Course: Weighted inequalities and dyadic harmonic analysis

Transcription

1 Course: Weighted inequalities and dyadic harmonic analysis María Cristina Pereyra University of New Mexico CIMPA2017 Research School - IX Escuela Santaló Harmonic Analysis, Geometric Measure Theory and Applications Buenos Aires, July 31 - August 11, 2017 María Cristina Pereyra (UNM) 1 / 34

2 Outline Lecture 1. Weighted Inequalities and Dyadic Harmonic Analysis. Model cases: Hilbert transform and Maximal function. Lecture 2. Brief Excursion into Spaces of Homogeneous Type. Simple Dyadic Operators and Weighted Inequalities à la Bellman. Lecture 3. Case Study: Commutators. Sparse Revolution. María Cristina Pereyra (UNM) 2 / 34

3 Outline Lecture 3 1 Case study: Commutator [H, b] Dyadic proof of quadratic estimate Transference theorem Coifman-Rochberg-Weiss argument Recent Progress 2 Sparse operators and families of dyadic cubes A 2 theorem for sparse operators Sparse vs Carleson families Domination by Sparse Operators Case study: Sparse operators vs commutators 3 Acknowledgements María Cristina Pereyra (UNM) 3 / 34

4 Case study: Commutator [H, b] Case study: Commutator [H, b] For b BMO, and H the Hilbert Transform, let [b, H]f := b(hf) H(bf). The commutator is bounded on L p for 1 < p < if and only if b BM O (Coifman, Rochberg, Weiss 76). Moreover [H, b]f p C p b BMO f p. Commutator is not of weak-type (1, 1) (Pérez 96). Commutator is more singular than H. bh and Hb are NOT necessarily bounded on L p when b BMO. The commutator introduces some key cancellation. This is very much connected to the celebrated H 1 - BMO duality by Feffferman, Stein 72. María Cristina Pereyra (UNM) 4 / 34

5 Case study: Commutator [H, b] Weighted Inequalities Theorem (Bloom 85) If u, v A 2 then [b, H] : L p (u) L p (v) is bounded if and only if b BMO µ where µ = u 1/p v 1/p and ˆ 1 b BMOµ := sup b(x) b I dx <. I R µ(i) Theorem (Alvarez, Bagby, Kurtz, Pérez 93) If w A p then [T, b]f L p (w) C p (w) b BMO f L p (w). Result valid for general linear operators T, and two-weight estimates. Proof used classical Coifman-Rochberg-Weiss 76 argument. Theorem (Daewon Chung 11 ) [H, b]f L 2 (w) C b BMO [w] 2 A 2 f L 2 (w). I María Cristina Pereyra (UNM) 5 / 34

6 Case study: Commutator [H, b] Dyadic proof of quadratic estimate Dyadic proof for commutator [H, b] Theorem (Daewon Chung 11) [H, b]f L 2 (w) C b BMO [w] 2 A 2 f L 2 (w). Daewon s "dyadic" proof is based on: (1) Use Petermichl s dyadic shift operator X instead of H, and prove uniform (on grids) quadratic estimates for its commutator [X, b]. (2) Decomposition of the product bf in terms of paraproducts bf = π b f + π b f + π f b the first two terms are bounded in L p (w) when b BMO and w A p, the enemy is the third term. Decomposing commutator [X, b]f = [X, π b ]f + [X, π b ]f + [ X(π f b) π Xf (b) ]. María Cristina Pereyra (UNM) 6 / 34

7 Case study: Commutator [H, b] Dyadic proof of quadratic estimate cont. "dyadic proof" commutator (3) Linear bounds for paraproducts π b, πb (Bez 08) and X (Pet 07) gives quadratic bounds for first two terms. [X, b]f = [X, π b ]f + [ X, π b ]f + [X(π f b) π Xf (b) ]. (4) Third term is better, it obeys a linear bound, and so do halves of the two commutators (using Bellman function techniques): X(π f b) π Xf (b) + Xπ b f + π b Xf C b BMO[w] A2 f. Providing uniform quadratic bounds for commutator [X, b] hence [H, b] L 2 (w) C b BMO [w] 2 A 2 f L 2 (w). Bad guys non-local terms π b X, Xπb. Estimate and extrapolated estimates are sharp! (Chung-P.-Pérez 12). María Cristina Pereyra (UNM) 7 / 34

