ o = [β o α o, β o + α o ], Θ = [β α, β + α] are two geometric parameter intervals such that ( b,b Θ o Θ and b is the parameter measuring the frequency support of our wave packets. It is convenient to use the notation T(I := { (u, t, η : t < I, u c(i < I t } for the upper-3 associated to the usual spatial tent over I. As usual, we denote by µ the outer measure generated by countable coverings by tents T(I, ξ, I R, ξ R via the premeasure T(I, ξ I.

6 6 FRANCESCO DI PLINIO, YEN Q. DO, AND GENNADY N. URALTSEV If s is a size [11] taking values on Borel functions F : X C and the corresponding outer-l p space on (X, µ is defined by the quasinorm ( 1 F L p (s := p λ p 1 p µ(s(f > λ dλ, 0 < p <, 0 { } µ(s(f > λ := inf µ(x \ E : E X, sup s (F1 E (T(I, ξ λ (I,ξ. We will work with outer L p spaces based on the sizes ( 1 s l (F (T := F (u, t, η 2 dudtdη I T l ( 1 s o (F (T := F (u, t, η 2 dudtdη I T sup F (u, t, η, (u,t,η T I T o F (u, t, η dudtdη where we wrote T for a generic tent T(I, ξ. The fact that the sizes s l, s o are dual, namely that for any two Borel functions F, A : X C F (u, t, ηa(u, t, η dudtdη 2s l (F (Ts o (A(T T yields the outer Hölder inequality [11, Proposition 3.6] (3.1 F (u, t, ηa(u, t, η dudtdη F L σ (s l A L τ (s o T whenever 1 < σ,τ < are Hölder dual exponents. The nature of the wave packet transforms f F (f,д A(д defined by (2.2, (2.3 is heavily exploited in the stopping-type outer L p -embedding theorems. We state the embedding theorems after some necessary definitions. For any interval Q R, we denote by KQ the concentric K-dilate of the interval Q. Let ζ C (R be a nonincreasing function which is equal to 1 on (, 5 2 and equals zero on [ 7 2,. We will use ζq in (x := ζ ( x c(q Q, ζ out Q := 1 ζ in Q as smooth approximations to 1 7Q, 1 R\5Q. Let Q be an interval, f L p (R. The corresponding good set is the subset of the upper 3-space defined as (3.2 G f,p,q := X \ T (3I where the union is being taken over all intervals I contained in the bad set (3.3 E f,p,q := { x R : M p (f ζ in Q (x > C 0 f 7Q,p }, Mp := M p M 1 1 < p <, M 1 p = 1 and C 0 is an absolute constant which we are free to choose later on. It is easy to see that a sufficiently large choice of C 0 ensures that 3 3 I 3 3 Q for all I E f,p,q. The first stopping embedding theorem, dealing with the wave packet transform f F (f of (2.2, is lifted from [7, Theorem 1].

7 SPARSE DOMINATION OF VARIATIONAL CARLESON OPERATORS 7 Proposition 3.1. Let f be a Schwartz function and 1 < p < 2. Then for any σ (p, there exists K = K (p, σ such that (3.4 in F (f ζq 1 G f,p,q L K Q σ (s l σ 1 f 7Q,p. The embedding theorem we use to treat the variationally truncated wave packet transform д A(д of (2.3, is the main result of [21]. Proposition 3.2. Let д be a Schwartz function. For any τ (r, there exists K = K (τ such that (3.5 in A(дζQ 1 G д,1,q L K Q τ (s o τ 1 д 7Q. In particular, the constant K does not depend on the parameters a, ξ, Ξ, N appearing in the definition (2.3 of the map A. 4. Proof of Proposition 2.1 Throughout this proof, the exponent p (r, is fixed and all the implicit constants are allowed to depend on r, p without explicit mention. Since the linearization parameters play no explicit role in the upcoming arguments we omit them from the notation, assume them fixed and simply write B(f,д for the form B ξ, a (f,д defined in (2.4. Given any interval Q, we introduce the localized version B Q (f,д := F (f (u, t, ηa(д(u, t, η dudtdη. T(3Q In our proof, we will make use of the three shifted dyadic grids D j = { 2 k [0, 1 + ( n + j 3 2 k : k, n Z }, j = 0, 1, 2 Unless otherwise mentioned, we call dyadic intervals the elements of the ensemble D = D 0 D 1 D 2. We define the enlargement of an interval I R as the unique interval Ĩ D with 9I Ĩ, Ĩ 6 9I, and c(ĩ is least possible The principal iteration. The main step of the proof of Proposition 2.1 is contained in the following lemma, which we will apply iteratively. Lemma 4.1. Let f,д be Schwartz functions, Q R be any interval. There is a collection of dyadic intervals I {f,д},q D 0 with the properties (4.1 (4.2 (4.3 and such that Ĩ 9Q I I {f,д},q, I, I I {f,д},q, I I = I = I (disjointness, I 2 12 Q (packing, I I {f,д },Q (4.4 B(f ζq in in,дζq Q f Q,p д Q + I I {f,д },Q B(f ζ in Q ζ in I,дζ in Q ζ in I.

