Sharp weak type estimates for a family of Soria bases
aa r X i v : . [ m a t h . C A ] J a n SHARP WEAK TYPE ESTIMATES FOR A FAMILY OF SORIA BASES
DMITRY DMITRISHIN, PAUL HAGELSTEIN, AND ALEX STOKOLOS
Abstract.
Let B be a collection of rectangular parallelepipeds in R whose sides are parallelto the coordinate axes and such that B contains parallelepipeds with side lengths of the form s, N s , t , where s, t > N lies in a nonempty subset S of the natural numbers. We showthat if S is an infinite set, then the associated geometric maximal operator M B satisfies theweak type estimate (cid:12)(cid:12)(cid:8) x ∈ R : M B f ( x ) > α (cid:9)(cid:12)(cid:12) ≤ C ˆ R | f | α (cid:18) + | f | α (cid:19) but does not satisfy an estimate of the form (cid:12)(cid:12)(cid:8) x ∈ R : M B f ( x ) > α (cid:9)(cid:12)(cid:12) ≤ C ˆ R φ (cid:18) | f | α (cid:19) for any convex increasing function φ : [0 , ∞ ) → [0 , ∞ ) satisfying the conditionlim x →∞ φ ( x ) x (log(1 + x )) = 0 . Introduction
This paper is concerned with sharp weak type estimates for a class of maximal operatorsnaturally arising from work surrounding the so-called Zygmund conjecture in multiparameterharmonic analysis. Let us recall that the strong maximal operator M is defined on L ( R n )by M f ( x ) = sup x ∈ R | R | ˆ R | f | , where the supremeum is over all rectangular parallelepipeds in R n containing x whose sidesare parallel to the coordinate axes. An important inequality associated to the strong maximaloperator is |{ x ∈ R n : M f ( x ) > α }| ≤ C n ˆ R n | f | α + | f | α ! n − . This inequality may be found in de Guzm´an [5, 6] (see also the related paper [3] of A. C´ordobaand R. Fefferman as well as the paper [1] of Capri and Fava) and may be used to provide
Mathematics Subject Classification.
Primary 42B25.
Key words and phrases. maximal functions, differentiation basis.P. H. is partially supported by a grant from the Simons Foundation ( a proof of the classical
Jessen-Marcinkiewicz-Zygmund Theorem [8], which tells us that theintegral of any function in L (log + L ) n − ( R n ) is strongly differentiable.Now, the strong maximal operator in R n is associated to an n -parameter basis of rectan-gular parallelepipeds. It is natural to consider weak type estimates for maximal operators in R n associated to k -parameter bases. The Zygmund Conjecture in this regard is the following:
Conjecture 1 (Zygmund Conjecture; now disproven) . Let B be a collection of rectangularparallelepipeds in R n whose sides are parallel to the coordinate axes and whose sidelengthsare of the form φ ( t , . . . , t k ) , . . . , φ n ( t , . . . , t k )where the functions φ i are nonnegative and increasing in each variable separately. Define theassociated maximal operator M B by M B f ( x ) = sup x ∈ R ∈B | R | ˆ R | f | . Then M B satisfies the weak type estimate(1.1) |{ x ∈ R n : M B f ( x ) > α }| ≤ C n ˆ R n | f | α + | f | α ! k − . This conjecture was disproven by Soria in [9]. That being said, it does hold in manyimportant cases. For example, A. C´ordoba proved in [2] that the Zygmund Conjecture holdsin the case that B consists of rectangular parallelepipeds in R with sides parallel to thecoordinate axes and whose sidelengths are of the form s, t, φ ( s, t ), where φ is nonnegativeand increasing in the variables s, t separately. Of particular interest to us in this paper is thefollowing extension of C´ordoba’s result due to Soria in [9]: Proposition 1.
Let B be a collection of rectangular parallelepipeds in R whose sides areparallel to the coordinate axes. Furthermore, suppose that, given a parallelepided R in B ofsidelengths r , r , r and another parallelepided R ′ in B of sidelengths r ′ , r ′ , r ′ , if r > r ′ ,then either r > r ′ or r > r ′ . Then (cid:12)(cid:12)(cid:12)n x ∈ R : M B f ( x ) > α o(cid:12)(cid:12)(cid:12) ≤ C ˆ R | f | α + | f | α ! . Note that this proposition encompasses bases that can be quite different in character thanthe ones consider by C´ordoba. In particular, in [9] Soria mentions as an example the basisof parallelepipeds with sidelengths of the form s, t, t .At this point we introduce another strand of research associated to Zygmund’s Conjecture.It is natural to consider, given a translation invariant basis B of rectangular parallelepipeds,whether or not the sharp weak type estimate associated to M B must be of the form |{ x ∈ R n : M B f ( x ) > α }| ≤ C n ˆ R n | f | α + | f | α ! k − HARP WEAK TYPE ESTIMATES FOR A FAMILY OF SORIA BASES 3 for some integer ≤ k ≤ n . In [10], Stokolos proved the following: Proposition 2.