8 Case study: Commutator [H, b] Dyadic proof of quadratic estimate Afterthoughts A posteriori one realizes the pieces that obey linear bounds are generalized Haar Shift operators and hence their linear bounds can be deduced from general results for those operators... As a byproduct of Chung s dyadic proof we get that Beznosova s extrapolated bounds for the paraproduct are optimal: π b f L p (w) C p [w] max{1, 1 A p p 1 } f L p (w) Proof: by contradiction, if not for some p then [H, b] will have better bound in L p (w) than the known optimal quadratic bound. María Cristina Pereyra (UNM) 8 / 34

9 Case study: Commutator [H, b] Transference theorem Transference theorem Theorem (Chung, P., Pérez 12, P. 13 ) Given linear operator T and 1 < r < if for all w A r there exists a C T,d > 0 such that for all f L r (w), T f L r (w) C T,d [w] α A r f L r (w). then its commutator with b BMO obeys the following bound [T, b]f L r (w) C r,t,d [w] α+max{1, 1 A r r 1 } b BMO f L r (w). Proof follows classical Coifman-Rochberg-Weiss 76 argument using (i) Cauchy integral formula; (ii) quantitative Coifman-Fefferman result: w A r implies w RH q with q = 1 + c d /[w] Ar and [w] RHq 2; (iii) quantitative version: b BMO implies e αb A r for α small enough with control on [e αb ] Ar. Higher-order-commutator T k b k 1 1 = [b, Tb ] (powers α + k max{1, r 1 }, k). María Cristina Pereyra (UNM) 9 / 34

10 Case study: Commutator [H, b] Transference theorem A 2 Conjecture (Now Theorem) Transference theorem for commutators are useless unless there are operators known to obey an initial L r (w) bound. Do they exist? Yes! Theorem (Hytönen, Annals 12) Let T be a Calderón-Zygmund operator, w A 2. Then there is a constant C T,d > 0 such that for all f L 2 (w), T f L 2 (w) C T,d [w] A2 f L 2 (w). We conclude that for all Calderón-Zygmund operators T their commutators obey a quadratic bound in L 2 (w). [T, b]f L 2 (w) C T,d [w] 2 A 2 b BMO f L 2 (w). [Tb k f L 2 (w) C T,d [w] 1+k A 2 b k BMO f L 2 (w). María Cristina Pereyra (UNM) 10 / 34

11 Case study: Commutator [H, b] Transference theorem Some generalizations Extensions to commutators with fractional integral operators, two-weight problem Cruz-Uribe, Moen 12 Extensions using [w] A1, A 1 p>1 A p, Ortiz-Caraballo 11. Mixed A 2 -A, A = p>1 A p, [w] A [w] A2, Hytönen, Pérez 13 [T, b] L 2 (w) C n [w] 1 2 A2 ( [w]a + [w 1 ] A ) 3 2 b BMO. See also Ortiz-Caraballo, Pérez, Rela 13. Matrix valued operators and BM O, Isralowitch, Kwon, Pott 15 Two weight setting (both weights in A p, à la Bloom) Holmes, Lacey, Wick 16. Also for biparameter Journé operators Holmes, Petermichl, Wick 17. Pointwise control by sparse operators adapted to commutator, improving weak-type, Orlicz bounds, and quantitative two weight Bloom bounds, Lerner, Ombrosi, Rivera-Ríos, arxiv 17. María Cristina Pereyra (UNM) 11 / 34

12 Case study: Commutator [H, b] Coifman-Rochberg-Weiss argument The Coifman-Rochberg-Weiss argument when r = 2 Conjugate operator as follows: for any z C define T z (f) = e zb T (e zb f). A computation + Cauchy integral theorem give (for "nice" functions), [b, T ](f) = d dz T z(f) z=0 = 1 ˆ T z (f) 2πi z 2 dz, ɛ > 0 Now, by Minkowski s inequality [b, T ](f) L 2 (w) 1 2π ɛ 2 ˆ z =ɛ z =ɛ T z (f) L 2 (w) dz, ɛ > 0. Key point is to find appropriate radius ɛ. Look at inner norm and try to find bounds depending on z. T z (f) L 2 (w) = T (e zb f) L 2 (w e 2Rez b ). Use main hypothesis: T L 2 (v) C[v] A2, for v = w e 2Rez b. Must check that if w A 2 then v A 2 for z small enough. María Cristina Pereyra (UNM) 12 / 34

