Zygmund type and flag type maximal functions, and sparse operators
ZZYGMUND TYPE AND FLAG TYPE MAXIMAL FUNCTIONS,AND SPARSE OPERATORS
GUILLERMO J. FLORES, JI LI AND LESLEY A. WARD
Abstract.
We prove that the maximal functions associated with a Zygmund dilationdyadic structure in three-dimensional Euclidean space, and with the flag dyadic structurein two-dimensional Euclidean space, cannot be bounded by multiparameter sparse opera-tors associated with the corresponding dyadic grid. We also obtain supplementary resultsabout the absence of sparse domination for the strong dyadic maximal function. Introduction and statement of main results
In recent years, it has been evidenced that
Sparse Operators play an important rolein the weighted bounds for many singular integrals, see for example [10, 11, 2, 3]. Suchtechniques have led to advances in sharp estimates within the Calder´on-Zygmund theory.The fundamental example is the sparse domination of the one-parameter dyadic maximalfunction M d f ( x ) := sup Q ∈ D n : x ∈ Q | Q | ˆ Q | f ( x ) | dx where the supremum is taken over all dyadic cubes in R n containing x , that is, M d f ( x ) ≤ C (cid:88) Q ∈ S (cid:18) | Q | ˆ Q f ( x ) dx (cid:19) χ Q ( x ) , where S is a sparse collection of dyadic cubes.Nevertheless, a remarkable recent result (Theorem A in [1]) shows that there is no sparsedomination in the tensor product setting R n × R m for the strong dyadic maximal function M sd f ( x, y ) := sup R ∈ D n × D m : ( x,y ) ∈ R | R | ˆ R | f ( x , y ) | dx dy , where the supremum is taken over all dyadic rectangles with sides parallel to the axescontaining ( x, y ). This result suggests that the sparse domination techniques in the one-parameter setting cannot be expected to work with the same approach in the multiparam-eter setting.The classical Calder´on–Zygmund singular integrals are related to the one-parameter di-lation structure on R n , defined by δ o ◦ ( x , x , . . . , x n ) := ( δx , . . . , δx n ), with x ∈ R n and Mathematics Subject Classification.
Key words and phrases.
Zygmund dilations, Maximal functions, Sparse domination.The authors are supported by ARC DP 160100153. a r X i v : . [ m a t h . C A ] M a r FLORES, G., LI, J. AND WARD, L. δ >
0. Meanwhile the product dilation structure is defined by δ p ◦ ( x , x , . . . , x n ) :=( δ x , . . . , δ n x n ), δ i > i = 1 , . . . , n . The key difference is that δ o maps cubes to cubes,while δ p maps cubes to rectangular prisms whose side-lengths are independent. Multipa-rameter dilations lie between these two extremes: the side-lengths need not be equal norbe completely independent of each other, but may be mutually dependent.With these dilation structures in mind, it is natural to wonder whether it is possibleto obtain certain sparse domination for multiparameter maximal functions which lie inbetween the two extreme cases M d and M sd .One of the most natural and interesting examples of a group of dilations in R thatlies in between the one-parameter and the full product setting is the so-called Zygmunddilation defined by ρ s,t ( x , x , x ) = ( sx , tx , stx ) for s, t > M z f ( x , x , x ) := sup R : ( x ,x ,x ) ∈ R | R | ˆ R | f ( u , u , u ) | du du du , (1.1)where the supremum above is taken over all rectangles in R with edges parallel to theaxes and side-lengths of the form s, t , and st (see [4]). See also [15] for a discussion of theZygmund conjecture about the differentiation properties of k -parameter bases of rectangu-lar prisms in R n , and [6] for the Zygmund type singular integrals and their commutators.The survey paper of R. Fefferman [7] has more information about research directions inthis setting.Another very important example in the multiparameter setting is the implicit flag struc-ture. To be precise, in [12, 13], M¨uller, Ricci and Stein studied Marcinkiewicz multiplierson the Heisenberg group H n associated with the sub-Laplacian on H n and the central in-variant vector field, and obtained the L p -boundedness for 1 < p < ∞ . This is surprisingsince these multipliers are invariant under a two-parameter group of dilations on C n × R ,while there is no two-parameter group of automorphic dilations on H n . Moreover, theyshowed that Marcinkiewicz multipliers can be characterized by a convolution operator ofthe form f ∗ K where K is a flag convolution kernel, which satisfies size and smooth-ness conditions lying in between the one-parameter and product singular integrals. Thecomplete flag Hardy space theory and the boundedness of the iterated commutator wasobtained only recently in [9] and [5], respectively. The fundamental tool in this setting isthe flag maximal function. We state the definition in R × R for the sake of simplicity: M flag f ( x , x ) := sup R : ( x ,x ) ∈ R | R | ˆ R | f ( u , u ) | du du , (1.2)where the supremum above is taken over all rectangles in R with edges parallel to theaxes and side-lengths of the form s and t satisfying s ≤ t . YGMUND TYPE AND FLAG TYPE MAXIMAL FUNCTIONS 3
The dyadic versions of the Zygmund maximal function and flag maximal function canbe defined easily by restricting to dyadic axis-parallel rectangles in (1.1) and (1.2). Wedenote them by M z ,d and M flag ,d , respectively.In this article we show that the maximal functions M z ,d and M flag ,d cannot be boundedby multiparameter sparse operators associated with the corresponding dyadic grid. Westate these results as Theorems 1.1 and 1.2, respectively. Theorem 1.1.
