# Two-parameter Hardy-Littlewood inequalities

Studia Mathematica (1996)

• Volume: 118, Issue: 2, page 175-184
• ISSN: 0039-3223

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## Abstract

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The inequality (*) $\left({\sum }_{|n|=1}^{\infty }{\sum }_{|m|=1}^{\infty }{|nm|}^{p-2}{|f̂\left(n,m\right)|}^{p}{\right)}^{1/p}\le {C}_{p}\parallel ƒ{\parallel }_{{H}_{p}}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space ${H}_{p}$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in ${L}_{p}$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series of an arbitrary function from ${H}_{1}$ converges a.e. and also in ${L}_{1}$ norm to that function.

## How to cite

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Weisz, Ferenc. "Two-parameter Hardy-Littlewood inequalities." Studia Mathematica 118.2 (1996): 175-184. <http://eudml.org/doc/216272>.

@article{Weisz1996,
abstract = {The inequality (*) $(∑_\{|n|=1\}^\{∞\} ∑_\{|m|=1\}^\{∞\} |nm|^\{p-2\} |f̂(n,m)|^p)^\{1/p\} ≤ C_p ∥ƒ∥_\{H_p\}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_p$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in $L_p$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series of an arbitrary function from $H_1$ converges a.e. and also in $L_1$ norm to that function.},
author = {Weisz, Ferenc},
journal = {Studia Mathematica},
keywords = {Hardy spaces; rectangle p-atom; atomic decomposition; Hardy-Littlewood inequalities; two-parameter trigonometric-Fourier coefficients; Hardy space; bidisc},
language = {eng},
number = {2},
pages = {175-184},
title = {Two-parameter Hardy-Littlewood inequalities},
url = {http://eudml.org/doc/216272},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Weisz, Ferenc
TI - Two-parameter Hardy-Littlewood inequalities
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 2
SP - 175
EP - 184
AB - The inequality (*) $(∑_{|n|=1}^{∞} ∑_{|m|=1}^{∞} |nm|^{p-2} |f̂(n,m)|^p)^{1/p} ≤ C_p ∥ƒ∥_{H_p}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_p$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in $L_p$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series of an arbitrary function from $H_1$ converges a.e. and also in $L_1$ norm to that function.
LA - eng
KW - Hardy spaces; rectangle p-atom; atomic decomposition; Hardy-Littlewood inequalities; two-parameter trigonometric-Fourier coefficients; Hardy space; bidisc
UR - http://eudml.org/doc/216272
ER -

## References

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