# 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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topWeisz, 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 -

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