Take r, s ≥ such that /r + 1 /s > . Then for every C > and η ∈ (0 , there exist integrable functions f and g , compactly supported and bounded, such that (cid:12)(cid:12) (cid:104)M z ,d f, g (cid:105) (cid:12)(cid:12) ≥ C (cid:88) R ∈ S z (cid:104)| f |(cid:105) R,r (cid:104)| g |(cid:105) R,s | R | , for all η -sparse collections S z of Zygmund dyadic edge-parallel rectangles. Here we are denoting the L r -average of a function f over a rectangle R by (cid:104)| f |(cid:105) R,r := (cid:18) | R | ˆ R | f | r (cid:19) /r , for r ≥ . Theorem 1.2.
Take r, s ≥ such that /r + 1 /s > . Then for every C > and η ∈ (0 , there exist integrable functions f and g , compactly supported and bounded, such that (cid:12)(cid:12) (cid:104)M flag ,d f, g (cid:105) (cid:12)(cid:12) ≥ C (cid:88) R ∈ S flag (cid:104)| f |(cid:105) R,r (cid:104)| g |(cid:105) R,s | R | , for all η -sparse collections S flag of flag dyadic edge-parallel rectangles. Also, we show that the strong dyadic maximal function M sd does not admit ( r, s )-sparsedomination for certain r and s , in the following result. Theorem 1.3.
Take r, s ≥ such that /r + 1 /s > . Then for every C > and η ∈ (0 , there exist integrable functions f and g , compactly supported and bounded, such that (cid:12)(cid:12) (cid:104) M sd f, g (cid:105) (cid:12)(cid:12) ≥ C (cid:88) R ∈ S (cid:104)| f |(cid:105) R,r (cid:104)| g |(cid:105) R,s | R | , for all η -sparse collections S of dyadic edge-parallel rectangles. This provides a supplementary explanation to the main result of [1].We remark that there are no direct implications among the previously stated theorems.For instance, in two-parameter setting M sd is greater than M flag ,d . But the sums overthe sparse collections of flag dyadic rectangles involved in Theorem 1.2 do not have adirect comparison with the sums over the sparse collections of dyadic rectangles involvedin Theorem 1.3. A similar situation occurs for M sd and M z ,d .The paper is organised as follows. In Section 2 we give notation and some key results,which we then use to prove Theorem 1.1. In Section 3 we prove Theorem 1.2, and inSection 4 we prove Theorem 1.3. FLORES, G., LI, J. AND WARD, L. Zygmund dilation dyadic structures
Notation and proof of Theorem 1.1.
