Two power-weight inequalities for the Hilbert transform on the cones of monotone functions

Степанов, В. Д.; Tikhonov, Sergey

doi:10.1080/17476933.2011.559546

Cited by 11 publications

(4 citation statements)

References 16 publications

(16 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…The subsequent attempts to extend this result to more general cases of summation parameters and weights in L p w have met with serious difficulties (see [42]- [44] for the case p = 2 and different weights). Some progress, however, has been achieved by narrowing down subclasses of L p w (see [45] and [46], for example).…”

Section: A Characterization Of Cesàro-andmentioning

confidence: 99%

Strong and weak associativity of weighted Sobolev spaces of the first order

Stepanov,

Ushakova

2023

Russian Math. Surveys

View full text Add to dashboard Cite

A brief overview of the recent results on the problem of characterization of associative and double associative spaces of function classes, including both ideal and non-ideal structures, is presented. The latter include two-weighted Sobolev spaces of the first order on the positive semi- axis. It is shown that, in contrast to the notion of duality, associativity can be ‘strong’ or ‘weak’. In addition, double associative spaces are further divided into three types. In this context it is established that a weighted Sobolev space of functions with compact support possesses weak associative reflexivity, while the strong associative space of a weak associative space is trivial. Weighted classes of Cesàro and Copson type have similar properties; for these classes the problem us fully investigated, and their connections with Sobolev spaces with power weights are established. As an application, the problem of boundedness of the Hilbert transform from a weighted Sobolev space to a weighted Lebesgue space is considered. Bibliography: 49 titles.

show abstract

Section: A Characterization Of Cesàro-andmentioning

confidence: 99%

Strong and weak associativity of weighted Sobolev spaces of the first order

Stepanov,

Ushakova

2023

Russian Math. Surveys

View full text Add to dashboard Cite

show abstract

“…One of the best sources for the theory of discrete Hilbert transforms and Hardy spaces is [BC98]. For weighted estimates for them, see [And77], [ST11a], [ST11b] and [Lif16b].…”

Section: Absolutely Convergent Fourier Seriesmentioning

confidence: 99%

Re-expansions on compact Lie groups

Akylzhanov¹,

Liflyand²,

Ruzhansky³

2019

Preprint

View full text Add to dashboard Cite

In this paper we consider the re-expansion problems on compact Lie groups. First, we establish weighted versions of classical re-expansion results in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general compact Lie groups can be formulated as follows: given a function on the maximal torus of a compact Lie group, what conditions on its (toroidal) Fourier coefficients are sufficient in order to have that the group Fourier coefficients of its central extension are summable. We derive the necessary and sufficient conditions for the above property to hold in terms of the root system of the group. Consequently, we show how this problem leads to the re-expansions of even/odd functions on compact Lie groups, giving a necessary and sufficient condition in terms of the discrete Hilbert transform and the root system. In the model case of the group SU(2) a simple sufficient condition is given.

show abstract

“…Это послужило мотивацией для широ-кого поля исследований весовых неравенств на конусах монотонных функций (см., например, [2], [5]- [14], [20], [22]- [37], [41]- [45], [47], [53], [55], [62], [66]- [68], [73], [76], [78], [88]- [91], [94], [99], [103], [104], [110], [111], [112], [117] и др. ).…”

Section: принципы двойственностиunclassified

Редукционные Теоремы Для Весовых Интегральных Неравенств На Конусе Монотонных Функций

Гогатишвили¹,

Степанов²

2013

Uspekhi Mat. Nauk

View full text Add to dashboard Cite

Редукционные теоремы для весовых интегральных неравенств на конусе монотонных функций А. Гогатишвили, В. Д. Степанов В работе дается обзор результатов, связанных с редукцией интеграль-ных неравенств с положительными операторами в весовых пространствах Лебега на вещественной полуоси на конусе монотонных функций к некото-рым неравенствам на конусе неотрицательных функций, для доказатель-ства которых имеется больше возможностей. При этом случай монотон-ных операторов является новым. В качестве приложения для ряда опе-раторов Вольтерра получена полная характеризация при всех возможных параметрах суммирования.Библиография: 118 названий.Ключевые слова: весовое пространство Лебега, конус монотонных функций, весовое интегральное неравенство, принцип двойственности, ограниченные операторы, редукционная теорема.

show abstract

Two power-weight inequalities for the Hilbert transform on the cones of monotone functions

Abstract: The two power-weight (L p u , L q v ) norm inequalities for the Hilbert transform on the cones of monotone functions are characterized in the case 05p q51.

Cited by 11 publications

References 16 publications

Strong and weak associativity of weighted Sobolev spaces of the first order

Strong and weak associativity of weighted Sobolev spaces of the first order

Re-expansions on compact Lie groups

Редукционные Теоремы Для Весовых Интегральных Неравенств На Конусе Монотонных Функций

Contact Info

Product

Resources

About