8 8 FRANCESCO DI PLINIO, YEN Q. DO, AND GENNADY N. URALTSEV The proof of the lemma consists of several steps, which we now begin. For simplicity, we write I Q in place of I {f,д},q until the end of the subsection Construction of I Q. Referring to the notation (3.3, we set E = E f,p,q E д,1,q. The collection I Q contains those I D 0 which are contained in E and are maximal with this property. Then (4.2 is obvious. Another consequence of the maximality of I is that (4.5 inf x 3I M 1f (x f Q,p, inf M 1д(x x 3I д Q. For further use we note that, if G Q := G f,p,q G д,1,q with reference to the notation (3.2, (4.6 NG Q := X \ G Q T(3I. I I Q It is easy to see that if C 0 is chosen large enough, I I Q I Q which readily yields (4.3. A fortiori, each I must intersect 7Q and be significantly shorter than Q. This guarantees that Ĩ 9Q for all I I Q, which is (4.1. This completes the construction of I Q Proof of estimate (4.4. We begin by using (4.6 to write (4.7 B(f ζq in in,дζq G Q F (f ζ in Q A(дζ in Q dudtdη + I I Q T(3I F (f ζq in in A(дζQ dudtdη Choosing τ (r,p, the dual exponent σ = τ (p,. We may apply Hölder s inequality and the embeddings Propositions 3.1 and 3.2 to control the first summand in (4.7 by an absolute constant times in F (f ζq 1 G f,p,q L in σ (s l A(дζQ 1 G f,p,q L Q f τ (s o 7Q,p д 7Q Q f Q,p д Q. We turn to the second summand in (4.7. which, inserting the cutoffs ζi in may be dominated by B(f ζ in Q ζi in,дζq in ζ I in + B I (f ζ in Q ζ a in I,дζQ ζ I b. I I Q I I Q (a,b {in,out} 2 (a,b(in,in The first term in the above display appears on the right hand side of (4.4, thus we are left with bounding the other three terms for which at least one of a, b is out by the first term on the right hand side of (4.4. Assuming a = in, b = out for the sake of definiteness, the other case being identical. Fix I I Q. We will show that (4.8 B I (f ζ in Q ζ I a in,дζq ζ I I f Q,p д Q Summing the above estimate over all I, which are contained in Q and are pairwise disjoint yields a factor of Q thus leading to the required estimate for (4.4.