Let B be a translation invariant basis of rectangles in R whose sides areparallel to the coordinate axes. If B does not satisfy the weak type (1 , estimate |{ x ∈ R : M B f ( x ) > α }| ≤ C ˆ R | f | α then M B satisfies the weak type estimate (cid:12)(cid:12)(cid:12)n x ∈ R : M B f ( x ) > α o(cid:12)(cid:12)(cid:12) ≤ C ˆ R | f | α + | f | α ! but does not satisfy a weak type estimate of the form |{ x ∈ R : M B f ( x ) > α }| ≤ C ˆ R φ | f | α ! for any nonnegative convex increasing function φ such that φ ( x ) = o ( x log x ) as x tends toinfinity. In essence, this proposition tells us that, if B is a translation invariant basis of rectanglesin R whose sides are parallel to the coordinate axes, then the optimal weak type estimatefor M B must be inequality 1.1 for k = 1 or k = 2. Optimal weak type estimates of this formwhen, say, k = are ruled out. The proof of Stokolos’ result is very delicate and involvesthe idea of crystallization that we will return to.It is of interest that Proposition 2 has at the present time never been extended to encom-pass translation invariant bases consisting of (some, but not all) rectangular parallelepipedsin dimensions 3 or higher. In particular, one might expect that the optimal weak type esti-mate for the maximal operator associated to such a basis of parallelepipeds in R would beinequality 1.1 when n = 3 and k is either 1, 2, or 3.The purpose of this paper is, motivated by Propositions 1 and 2 above, to considersharp weak type estimates associated to the translation invariant basis of rectangular paral-lelepipeds in R whose sides are parallel to the coordinate axes and whose sidelengths are ofthe form s, N s , t , where s, t > N lies in a nonempty subset S of the natural numbers.The end result, although not its proof, is strikingly straightforward and is stated as follows: Theorem 1.
Let B be a collection of rectangular parallelepipeds in R whose sides are parallelto the coordinate axes and such that B contains all parallelepipeds with side lengths of theform s, N s , t , where s, t > and N lies in a nonempty subset S of the natural numbers.If S is a finite set, then the associated geometric maximal operator M B satisfies the weaktype estimate of the form (1.2) (cid:12)(cid:12)(cid:12)n x ∈ R : M B f ( x ) > α o(cid:12)(cid:12)(cid:12) ≤ C ˆ R | f | α + | f | α ! DMITRY DMITRISHIN, PAUL HAGELSTEIN, AND ALEX STOKOLOS but does not satisfy an estimate of the form (cid:12)(cid:12)(cid:12)n x ∈ R : M B f ( x ) > α o(cid:12)(cid:12)(cid:12) ≤ C ˆ R φ | f | α ! for any convex increasing function φ : [0 , ∞ ) → [0 , ∞ ) satisfying the condition lim x →∞ φ ( x ) x (log(1 + x )) = 0 . If S is an infinite set, then the associated geometric maximal operator M B satisfies a weaktype estimate of the form (cid:12)(cid:12)(cid:12)n x ∈ R : M B f ( x ) > α o(cid:12)(cid:12)(cid:12) ≤ C ˆ R | f | α + | f | α ! but does not satisfy an estimate of the form (cid:12)(cid:12)(cid:12)n x ∈ R : M B f ( x ) > α o(cid:12)(cid:12)(cid:12) ≤ C ˆ R φ | f | α ! for any convex increasing function φ : [0 , ∞ ) → [0 , ∞ ) satisfying the condition lim x →∞ φ ( x ) x (log(1 + x )) = 0 . The remainder of the paper is devoted to a proof of this theorem. Note that for inequality1.2, it is easily seen that the constant C is at most linearly dependent on the number of ele-ments in S , although the sharp dependence of C on the number of elements of S is potentiallya quite difficult issue that we do not treat here. The primary content of the above theoremis the sharpness of the weak type estimate of M B in the case that S is infinite. In harmonicanalysis we typically show that an optimal weak type estimate on a maximal operator is sharpby testing the operator on a bump function or the characteristic function of a small intervalor rectangular parallelepiped. This can be done, for instance, with the Hardy-Littlewoodmaximal operator, the strong maximal operator, or even the maximal operator associated torectangles whose sides are parallel to the axes with sidelengths of the form t, t [9]. However,in dealing with maximal operators associated to rare bases of the type featured in Theorem1, such simple functions do not provide examples illustrating the sharpness of the optimalweak type results, and more delicate constructions such as will be seen here are needed.We remark that a recent paper of D’Aniello and Moonens [4] also treats the subject oftranslation invariant rare bases; in particular they provide sufficient conditions on a rarebasis B for the estimate 1.1 to be sharp when k = n . However, certain bases covered inTheorem 1 (such as when S = { m m : m ∈ N } ) do not fall into the scope of those considered HARP WEAK TYPE ESTIMATES FOR A FAMILY OF SORIA BASES 5 in their paper, although the interested reader is strongly encouraged to consult it.