13 Case study: Commutator [H, b] Coifman-Rochberg-Weiss argument For v = w e 2Rez b. Must check that if w A 2 then v A 2 for small z. ( ˆ ) ( ˆ ) 1 1 [v] A2 = sup w(x) e 2Rez b(x) dx w 1 (x) e 2Rez b(x) dx. If w A 2 w RH q for some q > 1 (Coifman, Fefferman 73). uantitative version: if q = d+5 [w] A2 then ( ˆ ) 1 ˆ 1 w q q 2 (x) dx w(x) dx, and similarly for w 1 A 2 (since [w] A2 = [w 1 ] A2 ), ( ˆ ) 1 ˆ 1 w q q 2 (x) dx w 1 (x) dx. In what follows q is as above. María Cristina Pereyra (UNM) 13 / 34

14 Case study: Commutator [H, b] Coifman-Rochberg-Weiss argument Using these and Holder s inequality we have for an arbitrary ( ˆ ) ( ˆ ) 1 1 [v] A2 = w(x)e 2Rez b(x) dx w(x) 1 e 2Rez b(x) dx 4 ( ˆ ) 1 ( ˆ ) 1 1 w q q 1 e ( ˆ ) 1 2Rez q q b 1 ( ˆ ) 1 w q q 1 e 2Rez q q b ˆ ˆ ˆ ˆ ( 1 )( 1 w w 1 )( 1 4 [w] A2 [e 2Rez q b ] ) 1 e ( 2Rez q q b 1 1 q A 2 e 2Rez q b Now, since b BMO there are 0 < α d < 1 and β d > 1 such that if 2Rez q α d b BMO then [e 2Rez q b ] A2 β d. Hence for these z, [v] A2 = [w e 2Rez b ] A2 4 [w] A2 βd 4 [w] A 2 β d. 1 q ) 1 q María Cristina Pereyra (UNM) 14 / 34

15 Case study: Commutator [H, b] Coifman-Rochberg-Weiss argument If z α d 2q b BMO then [v] A2 4[w] A2 β d and T z (f) L 2 (w) = T (e zb f) L 2 (v) [v] A2 f L 2 (w) 4[w] A2 β d f L 2 (w) (since e zb f L 2 (v) = e zb f L 2 (we 2Rez b ) = f L 2 (w)). Thus choose the radius ɛ := 1 2π ɛ 2 [b, T ](f) L 2 (w) 1 2π ɛ 2 ˆ z =ɛ α d 2q b BMO, and get ˆ z =ɛ T z (f) L 2 (w) dz 4[w] A2 β d f L 2 (w) dz = 1 ɛ 4[w] A 2 β d f L 2 (w), Note that ɛ 1 [w] A2 b BMO, because q = d+5 [w] A2 2 d [w] A2, [b, T ](f) L 2 (w) C d [w] 2 A 2 b BMO. María Cristina Pereyra (UNM) 15 / 34

16 Case study: Commutator [H, b] Recent Progress Recent progress Active area of research! Extensions to metric spaces with geometric doubling condition and spaces of homogeneous type. Generalizations to matrix valued operators (so far 3/2 estimates for paraproducts, linear for square function). Pointwise domination by sparse positive dyadic operators: Rough CZ operators and commutators, more next slides. Singular non-integral operators (Bernicot, Frey, Petermichl 15). Multilinear SIO (Culiuc, Di Plinio, Ou; Lerner, Nazarov ; K. Li 16. Benea, Muscalu 17). Non-homogeneous CZ operators (Conde-Alonso, Parcet 16). Uncentered variational operators (Franca Silva, Zorin-Kranich 16). Maximally truncated oscillatory SIO (Krause, Lacey 17). Spherical maximal function (Lacey 17). Radon transform (Oberlin 17). Hilbert transform along curves (Cladek, Ou 17). Convex body domination (Nazarov, Petermichl, Treil, Volberg 17). María Cristina Pereyra (UNM) 16 / 34

