On differentiation of integrals\cr in the infinite-dimensional torus

Kosz, Dariusz

doi:10.4064/sm191001-10-2

Cited by 10 publications

(7 citation statements)

References 13 publications

(20 reference statements)

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“…Precisely, M R0 is not of strong-type (1, 1) but it is of weak-type (1, 1), as was proven in [FR20] using the classical martingaletype argument, hence it is of strong-type (p, p) for all p ∈ (1, ∞]. On the other hand, M R does not satisfy any strong-type (or weak-type) (p, p) inequality for any finite p, as was shown in [Ko21].…”

Section: Introductionmentioning

confidence: 68%

“…By defining a suitable "dyadic" basis (the restricted Rubio de Francia basis, denoted by R 0 ), they proved that the corresponding "dyadic" Hardy-Littlewood maximal function satisfies the weak type (1, 1) inequality. Soon thereafter, the first author [Kos21] showed that the maximal function associated with a wider natural basis (the Rubio de Francia basis, denoted by R) is not of weak type (1, 1). See Subsection 2.3 for the definitions of R 0 and R.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…We will call their elements cubes, since they will play the same role as the cubes in the Euclidean setting. Although the maximal function related to this basis R, as recently showed by the first author [Kos21], is not of weak type (1, 1), we can find inside this class R a subclass R 0 (the restricted Rubio de Francia basis) of base sets which play the role of dyadic cubes, and so we will give the name of dyadic cubes to its elements. More precisely, we will consider the family of dyadic intervals in T ω introduced by Rubio de Francia and defined in [FR20], from which we will borrow the notation (see also the Ph.D. dissertation [Fer19]).…”

Section: Maximal Operator and Weightsmentioning

confidence: 88%

“…The infinite-dimensional torus T ω is the product of countably many copies of the one-dimensional torus T. Motivated by the interest in the study of Fourier analysis on T ω pointed out by Rubio de Francia [RdF78,RdF80], in recent times there has been an effort to understand how the fundamental tools of classical harmonic analysis look like in the setting of T ω [FR20, Ko21,Ko22,KMPRR22]. As noticed in [Ko22, Theorem 1.1], by the results of [CW71], [PS09], and [Ed98], topologically infinite-dimensional spaces cannot satisfy the doubling condition, which rules out the application of the general theory of spaces of homogeneous type in the Coifman-Weiss [CW71] sense to the objects defined on T ω .…”

Section: Introductionmentioning

confidence: 99%

“…Recent research has focused on obtaining an analogue of the Calderón-Zygmund decomposition and the introduction of a suitable "dyadic" basis R 0 and a "dyadic" Hardy-Littlewood maximal function M R0 [FR20] which satisfies the weak-type (1, 1) inequality. In [Ko21] it was shown that M R does not satisfy any restricted weak-type (p, p) estimate, for p ∈ [1, ∞), being R the wider natural basis, also introduced in [FR20]. The intermediate case was studied in [KMPRR22], where bases R 0 ⊂ S ⊂ R were found for which the corresponding maximal functions present a satisfactory control on Lebesgue spaces.…”

Section: Introductionmentioning

confidence: 99%

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Maximal operators on the infinite-dimensional torus

et al. 2022

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We study maximal operators related to bases on the infinitedimensional torus T ω . For the normalized Haar measure dx on T ω it is known that M R 0 , the maximal operator associated with the dyadic basis R 0 , is of weak type (1, 1), but M R , the operator associated with the natural general basis R, is not. We extend the latter result to all q ∈ [1, ∞). Then we find a wide class of intermediate bases R 0 ⊂ R ⊂ R, for which maximal functions have controlled, but sometimes very peculiar behavior. Precisely, for given q 0 ∈ [1, ∞) we construct R such that M R is of restricted weak type (q, q) if and only if q belongs to a predetermined range of the form (q 0 , ∞] or [q 0 , ∞]. Finally, we study the weighted setting, considering the Muckenhoupt A R p (T ω ) and reverse Hölder RH R r (T ω ) classes of weights associated with R. For each p ∈ (1, ∞) and each w ∈ A R p (T ω ) we obtain that M R is not bounded on L q (w) in the whole range q ∈ [1, ∞). Since we are able to show thatthe unboundedness result applies also to all reverse Hölder weights.

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Section: Introductionmentioning

confidence: 68%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Maximal Operator and Weightsmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maximal operators on the infinite-dimensional torus

et al. 2022

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Weak-type maximal function estimates on the infinite-dimensional torus

2023

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We prove necessary and sufficient conditions for the weak-$$L^p$$ L p boundedness, for $$p \in (1,\infty )$$ p ∈ ( 1 , ∞ ) , of a maximal operator on the infinite-dimensional torus. In the endpoint case $$p=1$$ p = 1 we obtain the same weak-type inequality enjoyed by the strong maximal function in dimension two. Our results are quantitatively sharp.

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