Acknowledgment:
We wish to thank Ioannis Parissis as well as the referees for theirhelpful comments and suggestions regarding this paper.2.
Crystallization and Preliminary Weak Type Estimates
In this section, we shall introduce a collection of two-dimensional “crystals” that we willuse to prove Theorem 1. We remark that similar types of crystalline structures were used byStokolos in [10, 11, 12] as well as by Hagelstein and Stokolos in [7].Let m < m < · · · be an increasing sequence of natural numbers. We may associate tothis sequence and any k ∈ N a set in [0 , m k ] denoted by Y { m j } kj =1 defined by Y { m j } kj =1 = t ∈ [0 , m k ] : k X j =1 r (cid:18) t m j (cid:19) = k . Here r ( t ) denotes the standard Rademacher function defined on [0 ,
1) by r ( t ) = χ [0 , ] ( t ) − χ ( , ( t )and extended to be 1-periodic on R .Note that µ ( Y { m j } kj =1 ) = 2 − k m k . Associated to the set Y { m j } kj =1 is the crystal Q { m j } kj =1 ⊂ [0 , m k ] × [0 , m k ] defined by Q { m j } kj =1 = Y { m j } kj =1 × Y { m j } kj =1 . Note µ ( Q { m j } kj =1 ) = 2 − k m k . Here µ j refers to the Lebesgue measure on R j .We also associate to { m j } kj =1 the geometric maximal operator M { m j } kj =1 defined on L loc ( R )by M { m j } kj =1 f ( x ) = sup x ∈ R | R | ˆ R | f | , where the supremum is over all rectangles in R containing x whose sides are parallel to thecoordinate axes with areas in the set { m , . . . , m k } .In the case that the context is clear, we may refer to the set Y { m j } kj =1 simply as Y k , the set Q { m j } kj =1 simply as Q k , and the maximal operator M { m j } kj =1 simply as M k .A few basic observations regarding the sets Y k and Q k are in order. DMITRY DMITRISHIN, PAUL HAGELSTEIN, AND ALEX STOKOLOS
First, note that Y k +1 is a disjoint union of mk +1 − mk copies of Y k . In fact, defining thetranslation τ s E of a set E in R by χ τ s E ( x ) = χ E ( x − s ), we have Y k +1 = mk +1 − mk − [ l =0 τ l mk Y k . Furthermore, by induction we see that if 1 ≤ r ≤ k we have Y k +1 is a disjoint union of2 m k +1 − m k · m k − m k − · · · m r +1 − m r = 2 m k +1 − m r − k + r − copies of Y r , with Y k +1 = [ ( lr,...,lk )0 ≤ li ≤ mi +1 − mi − − τ l r mr τ l r +1 mr +1 · · · τ l k mk Y r . We also remark that the average of χ Y k over [0 , m j ] is exactly 2 − j , and moreover the averageof χ Y k over any translate τ l j mj τ l j +1 mj +1 · · · τ l k − mk − [0 , m j ] with 0 ≤ l i ≤ m i +1 − m i − − − j . Observe that the number of such translates is2 m j +1 − m j − · m j +2 − m j +1 − · · · m k − m k − − = 2 m k − m j + j − k . We now consider how M k acts on χ Q k . We will do so in the special case that, for 1 ≤ j ≤ k we have that m k − j ≤ m k − j +1 − m j . (This will be the case if the m j increase rapidly in j , forexample if m j +1 ≥ m j for all j .)Fix now 1 ≤ j ≤ k . We are going to show that there exist2 m k − m k − j +1 + m j − j · m k − m j − k + j = 2 m k − m k − j +1 − k pairwise a.e. disjoint rectangles with sides parallel to the coordinate axes in [0 , m k ] × [0 , m k ]whose areas are all 2 m k − j +1 and such that the average of χ Q k over each of these rectanglesis 2 − k . Moreover, each of these rectangles will be a translate of [0 , m j ] × [0 , m k − j +1 − m j ].Accordingly, the measure of the union of these rectangles will be 2 m k − k .We have already indicated above that the average of χ Y k over each of 2 m k − m j − k + j pair-wise a.e. disjoint translates of [0 , m