Two-Dimensional Martingale Transforms and Their Applications in Summability of Walsh–Fourier Series

Ushangi Goginava; Károly Nagy

doi:10.1007/s12220-023-01307-9

Two-Dimensional Martingale Transforms and Their Applications in Summability of Walsh–Fourier Series

Ushangi Goginava
, Károly Nagy

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

In the paper, we are going to prove that the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh–Fourier series is bounded from Llog L((0 , 1] × (0 , 1]) to L₁_,_∞((0 , 1] × (0 , 1]) . As a consequence, it can be obtained that Llog L((0 , 1] × (0 , 1]) is maximal Orlicz space, where the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh–Fourier series for the functions from this space converge in measure.

Original language	English
Article number	245
Journal	Journal of Geometric Analysis
Volume	33
Issue number	8
DOIs	https://doi.org/10.1007/s12220-023-01307-9
Publication status	Published - Aug 2023

Keywords

Dyadic martingale Hardy space
Marcinkiewicz means
Martingale transform
Nörlund logarithmic mean
Walsh system

ASJC Scopus subject areas

Geometry and Topology

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10.1007/s12220-023-01307-9

Cite this

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title = "Two-Dimensional Martingale Transforms and Their Applications in Summability of Walsh{\textendash}Fourier Series",

abstract = "In the paper, we are going to prove that the N{\"o}rlund logarithmic means of quadratic partial sums of two-dimensional Walsh{\textendash}Fourier series is bounded from Llog L((0 , 1] {\texttimes} (0 , 1]) to L1,∞((0 , 1] {\texttimes} (0 , 1]) . As a consequence, it can be obtained that Llog L((0 , 1] {\texttimes} (0 , 1]) is maximal Orlicz space, where the N{\"o}rlund logarithmic means of quadratic partial sums of two-dimensional Walsh{\textendash}Fourier series for the functions from this space converge in measure.",

keywords = "Dyadic martingale Hardy space, Marcinkiewicz means, Martingale transform, N{\"o}rlund logarithmic mean, Walsh system",

author = "Ushangi Goginava and K{\'a}roly Nagy",

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year = "2023",

month = aug,

doi = "10.1007/s12220-023-01307-9",

language = "English",

volume = "33",

journal = "Journal of Geometric Analysis",

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T1 - Two-Dimensional Martingale Transforms and Their Applications in Summability of Walsh–Fourier Series

AU - Goginava, Ushangi

AU - Nagy, Károly

PY - 2023/8

Y1 - 2023/8

N2 - In the paper, we are going to prove that the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh–Fourier series is bounded from Llog L((0 , 1] × (0 , 1]) to L1,∞((0 , 1] × (0 , 1]) . As a consequence, it can be obtained that Llog L((0 , 1] × (0 , 1]) is maximal Orlicz space, where the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh–Fourier series for the functions from this space converge in measure.

AB - In the paper, we are going to prove that the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh–Fourier series is bounded from Llog L((0 , 1] × (0 , 1]) to L1,∞((0 , 1] × (0 , 1]) . As a consequence, it can be obtained that Llog L((0 , 1] × (0 , 1]) is maximal Orlicz space, where the Nörlund logarithmic means of quadratic partial sums of two-dimensional Walsh–Fourier series for the functions from this space converge in measure.

KW - Dyadic martingale Hardy space

KW - Marcinkiewicz means

KW - Martingale transform

KW - Nörlund logarithmic mean

KW - Walsh system

UR - https://www.scopus.com/pages/publications/85159708955

UR - https://www.scopus.com/pages/publications/85159708955#tab=citedBy

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DO - 10.1007/s12220-023-01307-9

M3 - Article

AN - SCOPUS:85159708955

SN - 1050-6926

VL - 33

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 8

M1 - 245

ER -

Two-Dimensional Martingale Transforms and Their Applications in Summability of Walsh–Fourier Series

Abstract

Keywords

ASJC Scopus subject areas

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