17 Sparse operators and families of dyadic cubes Sparse positive dyadic operators Cruz-Uribe, Martell, Pérez 10 showed the A 2 -conjecture in a few lines for sparse operators A S, where S is a sparse collection of dyadic cubes, defined as follows A S f(x) = S m f 1 (x). Definition A collection of dyadic cubes S in R d is η-sparse, 0 < η < 1 if there are pairwise disjoint measurable sets E with E η S. (Rough) CZ operators are pointwise dominated by a finite number of sparse operators Lerner 10, 13, Conde-Alonso, Rey 14, Lerner, Nazarov 14, Lacey 15, quantitative form Lerner 15, Hytönen, Roncal, Tapiola 15. María Cristina Pereyra (UNM) 17 / 34

18 Sparse operators and families of dyadic cubes A 2 theorem for sparse operators A 2 theorem for A S f(x) = S m f 1 (x) For w A 2, S sparse family, to show that A S f L 2 (w) C[w] A2 f L 2 (w) is equivalent by duality to show f L 2 (w), g L 2 (w 1 ) A S f, g C[w] A2 f L 2 (w) g L 2 (w 1 ). By CS inequality E = E w 1 2 w 1 2 (w(e )) 1 2 (w 1 (E )) 1 2 and A S f, g S f g 1 η S [w] A 2 η f ww 1 w 1 g w 1 w w w w 1 E S f ww 1 w 1 (w 1 (E )) 1 g w 1 w 2 (w(e )) 1 2 w María Cristina Pereyra (UNM) 18 / 34

19 Sparse operators and families of dyadic cubes A 2 theorem for sparse operators cont. A 2 theorem for sparse operators A S f, g [w] A 2 η [w] A 2 η [w] A 2 η S f ww 1 w 1 (w 1 (E )) 1 g w 1 w 2 (w(e )) 1 2 w [ f ww 1 2 w 1 2 S [ ˆ S ] 1 [ w 1 2 (E ) E M 2 w 1 (fw)w 1 dx S ] 1 2 [ S g w 1 w 2 ˆ w 2 ] 1 2 w(e ) E M 2 w(gw 1 )w dx ] 1 2 [w] A 2 M η w 1(fw) L 2 (w 1 ) M w (gw 1 ) L 2 (w) C[w] A2 fw L 2 (w 1 ) gw 1 L 2 (w) = C[w] A2 f L 2 (w) g L 2 (w 1 ). Similar argument yields linear bounds in L p (w) for p > 2 and by duality get [w] 1 p 1 A p = [w 1 p 1 ] Ap when 1 < p < 2 (Moen 12). María Cristina Pereyra (UNM) 19 / 34

20 Sparse operators and families of dyadic cubes Sparse vs Carleson families Sparse vs Carleson families of dyadic cubes Definition A family of dyadic cubes S in R d is called Λ-Carleson, Λ > 1 if P Λ D. P S,P Equivalent to: sequence { P 1 S (P )} P D is Carleson with intensity Λ. Lemma (Lerner-Nazarov 14 in Intuitive Dyadic Calculus) S is Λ-Carleson iff S is 1/Λ-sparse. Proof ( ). S a 1/Λ-sparse means for all P S there are E P P pairwise disjoint subsets such that Λ E P P. Hence P Λ E P Λ. P S,P P S,P María Cristina Pereyra (UNM) 20 / 34

21 Sparse operators and families of dyadic cubes Sparse vs Carleson families Λ-Carleson 1/Λ-sparse Proof ( ) (Lemma 6.3 in Lerner, Nazarov 14). If S had a bottom layer D K, then consider all S D K, choose any sets E with E = 1 Λ. Then go up layer by layer, for each D k, k K, choose any E \ R S,R E R with E = 1 Λ. Choice always possible because for every S we have R S,R E R 1 Λ R S,R R Λ 1 (1 Λ = 1 ), Λ Where we used in ( ) the Λ-Carleson hypothesis. So \ R S,R E R 1 Λ, and by construction the sets E are pairwise disjoint, and we are done. But, what if there is no bottom layer? Run construction for each K 0 and pass to the limit! Have to be a bit careful! All we have to do is replace free choice with canonical choice. from Lerner, Nazarov 14 María Cristina Pereyra (UNM) 21 / 34