j ] is 2 − j . Somewhat more technically, we now need toprove that the average of χ Y k over 2 m k − m k − j +1 + m j − j pairwise a.e. disjoint intervals of length2 m k − j +1 − m j is equal to 2 j − k .Note that the average of χ Y k over [0 , m k − j ] is 2 j − k as well as any translate τ [0 , m k − j ] ofthis interval where τ is of the form l · m k − j for 0 ≤ l ≤ m k − j +1 − m j − m k − j − I := [0 , m k − j +1 − m j ] over which the average of χ Y k is 2 j − k . It isespecially important to recognize here that Y k ∩ [0 , m k − j +1 − ] = mk − j +1 − mk − j − [ i =0 τ i mk − j Y k − j , HARP WEAK TYPE ESTIMATES FOR A FAMILY OF SORIA BASES 7 where the latter is a pairwise a.e. disjoint union. It is here that we need the condition that m k − j ≤ m k − j +1 − m j , so that [0 , m k − j +1 − m j ] can be tiled by pairwise a.e. disjoint intervalsof length 2 m k − j over which the average of χ Y k − j is 2 j − k .Now, [0 , m k ] contains many pairwise a.e. disjoint translates of I ∩ Y k , each of whom beingcontained in a collection of translates of I that are themselves pairwise a.e. disjoint; we countthem here. The number of translates is the number of pairwise a.e. disjoint translates of I whose union is the left half of [0 , m k − j +1 ] (which is 2 m k − j +1 − − m k − j +1 + m j = 2 m j − ) timesthe number of translates of Y k − j +1 needed to form Y k (which is 2 m k − m k − j +1 − k +( k − j +1) =2 m k − m k − j +1 − j +1 .) Hence the total number of translates is2 m j − · m k − m k − j +1 − j +1 = 2 m j + m k − m k − j +1 − j . Hence, Y k contains 2 m j + m k − m k − j +1 − j pairwise a.e. disjoint intervals of length 2 m k − j +1 − m j over each of which the average of χ Y k is 2 j − k . As we have already shown that the averageof χ Y k over each of 2 m k − m j − k + j pairwise a.e. disjoint translates of [0 , m j ] is 2 − j , we havethen that there exist 2 m j + m k − m k − j +1 − j · m k − m j − k + j = 2 m k − m k − j +1 − k pairwise a.e. disjointrectangles in [0 , m k ] × [0 , m k ] of size 2 m k − j +1 − m j · m j = 2 m k − j +1 over each of which theaverage of χ Q k is 2 − j · j − k = 2 − k . Note the measure of the union of these rectangles is2 m k − m k − j +1 − k · m k − j +1 = 2 m k − k . We come now to a crucial observation. By the construction of Y k , any dyadic intervalof length 2 m j is at most only half filled by the translates of intervals of length 2 m j − suchthat the union of those translates acting on Y j − is Y j . Accordingly, the union of the above2 m k − m k − j +1 − k pairwise a.e. disjoint rectangles in [0 , m k ] × [0 , m k ] of size 2 m k − j +1 over each ofwhich the average of χ Q k is 2 − k is at most only half filled by the corresponding set of rectanglesof size 2 m k − ( j − . Hence the union of all the rectangles R in [0 , m k ] × [0 , m k ] whose sides areparallel to the coordinate axes and of area in the set { m k − j : j = 1 , . . . , ⌈ k ⌉} and such thatthe average of χ Q k over R is greater than or equal to 2 − k must exceed · k · m k − k = k m k − k . This series of observations leads to the proof of the following lemma.
Lemma 1.
Let the geometric maximal operator M { m j } kj =1 and the set Q { m j } kj =1 be defined asabove. Suppose for ≤ j ≤ k we have that m k − j ≤ m k − j +1 − m j . Then µ x ∈ [0 , m k ] × [0 , m k ] : M { m j } kj =1 χ Q { mj } kj =1 ( x ) ≥ − k ≥ k m k − k = 18 k − k µ (cid:18) Q { m j } kj =1 (cid:19) . Proof of Theorem 1
Proof of Theorem 1.