9 SPARSE DOMINATION OF VARIATIONAL CARLESON OPERATORS Proof of 4.8. We introduce the Carleson box over the interval P box(p = { (u, t, η X : u P, 1 3 P < t < 4 3 P } Fix I I Q. At the root of our argument for (4.8 is the fact that supp ζ out I lies outside 5I. This leads to the exploitation of the following lemma, whose proof is given at the end of the paragraph. Lemma 4.2. Let P be any interval and h L 1 (R. Then loc ( dist(p, supph 100 (4.9 sup F (h(u, t, η 1 + inf (u,t,η box(p P M 1h(x, x 3P ( dist(p, supph 100 (4.10 A(h(u, t, η dudtdη P 1 + inf P M 1h(x. x 3P box(p Now let P P k (I be the collection of dyadic subintervals of I with P = 2 k I. If P P k (I there holds dist(p, supp ζi out I 2 k P. Moreover P I, inf M 1h(x 2 k inf M 1h(x x 3P x 3J P P k (I for all locally integrable h. Since the union of {box(p : P P k (I, k = 0, 1...} covers T(3I, we obtain B I (f ζq in ζ I in,дζq in ζ out F (f ζ in A(дζ in dudtdη k 0 P P k (I k 0 P P k (I ( I k 0 P P k (I box(p sup F (f ζq in ζ I in (u, t, η (u,t,η box(p P 2 100k k 0 P P k (I ( inf x 3P M 1(f ζ in P Q ζ I in Q ζ I in ( box(p A(дζ in ( (x 2 100k inf M 1(дζ in x 3P ( ( inf 1(f (x x 3P inf 1(д(x x 3P which, by virtue of (4.5, complies with (4.8. Q ζ I out Q ζ I out Q ζ I in I dudtdη (x ( ( inf 1(f (x x 3I inf 1(д(x x 3I Proof of Lemma 4.2. Estimate (4.9 is immediate from the fast decay of ψ u,t,η. For estimate (4.10, let x Ψ c,c + u,t,η be a choice of truncated wave packets which approximately extremizes the supremum in A(h. Then Ψ c,c + u,t,η (x := ( 1 + x c(p 120 P Ψ c,c + u,t,η (x are adapted truncated wave packets as well since multiplying by a polynomial does not change the frequency support. It is easy to see that the left hand side of (4.10 is bounded by the right

10 10 FRANCESCO DI PLINIO, YEN Q. DO, AND GENNADY N. URALTSEV hand side times (4.11 sup x R box(p N j=1 a j (x Ψ ξ j (x,ξ j+1 (x u,t,η dudtdη. For each x R at most one of the N summands in the inner term is nonzero, when η ξ j (x + η ξ j+1 (x t 1, and is P 1. Since the dudtdη-measure of the set {(u, t, η box(p : η ξ j (x + η ξ j+1 (x t 1 } is bounded by P, the factor (4.11 is bounded by an absolute constant, and (4.10 follows. This completes the proof of the lemma The iteration argument. With Lemma 4.1 in hand, we proceed to the construction of the sparse collection S of Proposition 2.1 and to the proof of the associated estimate. Fixing f,д C0 (R, we choose a cube Q 0 D 0 large enough so that the supports of f and д are both contained in 5Q 0. The base case for the iteration is R 0 = {Q 0 }. We also define f Q0 = f,д Q0 = д For k 1, assume that R k 1 has been constructed and f Q,д Q defined for all Q R k 1. We apply Lemma 4.1 to each Q R k 1 with f = f Q,д = д Q, and define R k := Q R k 1 I {fq,д Q },Q, and for I R k I {fq,д Q },Q set Q (I := Q, f I := f Q(I ζ in Q(I. Note that f Q f at every step of the construction. Since f and д are bounded functions, for k = k sufficiently large all the collections {I {fq,д Q },Q } for Q R k 1 will be empty. The iterative procedure stops at k = k, and we set R := k kr k and S := { Q : Q R} Making use of estimate (4.4 along the iteration of Lemma 4.1 we readily obtain (4.12 B(f,д = B(f ζ in Q 0,дζ in Q 0 Q f Q Q,p д Q Q Q f Q,p д Q Q R k Q R k 0 where the first step is allowed by the support condition on f,д and the second inequality follows from f Q f. Finally, we have shown that (4.13 B(f,д Λ S (f,д Λ S (f,д := L f L,p д L We claim that R is a η-sparse collection with an absolute small constant η: thus, since dilates of a sparse collection are sparse as well, S is sparse with slightly different but still absolute sparsity constant. Noticing that neither the implicit constant in the above inequality nor the construction of S depend in any way on the parameters a, ξ, Ξ, N, this concludes the proof of the Proposition, up to the proof of the claim. The proof of the claim is very similar to the argument in [15, Proof of Theorem 4.2]. A useful fact from [18, Section 6] is that it suffices to prove sparsity of each of the O (1 subsets G R with the property that (4.14 Q, Q G, Q Q, 9Q 9Q = Q Q. Fix such a G and take Q G. It suffices, according to [18, Lemma 6.3], to prove the estimate Q 2 4 Q. Q G,Q Q L S k 0