22 Sparse operators and families of dyadic cubes Sparse vs Carleson families Λ-Carleson 1/Λ-sparse Cont. proof ( ) (Lemma 6.3 in Lerner, Nazarov 14). Fix K 0, for S ( k K D k ) define Ê(K) inductively as follows: if S D K then Ê(K) is cube with same "SW corner" x as, and Ê(K) = 1 Λ, namely Ê(K) := x + Λ 1 d ( x ). if S D k, k < K then Ê(K) R Ê (K) are defined for R S, R, set := ( x + t( x ) ) F (K), F (K) := R S,R Ê(K) R, and t [0, 1] is the largest number such that E (K) 1 Λ where E (K) = ( x + t( x ) ) \ F (K). Such t [0, 1] exists, moreover E (K) = 1 Λ by monotonicity and continuity of the function t ( x + t( x ) ) \ F (K). María Cristina Pereyra (UNM) 22 / 34

23 Sparse operators and families of dyadic cubes Sparse vs Carleson families Figure 11 from Intuitive dyadic calculus: the basics, by A. K. Lerner, F. Nazarov 14 María Cristina Pereyra (UNM) 23 / 34

24 Sparse operators and families of dyadic cubes Sparse vs Carleson families Λ-Carleson 1/Λ-sparse Cont. proof ( ) (Lemma 6.3 in Lerner, Nazarov 14). Claim: Ê (K) R Ê(K+1) R for every S ( ) k K D k. Proof by backward induction. Let Ê = lim Ê (K) K = K=0Ê(K). Note that E (K) = Ê(K) \ F (K) = (1/Λ), and F (K) F (K+1). E := lim K E(K) = Ê \ ( lim F (K) ) K = Ê \ ( ) R S,R Ê R is a well defined subset of with E = 1 Λ. Sets E with S are pairwise disjoint by construction. María Cristina Pereyra (UNM) 24 / 34

25 Sparse operators and families of dyadic cubes Domination by Sparse Operators Lemma (Rey, Reznikov 15) Let {α } I D be a Carleson sequence, then the positive dyadic operator T 0 f(x) := D α f 1 (x) is bounded in L 2 (w) for all w A 2, moreover T 0 f L 2 (w) C[w] A2 f L 2 (w). Proof. Done if we can dominate T 0 with sparse operators. Rey, Reznikov 15 showed that localized positive dyadic operators of complexity m 1 defined for {α I} Carleson, T 0 m f(x) := D( 0 ) R D m() α R R f 1R(x) are pointwise bounded by localized sparse operators. Lerner, Nazarov 14 removed the localization. Finally T 0 is a sum of T 1s simply because 1 = R D 1 () 1R. María Cristina Pereyra (UNM) 25 / 34

26 Sparse operators and families of dyadic cubes Domination by Sparse Operators Domination by sparse operators S, S i are sparse families. Martingale transform: 1 0 T σ f A S f. Same holds for maximal truncations (Lacey 15). Paraproduct: 1 0 π b f A S f (Lacey 15). CZ operators T f N d i=1 A S i f. Square function S d f 2 N d i=1 I S i f 2 I 1 I (Lacey, K. Li 16). Commutator [b, T ] for T an ω-cz operator with ω satisfying a Dini condition, b L 1 loc can be pointwise dominated by finitely many sparse-like operators and their adjoints (Lerner, Ombrosi, Rivera-Ríos 17). María Cristina Pereyra (UNM) 26 / 34

27 Sparse operators and families of dyadic cubes Case study: Sparse operators vs commutators Case study: Sparse operators vs commutators Pérez, Rivera-Ríos 17. The following L log L-sparse operator cannot bound pointwise [T, b] B S f(x) = S f L log L, 1 (x). (M 2 M L log L is correct maximal function for commutator). Lerner, Ombrosi, Rivera-Ríos 17. Adapted sparse operator and its adjoint provide pointwise estimates for [T, b]: T S,b f(x) := S b(x) b f 1 (x), TS,b f(x) := b b f 1 (x). S María Cristina Pereyra (UNM) 27 / 34

28 Sparse operators and families of dyadic cubes Case study: Sparse operators vs commutators Sparse domination for commutator Theorem (Lerner, Ombrosi, Rivera-Ríos 17) Let T an ω-cz operator with ω satisfying a Dini condition, b L 1 loc. For every compactly supported f L (R n ), there are 3 n dyadic lattices D (k) and n -sparse families S k D (k) such that for a.e. x R n 3n ( [b, T ]f(x) c n C T TSk,b f (x) + TS k,b f (x)). k=1 uadratic bounds on L 2 (w) for [b, T ] follow from quadratic bounds for this adapted sparse operators. uadratic bounds on L 2 (w) for T S,b, T S,b, T S,b f L 2 (w) + T S,b f L 2 (w) C b BMO [w] 2 A 2 f L 2 (w), and much more follow from a key lemma. María Cristina Pereyra (UNM) 28 / 34