Let B be a collection of rectangular parallelepipeds in R whose sidesare parallel to the coordinate axes and such that B contains parallelepipeds with side lengthsof the form s, N s , t , where t > S is a nonempty set consisting of natural numbers. DMITRY DMITRISHIN, PAUL HAGELSTEIN, AND ALEX STOKOLOS If S is a finite set, then the associated geometric maximal operator M B is comparableto the maximal operator averaging over rectangular parallelepipeds with side lengths of theform s, s , t . In [9], Soria showed that this operator maps L (1 + log + L )( R ) continuously intoweak L ( R ) but does not map any larger Orlicz class into weak L ( R ). So Theorem 1 holdsin this case.Suppose now S is an infinite set. Note that the maximal operator M B is dominated by thestrong maximal operator in R , so the weak type estimate (cid:12)(cid:12)(cid:12)n x ∈ R : M B f ( x ) > α o(cid:12)(cid:12)(cid:12) ≤ C ˆ R | f | α + | f | α ! automatically holds.Since S is an infinite set, there exists a subset { m j } ∞ j =1 of S satisfying the condition that2 m j ≤ m j +1 for all j . So the hypothesis of Lemma 1 holds for { m j } kj =1 for all k .For each natural number k , we let Z k ⊂ [0 , m k ] × [0 , m k ] × [0 , k ] be defined by Z k = Q k × [0 , . To show the estimate (cid:12)(cid:12)(cid:12)n x ∈ R : M B f ( x ) > α o(cid:12)(cid:12)(cid:12) ≤ C ˆ R φ | f | α ! does not hold for any convex increasing function φ : [0 , ∞ ) → [0 , ∞ ) satisfying the conditionlim x →∞ φ ( x ) x (log(1 + x )) = 0 , it suffices to show that µ (cid:16)n x ∈ [0 , m k ] × [0 , m k ] × [0 , k ] : M B χ Z k ( x ) ≥ − k o(cid:17) ≥ k − k µ ( Z k ) . Fix 1 ≤ r ≤ k . Note that, just as Y k is a disjoint union of 2 m k − m r − k + r copies of Y r , wehave that Q k is a disjoint union of 2 m k − m r − k + r ) copies of Q r , with each of these copies beingcontained in pairwise a.e. disjoint squares of sidelength 2 m r . By Lemma 1, for each one ofthese squares ˜ Q , µ (cid:16)n x ∈ ˜ Q : M r χ ˜ Q ∩ Q k ( x ) ≥ − r o(cid:17) ≥ r m r − r . Note each of the rectangles associated to the maximal operator M r has sidelength in theset { m , . . . , m r } ⊂ { m , . . . , m k } and hence for any of these rectangles R the associatedparallelepiped R × [0 , k − r ] lies in the basis B . Note that if1 µ ( R ) ˆ R χ ˜ Q ∩ Q k ≥ − r , HARP WEAK TYPE ESTIMATES FOR A FAMILY OF SORIA BASES 9 then 1 µ ( R × [0 , k − r ]) ˆ R × [0 , k − r ] χ Q k × [0 , ≥ − r r − k = 2 − k . Taking into account only the top half of these parallelepipeds, for any one of the abovesquares ˜ Q we obtain µ (cid:16)n x ∈ [0 , m k ] × [0 , m k ] × [2 k − r − , k − r ] : M B χ Z k ( x ) ≥ − k o(cid:17) ≥ m k − m r − k + r ) µ (cid:16)n x ∈ ˜ Q : M r χ ˜ Q ∩ Q k ( x ) ≥ − r o(cid:17) · k − r − ≥ m k − m r − k + r ) r m r − r · k − r − = r
16 2 m k − k . We now take advantage of the fact that, for different values of r , the sets[0 , m k ] × [0 , m k ] × [2 k − r − , k − r ] are pairwise a.e. disjoint. In particular, we have µ (cid:16)n x ∈ [0 , m k ] × [0 , m k ] × [0 , k ] : M B χ Z k ( x ) ≥ − k o(cid:17) ≥ k X r =1 µ (cid:16)n x ∈ [0 , m k ] × [0 , m k ] × [2 k − r − , k − r ] : M B χ Z k ( x ) ≥ − k o(cid:17) ≥ k X r =1 r
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Email address : [email protected] P. H.: Department of Mathematics, Baylor University, Waco, Texas 76798
Email address : paul [email protected] A. S.: Department of Mathematical Sciences, Georgia Southern University, Statesboro,Georgia 30460
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