As usual, the collection D of dyadic intervalsin R is defined by D = (cid:8) R ⊂ R : R = (cid:2) k − j , ( k + 1)2 − j (cid:1) for k, j ∈ Z (cid:9) . Then, we shall denote by D z the collection of all Zygmund dyadic rectangles R = I × J × S in D , that is, those R ∈ D such that | S | = | I | · | J | .We shall evaluate M z ,d on finite sums of special point masses.In general, given a locally integrable function f we can define the associated measure µ f by µ f ( E ) := ´ E | f | for every measurable set E . Then M z ,d f ( x ) := sup R ∈ D z : x ∈ R | R | ˆ R | f | = sup R ∈ D z : x ∈ R µ f ( R ) | R | =: M z ,d µ f ( x ) . Now, if F is a finite set in R n we define the finite sum µ of point masses associated with F by µ := 1 (cid:93) F (cid:88) p ∈F δ p , where (cid:93) F denotes the number of points in F and δ p denotes the single point mass concen-trated at p . Then we naturally make sense of M z ,d µ ( x ) := 1 (cid:93) F sup R ∈ D z : x ∈ R | R | (cid:88) p ∈F δ p ( R ) , (cid:104) f, µ (cid:105) := ˆ f dµ = 1 (cid:93) F (cid:88) p ∈F f ( p ) , and (cid:104) µ (cid:105) R,r := 1 (cid:93) F (cid:18) | R | (cid:88) p ∈F δ p ( R ) (cid:19) /r for every rectangle R and 1 ≤ r < ∞ .Next, given η with 0 < η <
1, we shall say that a collection S of sets of finite measure(usually, rectangles or even dyadic edge-parallel rectangles) is called η - sparse , if for each R ∈ S there is a subset E R ⊂ R such that | E R | ≥ η | R | , and the collection { E R } is pairwisedisjoint. Then, for r, s ≥
1, we say that an operator T admits an ( r, s ) η -sparse domination if (cid:12)(cid:12) (cid:104) T f, g (cid:105) (cid:12)(cid:12) ≤ C (cid:88) R ∈S (cid:104)| f |(cid:105) R,r (cid:104)| g |(cid:105) R,s | R | , for every pair of functions f and g sufficiently nice in the given context.We are now in position to state the following key result from which we will deduceTheorem 1.1 as a corollary. We are denoting by C and c positive constants, not necessarilythe same at each occurrence. YGMUND TYPE AND FLAG TYPE MAXIMAL FUNCTIONS 5
Theorem 2.1.
Let r, s ≥ such that /r + 1 /s > . Then for every natural number k andfor every η ∈ (0 , there exist finite sums µ k and ν k of point masses in R such that (a) (cid:104)M z ,d µ k , ν k (cid:105) ≥ c k , and (b) (cid:88) R ∈ S z (cid:104) µ k (cid:105) R,r (cid:104) ν k (cid:105) R,s | R | ≤ Cη k /s (cid:18) k (cid:19) for all η -sparse collections S z of Zygmunddyadic rectangles. We can deduce Theorem 1.1 as follows. If we assume that M z ,d admits an ( r, s ) η -sparsedomination with 1 /r + 1 /s >
1, then for each k ∈ N we have (cid:104)M z ,d µ k , ν k (cid:105) ≤ C (cid:88) R ∈ S z (cid:104) µ k (cid:105) R,r (cid:104) ν k (cid:105) R,s | R | , for µ k and ν k as in Theorem 2.1. But the latter forces η = 0 by (a) and (b) of the previoustheorem, which leads to a contradiction. Therefore, by using a limiting and approximationargument, we obtain a proof of Theorem 1.1.2.2. Construction of special finite sums of point masses.
Now we shall give theexplicit formulas of µ k and ν k in Theorem 2.1. For brevity we drop the subscript k from µ k and ν k . Also, for our proof below we need one more auxiliary result.The authors in [1] introduced a dyadic distance function given by d D ( p, q ) := inf (cid:8) | R | / : R ∈ D n and p, q ∈ R (cid:9) , for every pair of points p and q in the cube [0 , n . The function d D turns out to beintuitive in terms of the geometry in the dyadic size-parallel rectangles setting. We notethat this function does not satisfy the conditions of a true distance, as remarked in [1], butnevertheless we shall refer to d D as the dyadic distance between two points in [0 , n .Next, associated with the Zygmund dilation structure , let d D z be the Zygmund dyadicdistance given by d D z ( p, q ) := inf (cid:8) | R | / : R ∈ D z and p, q ∈ R (cid:9) , for every pair of points p and q in the cube [0 , . Lemma 2.2.
