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On the maximal operators of weighted Marcinkiewicz type means of two-dimensional Walsh–Fourier series

  • Károly Nagy EMAIL logo and Mohamed Salim
Published/Copyright: October 10, 2021
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 29 Issue 1

Abstract

Goginava proved that the maximal operator σ α , * ( 0 < α < 1 ) of two-dimensional Marcinkiewicz type ( C , α ) means is bounded from the two-dimensional dyadic martingale Hardy space H p ⁢ ( G 2 ) to the space L p ⁢ ( G 2 ) for p > 2 2 + α . Moreover, he showed that assumption p > 2 2 + α is essential for the boundedness of the maximal operator σ α , * . It was shown that at the point p 0 = 2 2 + α the maximal operator σ α , * is bounded from the dyadic Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) to the space weak- L 2 / ( 2 + α ) ⁢ ( G 2 ) . The main aim of this paper is to investigate the behaviour of the maximal operators of weighted Marcinkiewicz type σ α , * means ( 0 < α < 1 ) in the endpoint case p 0 = 2 2 + α . In particular, the optimal condition on the weights is given which provides the boundedness from H 2 / ( 2 + α ) ⁢ ( G 2 ) to L 2 / ( 2 + α ) ⁢ ( G 2 ) . Furthermore, a strong summation theorem is stated for functions in the dyadic martingale Hardy space H 2 / ( 2 + α ) ⁢ ( G 2 ) .

Keywords: Walsh–Paley system; maximal operator; Cesàro mean; Marcinkiewicz mean; Hardy space; bounded operator; strong summation
MSC 2010: 42C10; 42C10

Funding source: United Arab Emirates University

Award Identifier / Grant number: G00002599

Funding statement: The research was supported by project UAEU UPAR 2017 Grant G00002599.

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Received: 2020-06-25
Revised: 2020-11-02
Accepted: 2020-11-09
Published Online: 2021-10-10
Published in Print: 2022-02-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

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  3. On a Robin type problem involving 𝑝(𝑥)-Laplacian operator
  4. On the splitting type of holomorphic vector bundles induced from regular systems of differential equation
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  8. On the maximal operators of weighted Marcinkiewicz type means of two-dimensional Walsh–Fourier series
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https://doi.org/10.1515/gmj-2021-2109
Keywords for this article
Walsh–Paley system; maximal operator; Cesàro mean; Marcinkiewicz mean; Hardy space; bounded operator; strong summation

Articles in the same Issue

  1. Frontmatter
  2. The local formula of representation of a solution for a functional differential equation with the mixed initial condition considering perturbations of delays containing in the phase coordinates and in controls
  3. On a Robin type problem involving 𝑝(𝑥)-Laplacian operator
  4. On the splitting type of holomorphic vector bundles induced from regular systems of differential equation
  5. On inclusion properties of discrete Morrey spaces
  6. Infinitely many solutions for nonlocal elliptic systems in Orlicz–Sobolev spaces
  7. Critical points approaches to a nonlocal elliptic problem driven by 𝑝(𝑥)-biharmonic operator
  8. On the maximal operators of weighted Marcinkiewicz type means of two-dimensional Walsh–Fourier series
  9. Several characterizations of Bessel functions and their applications
  10. The incomplete exponential pRp (α,β;z) function with applications
  11. Higher integrability and reverse Hölder inequalities in the limit cases
  12. When are multiplicative semi-derivations additive?
  13. On a structure of the set of positive solutions to second-order equations with a super-linear non-linearity
  14. Modulus of continuity and convergence of subsequences of Vilenkin–Fejér means in martingale Hardy spaces
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