11 SPARSE DOMINATION OF VARIATIONAL CARLESON OPERATORS 11 Let Q G, Q Q. Call R = Q (Q. We claim that R Q, which follows from the grid property (4.14, the obvious 9Q 9R which is a consequence of (4.1, and the less obvious fact that if the reverse inclusion were true Q would have been contained in the set E R at the corresponding application of Lemma 4.1, which violates the maximality of Q. It follows in particular that R must be contained in Ĩ for some I with Q (I = Q. But according to (4.3 the measure of the union of such Ĩ is less than 2 4 Q. This completes the proof of the sparsity claim and in turn the proof of Proposition 2.1. References [1] David Beltran and Jonathan Bennett, Subdyadic square functions and applications to weighted harmonic analysis, preprint arxiv: [2] Jose M. Conde-Alonso and Guillermo Rey, A pointwise estimate for positive dyadic shifts and some applications, preprint arxiv: , to appear in Math. Ann. (2014. [3] David V. Cruz-Uribe, José Maria Martell, and Carlos Pérez, Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkhäuser/Springer Basel AG, Basel, MR [4] Amalia Culiuc, Francesco Di Plinio, and Yumeng Ou, Domination of multilinear singular integrals by positive sparse forms, preprint arxiv: [5] Francesco Di Plinio, Weak-L p bounds for the Carleson and Walsh-Carleson operators, C. R. Math. Acad. Sci. Paris 352 (2014, no. 4, MR [6] Francesco Di Plinio and Andrei K. Lerner, On weighted norm inequalities for the Carleson and Walsh-Carleson operator, J. Lond. Math. Soc. (2 90 (2014, no. 3, MR [7] Francesco Di Plinio and Yumeng Ou, A modulation invariant Carleson embedding theorem outside local L 2, preprint arxiv: , to appear in J. Anal. Math. [8] Yen Do and Michael Lacey, Weighted bounds for variational Fourier series, Studia Math. 211 (2012, no. 2, MR [9], Weighted bounds for variational Walsh-Fourier series, J. Fourier Anal. Appl. 18 (2012, no. 6, MR [10] Yen Do, Camil Muscalu, and Christoph Thiele, Variational estimates for the bilinear iterated Fourier integral, preprint arxiv: [11] Yen Do and Christoph Thiele, L p theory for outer measures and two themes of Lennart Carleson united, Bull. Amer. Math. Soc. (N.S. 52 (2015, no. 2, MR [12] Loukas Grafakos, José María Martell, and Fernando Soria, Weighted norm inequalities for maximally modulated singular integral operators, Math. Ann. 331 (2005, no. 2, MR (2005k:42037 [13] Richard A. Hunt and Wo Sang Young, A weighted norm inequality for Fourier series, Bull. Amer. Math. Soc. 80 (1974, MR (49 #3419 [14] Tuomas P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2 175 (2012, no. 3, MR [15] Michael T. Lacey, An elementary proof of the A 2 Bound, preprint arxiv: , to appear in Israel J. Math. [16], Oscillatory singular integrals and sparse operators, preprint arxiv: [17] Andrei Lerner, On pointwise estimates involving sparse operators, New York J. Math. 22 (2016, [18] Andrei Lerner and Fedor Nazarov, Intuitive dyadic calculus: the basics, preprint arxiv: (2015. [19] Andrei K. Lerner, A simple proof of the A 2 conjecture, Int. Math. Res. Not. IMRN (2013, no. 14, MR [20] Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, and James Wright, A variation norm Carleson theorem, J. Eur. Math. Soc. (JEMS 14 (2012, no. 2, MR [21] Gennady Uraltsev, Variational Carleson embeddings into the upper 3-space, preprint arxiv:xxxx.xxxx.

12 12 FRANCESCO DI PLINIO, YEN Q. DO, AND GENNADY N. URALTSEV Brown University Mathematics Department, Box 1917, Providence, RI 02912, USA address, F. Di Plinio: Department of Mathematics, The University of Virginia, Charlottesville, VA , USA address, Y. Do: Mathematics Department, Universität Bonn, Endenicher Allee 60, D Bonn, Germany address, G. Uraltsev: gennady.uraltsev@math.uni-bonn.de