For every natural number k , set m = k k . Then there exist two sets of points P z and Z z contained in the cube [0 , , linked closely to the Zygmund dilation previouslyintroduced, satisfying the following properties. (a) (cid:93) P z = 2 m +1 and d D z ( p, q ) ≥ m for every pair of points p, q ∈ P z . (b) (cid:93) Z z ≥ Cm m . (c) For each z ∈ Z z there is exactly one point p ∈ P z such that d D z ( p, z ) = C m + k . (d) Let R z be a Zygmund dyadic rectangle and let R = R z ∩ [0 , . Then we have Note that R is a dyadic rectangle in D and is not necessarily a Zygmund dyadic rectangle. FLORES, G., LI, J. AND WARD, L. ( i ) if | R | ≥ m +2 , then (cid:93) ( R ∩ P z ) = (cid:93) P z · | R | and (cid:93) ( R ∩ Z z ) ≤ Ckm m | R | ; ( ii ) if | R | < m +2 and R contains at least one point of P z and one point of Z z ,then (cid:93) ( R ∩ P z ) ≤ C k , (cid:93) ( R ∩ Z z ) ≤ Ck k and | R | ≥ C (2 m + k ) . See Figure 1 for a schematic diagram of P z . Figure 1.
Schematic diagram indicating one of the examples of the loca-tions of the points in P z , which lie in identical copies of the intersection P with the xy − plane. These copies are placed at discrete equally spacedheights, as if laid out on the floors of a multi-storey building.In order to prove that there is no sparse domination in the tensor product setting R n × R m for the strong dyadic maximal function, the authors in [1] introduced two fundamental sets P and Z in R (see [1, Theorem 2.1] for P and [1, Theorem 2.2] for Z ). We have thenmade a construction adapted to the Zygmund dilatation structure from these ones. In orderto prove Lemma 2.2 and for the convenience of the reader, we state next the previouslymentioned theorems.
YGMUND TYPE AND FLAG TYPE MAXIMAL FUNCTIONS 7
Theorem 2.3 (Theorem 2.1, Theorem 2.2 and Remark 2.3 of [1]) . For every naturalnumber m and every natural number k (cid:28) m there exist two sets of points P and Z contained in [0 , satisfying the following properties. (a) (cid:93) P = 2 m +1 and d D ( p, q ) ≥ m for every pair of points p, q ∈ P . (b) (cid:93) Z ≥ Cm m . (c) For each z ∈ Z there is exactly one point p ∈ P such that (cid:0) d D ( p, z ) (cid:1) = C m + k . (d) Let R be a dyadic rectangle in D . Then ( i ) if | R | ≥ m +1 , we have (cid:93) ( R ∩ P ) = (cid:93)P · | R | and (cid:93) ( R ∩ Z ) ≤ Ckm m | R | ; ( ii ) if | R | < m +1 and R contains one point of P , we have (cid:93) ( R ∩ Z ) ≤ Ck .The implied constants are independent of k and m . See Figure 2 and 3 below for schematic diagrams of P . See Figure 5 next section forschematic diagrams of Z . Figure 2.
Schematic diagram indicating one of the examples of the loca-tions of the points in P for m = 0 and m = 1. Proof of Lemma 2.2.
Let P z be the union of the level sets P×{ j − m } for j = 0 , , . . . , m − Z z . In particular, the j th level set P × { j − m } lies on the j th floor inFigure 1. Thus, the items (a), (b) and (c) are an immediate consequence of the properties(a) for P , (b)-(c) for Z and that the height of the Zygmund dyadic rectangles strictly con-tained in [0 , is low enough, see Figure 4 below for indications of the Zygmund dyadicrectangles.A point count and a pigeonholing argument allow us to obtain the item (d)-( i ), as notedin Remark 2.3 of [1]. FLORES, G., LI, J. AND WARD, L.
Figure 3.
