# Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

Studia Mathematica (1997)

• Volume: 125, Issue: 3, page 231-246
• ISSN: 0039-3223

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## Abstract

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Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $\left({\sum }_{k=1}^{\infty }{\sum }_{j=1}^{\infty }{|f̂\left(k,j\right)|}^{p}{\left(kj\right)}^{p-2}{\right)}^{1/p}\le {C}_{p}\parallel f{\parallel }_{{H}_{**}^{p}}$ (1/2 < p≤2) where f belongs to the Hardy space ${H}_{**}^{p}\left({G}_{m}×{G}_{s}\right)$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.

## How to cite

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Simon, Péter, and Weisz, Ferenc. "Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients." Studia Mathematica 125.3 (1997): 231-246. <http://eudml.org/doc/216435>.

@article{Simon1997,
abstract = {Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $(∑_\{k=1\}^∞ ∑_\{j=1\}^∞ |f̂(k,j)|^\{p\}(kj)^\{p-2\})^\{1/p\} ≤ C_p∥f∥_\{H^p_\{**\}\}$ (1/2 < p≤2) where f belongs to the Hardy space $H_\{**\}^p (G_m × G_s)$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.},
author = {Simon, Péter, Weisz, Ferenc},
journal = {Studia Mathematica},
keywords = {two-parameter martingales and Hardy spaces; rectangle p-atoms; Vilenkin functions; Hardy-Littlewood inequality; Hardy spaces; double Vilenkin-Fourier series; multiplicative Vilenkin groups; Hardy-Littlewood type inequality; Vilenkin systems},
language = {eng},
number = {3},
pages = {231-246},
title = {Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients},
url = {http://eudml.org/doc/216435},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Simon, Péter
AU - Weisz, Ferenc
TI - Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 3
SP - 231
EP - 246
AB - Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $(∑_{k=1}^∞ ∑_{j=1}^∞ |f̂(k,j)|^{p}(kj)^{p-2})^{1/p} ≤ C_p∥f∥_{H^p_{**}}$ (1/2 < p≤2) where f belongs to the Hardy space $H_{**}^p (G_m × G_s)$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.
LA - eng
KW - two-parameter martingales and Hardy spaces; rectangle p-atoms; Vilenkin functions; Hardy-Littlewood inequality; Hardy spaces; double Vilenkin-Fourier series; multiplicative Vilenkin groups; Hardy-Littlewood type inequality; Vilenkin systems
UR - http://eudml.org/doc/216435
ER -

## References

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