# Riesz means of Fourier transforms and Fourier series on Hardy spaces

Studia Mathematica (1998)

• Volume: 131, Issue: 3, page 253-270
• ISSN: 0039-3223

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## Abstract

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Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from ${H}_{p}\left(ℝ\right)$ to ${L}_{p}\left(ℝ\right)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where ${H}_{p}\left(ℝ\right)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍\in {L}_{1}\left(ℝ\right)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on ${H}_{p}\left(ℝ\right)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍\in {H}_{p}\left(ℝ\right)$, the Riesz means converge to ⨍ in ${H}_{p}\left(ℝ\right)$ norm (1/(α+1) < p < ∞). The same results are proved for the conjugate Riesz means and for Fourier series of distributions.

## How to cite

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Weisz, Ferenc. "Riesz means of Fourier transforms and Fourier series on Hardy spaces." Studia Mathematica 131.3 (1998): 253-270. <http://eudml.org/doc/216579>.

@article{Weisz1998,
abstract = {Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p(ℝ)$ to $L_p(ℝ)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p(ℝ)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍ ∈ L_1(ℝ)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p(ℝ)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍ ∈ H_p(ℝ)$, the Riesz means converge to ⨍ in $H_p(ℝ)$ norm (1/(α+1) < p < ∞). The same results are proved for the conjugate Riesz means and for Fourier series of distributions.},
author = {Weisz, Ferenc},
journal = {Studia Mathematica},
keywords = {Hardy spaces; p-atom; atomic decomposition; interpolation; Fourier transforms; Riesz means; atom; Fourier transform},
language = {eng},
number = {3},
pages = {253-270},
title = {Riesz means of Fourier transforms and Fourier series on Hardy spaces},
url = {http://eudml.org/doc/216579},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Weisz, Ferenc
TI - Riesz means of Fourier transforms and Fourier series on Hardy spaces
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 3
SP - 253
EP - 270
AB - Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p(ℝ)$ to $L_p(ℝ)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p(ℝ)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍ ∈ L_1(ℝ)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p(ℝ)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍ ∈ H_p(ℝ)$, the Riesz means converge to ⨍ in $H_p(ℝ)$ norm (1/(α+1) < p < ∞). The same results are proved for the conjugate Riesz means and for Fourier series of distributions.
LA - eng
KW - Hardy spaces; p-atom; atomic decomposition; interpolation; Fourier transforms; Riesz means; atom; Fourier transform
UR - http://eudml.org/doc/216579
ER -

## References

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13. [13] F. Weisz, Cesàro summability of one- and two-dimensional trigonometric-Fourier series, Colloq. Math. 74 (1997), 123-133. Zbl0891.42006
14. [14] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994. Zbl0796.60049
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