Schematic diagram indicating one of the examples of the loca-tions of the points in P for m = 2.Now, let R be as in (d)-( ii ) and set R = I × J × S . Then | I × J | ≤ | S | because R z is a Zygmund dyadic rectangle and S is the minimum between 1 and the height of R z .Furthermore | I × J | < / m +1 (otherwise we have a contradiction with | R | < / m +2 ).Also, set p ∈ R ∩ P z and z ∈ R ∩ Z z . Without loss of generality we can suppose that p and z have the same height, from the definitions of P z and Z z . Then, | I × J | = C/ m + k by properties (a) for P and (c) for Z . Therefore | R | ≥ C (2 m + k ) and | S | < C k m . Finally, by property (d)-( ii ) for Z we have that (cid:93) ( R ∩ P z ) ≤ C k and (cid:93) ( R ∩ Z z ) ≤ Ck k . The proof of Lemma 2.2 is complete. (cid:3)
Proof of Theorem 2.1.
Let µ and ν be the finite sums of point masses associated with P z and Z z of Lemma 2.2, respectively. Again, we have dropped the subscript k .For each z ∈ Z z there is only one point p ∈ P z such that d D z ( p, z ) = C/ m + k , by Lemma2.2 (c). So there is a Zygmund dyadic rectangle R z containing both p and z such that | R z | = C/ (2 m + k ) . Then, from the definition of µ , M z ,d µ ( z ), and by Lemma 2.2 (a), we YGMUND TYPE AND FLAG TYPE MAXIMAL FUNCTIONS 9
Figure 4.
Schematic diagram indicating some examples of the Zygmunddyadic rectangles in each layer.have M z ,d µ ( z ) ≥ µ ( R z ) | R z | = C (2 m + k ) m +1 = C k . Therefore, from the definition of ν , (cid:104)M z ,d µ , ν (cid:105) = 1 (cid:93) Z z (cid:88) z ∈Z z M z ,d µ ( z ) ≥ C k and hence Theorem 2.1 (a) is proved.We next show (b) of Theorem 2.1. Let R be a Zygmund dyadic rectangle in S z and let R = R ∩ [0 , . Suppose now that R = I × J × S , so we shall consider the two casescorresponding to the items ( i ) and ( ii ) of Lemma 2.2 (d).First, suppose that | R | ≥ m +2 . Then by (d)-( i ) and (a) of Lemma 2.2, we have (cid:104) µ (cid:105) R,r = 1 (cid:93) P z (cid:18) (cid:93) ( R ∩ P z ) | R | (cid:19) /r = 12 (4 m +1)(1 − /r ) (cid:18) | R || R | (cid:19) /r ≤ (cid:18) | R || R | (cid:19) /r . Next, by (d)-( i ) and (b) of Lemma 2.2, we have (cid:104) ν (cid:105) R,s = 1 (cid:93) Z z (cid:18) (cid:93) ( R ∩ Z z ) | R | (cid:19) /s ≤ C ( km m ) /s m m (cid:18) | R || R | (cid:19) /s ≤ Ck /s (cid:18) | R || R | (cid:19) /s . Thus, we get (cid:104) µ (cid:105) R,r (cid:104) ν (cid:105) R,s ≤ Ck /s (cid:18) | R || R | (cid:19) /r + 1 /s . (2.1)Now, suppose that | R | < m +2 and (cid:104) µ (cid:105) R,r (cid:104) ν (cid:105) R,s > (cid:104) µ (cid:105) R,r (cid:104) ν (cid:105) R,s =0 contribute nothing to the sum on the left-hand side of the inequality of Theorem 2.1 (b)).So, R contains at least one point of P z and one point of Z z . Then by (d)-( ii ) and (a) ofLemma 2.2, we have (cid:104) µ (cid:105) R ,r = 1 (cid:93) P z (cid:18) (cid:93) ( R ∩ P z ) | R | (cid:19) /r ≤ C k/r (2 m + k ) /r m +1 ≤ C k/r . Also, by (d)-( ii ) and (b) of Lemma 2.2, we have (cid:104) ν (cid:105) R ,s = 1 (cid:93) Z z (cid:18) (cid:93) ( R ∩ Z z ) | R | (cid:19) /s ≤ C ( k k ) /s (2 m + k ) /s m m ≤ Ck /s k/s m . As a consequence, we get that (cid:104) µ (cid:105) R,r (cid:104) ν (cid:105) R,s = (cid:104) µ (cid:105) R ,r (cid:104) ν (cid:105) R ,s (cid:18) | R || R | (cid:19) /r + 1 /s ≤ C k /s k (1 /r + 1 /s ) m (cid:18) | R || R | (cid:19) /r + 1 /s . (2.2)Since m = k k , by (2.1) and (2.2) we have (cid:104) µ (cid:105) R,r (cid:104) ν (cid:105) R,s ≤ C k /s (cid:18) k (cid:19) (cid:18) | R || R | (cid:19) /r + 1 /s . (2.3)We next split S z into