29 Sparse operators and families of dyadic cubes Case study: Sparse operators vs commutators Key lemma T S,b f(x) = S b b f 1 (x) Lemma (Lerner, Ombrosi, Rivera-Ríos 17) η 2(1+η) -sparse Given S η-sparse family in D, b L 1 loc then S D a family, S S, such that S, with Ω(b; R) := 1 R R b(x) b R dx, b(x) b 2 n+2 Ω(b; R)1 R (x), a.e. x, R S,R Corollary (uantitative Bloom, LOR 17) Let u, v A p, µ = u 1/p v 1/p, b BMOµ = sup Ω(b; )/µ(), then T S,b f (x) c n b BMOµ A S( A S( f )µ ) (x). Hence T S,b f L p (v) c n,p b BMOµ A S L p (v) A S L p (u) f L p (u) c n,p b BMOµ ( [v]ap [u] Ap ) max{1, 1 p 1 } f L p (u). María Cristina Pereyra (UNM) 29 / 34

30 Sparse operators and families of dyadic cubes Case study: Sparse operators vs commutators For u, v A p, µ = u 1/p v 1/p and b BMO µ that T S,b f L p (v) c n,p b BMOµ ( [v]ap [u] Ap ) max{1, 1 p 1 } f L p (u). Set now u = v = w A p, then µ 1 and b BMO TS,b f L p (w) c n,p b BMO [w] 2 max{1, 1 A p p 1 } f L p (w). María Cristina Pereyra (UNM) 30 / 34

31 Acknowledgements ;-) Gracias Úrsula y todo el comité organizador por darme la oportunidad de dar este curso!!!! Y por supuesto gracias a los estudiantes que sin ustedes no hay curso!!! María Cristina Pereyra (UNM) 31 / 34

32 Acknowledgements Domination of martingale transform d après Lacey Given I 0 D, need to find sparse S D such that 1 I0 T σ f CA S f. Sharp truncation T σ is of weak-type (1, 1) (Burkholder 66), sup λ {x R : T σ f(x) > λ} C f L 1 (R). λ>0 Maximal function M is also of weak-type (1, 1). So C 0 > 0 s.t. F I0 := {x I 0 : max{mf, Tσf}(x) > 1 2 C 0 f I0 } satisfies F I0 1 2 I 0. Where Tσf = sup I D σ I f, h I h I I D,I I Let E I0 = {I D : maximal intervals I contained in F I0 }, then T σ f(x) 1 I0 (x) C 0 f I0 + Tσ I f(x) (1) I E I0 where T I σ f := σĩ f I 1 I + J:J I. σ J f, h J h J, Ĩ is the parent of I. María Cristina Pereyra (UNM) 32 / 34

33 Acknowledgements Domination of martingale transform d après Lacey Repeat for each I E I0, then for each I E I, etc. Let S 0 = {I 0 }, and S j := I Sj 1 E I. Finally let S := j=0 S j. For each I S, let E I = I \ F I, by construction E I 1 2 I and S is a 1 2-sparse family. This is an example of a stopping time illustrated below using the house/roof metaphor Figure 8 from Intuitive dyadic calculus: the basics, by A. K. Lerner, F. Nazarov 14 María Cristina Pereyra (UNM) 33 / 34

34 Acknowledgements Domination of martingale transform d après Lacey Claim (1): T σ f(x) 1 I0 (x) C 0 f I0 + I E I0 T I σ f(x). Note that T σ f(x) Tσf(x). Thus, if x I 0 \ F I0 then T σ f(x) 1 2 C 0 f I0, and (1) is satisfied. If x F I0 then there is unique I S 1 with x I, and T σ f(x) = J Ĩ σ J f, h J h J (x) + J Ĩ σ J f, h J h J (x) = J Ĩ σ J f, h J h J (x) σĩ f Ĩ + T I σ f(x). where T I σ f := σĩ f I1 I + J I σ J f, h J h J, and f, hĩ hĩ(x) = f I f Ĩ. T I σ σĩ f I 1 I has a similar estimate to (1), we can then recursively get the sparse domination. María Cristina Pereyra (UNM) 34 / 34