the disjoint union of the subcollections S z ,j = (cid:8) R ∈ S z : 2 − j − | R | ≤ | R | < − j | R | (cid:9) for j = 0 , , . . . We note that each of the rectangles R in S z ,j is contained inΩ j := (cid:26) z ∈ R : M z ,d ( χ [0 , )( z ) > j (cid:27) . The weak-type estimate of the Zygmund maximal function ( L log L → L , ∞ , see [4]) gives | Ω j | ≤ C j j . Finally, by (2.3) and 1 /r + 1 /s > (cid:88) R ∈ S z (cid:104) µ (cid:105) R,r (cid:104) ν (cid:105) R,s | R | = ∞ (cid:88) j =0 (cid:88) R ∈ S z ,j (cid:104) µ (cid:105) R,r (cid:104) ν (cid:105) R,s | R |≤ C k /s (cid:18) k (cid:19) ∞ (cid:88) j =0 j (1 /r + 1 /s ) (cid:88) R ∈ S z ,j | R | YGMUND TYPE AND FLAG TYPE MAXIMAL FUNCTIONS 11 ≤ C k /s (cid:18) k (cid:19) ∞ (cid:88) j =0 | Ω j | η j (1 /r + 1 /s ) ≤ Cη k /s (cid:18) k (cid:19) ∞ (cid:88) j =0 j j j (1 /r + 1 /s ) ≤ Cη k /s (cid:18) k (cid:19) , as required.The proof of Theorem 2.1 is complete. (cid:3) Flag dyadic structure
Now we shall make some observations about the construction of the sets P and Z in [1],which, together with the appropriate modifications regarding the exponents r and s , willlead us to an immediate proof of Theorem 1.2.The construction of the set P is based on dyadic cubes in the plane and is compatiblewith the flag dyadic structure considered in this paper. So, we pick up the finite sum ofpoint masses µ associated with this same P .In order to get the set Z for fixed k and m , the authors of [1] first consider dyadicrectangles R of measure 2 − m − . Then they choose special points of these rectangles toassemble Z (see Lemma 3.2 in [1]) and then prove that (cid:93) ( R ∩ Z ) ≤ Ck (see Lemma 3.3 in[1]). Now, we can keep only those rectangles R compatible with the flag dyadic structure Figure 5.
Schematic diagram indicating possible examples of the locationsof the points in Z fixed a point in P .considered here and so we can build the set Z flag with the obvious modifications on the constants involved. After that, we take the finite sum of point masses ν associated withthis new set Z flag . Finally, the proof of Theorem 1.2 can be deduced following the stepspreviously carried out for the dyadic maximal function M z ,d in Section 2.4. The strong dyadic maximal function and Sparse domination
In this section, we give a sketch of the proof of Theorem 1.3. First, we modify theproof of Proposition 2.6 in [1], properly introducing the L r -average and L s -average as wehave done in the proof of Theorem 2.1 (b). Then, following the procedure of the proof ofTheorem 1.1, one can conclude Theorem 1.3 in the biparameter setting.Next, we modify the proof of Theorem 4.5 in [1], properly introducing again the L r -average and L s -average. Then, using the previous step, one can conclude Theorem 1.3 forthe full multiparameter setting. Acknowledgement
The authors would like to thank Jill Pipher, Guillermo Rey, andYumeng Ou for introducing us to the paper [1] and for many helpful discussions.
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Guillermo J. Flores, FI Univ. Cat´olica de C´ordoba, Av. Armada Argentina 3555 CP:X5016DHK & CIEM-FAMAF Univ. Nacional de C´ordoba, Av. Medina Allende s/n CiudadUniv. CP:X5000HUA, C´ordoba, Argentina.
E-mail address : [email protected] Ji Li, Department of Mathematics, Macquarie University, NSW, 2109, Australia.
E-mail address : [email protected] Lesley A. Ward, School of Information Technology and Mathematical Sciences, Uni-versity of South Australia, Mawson Lakes SA 5095, Australia.
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