Geodesic
Riesz means of Fourier transforms and Fourier series on Hardy spaces
Studia Mathematica, Tome 131 (1998) no. 3, pp. 253-270

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/sm-131-3-253-270
Keywords: Hardy spaces, p-atom, atomic decomposition, interpolation, Fourier transforms, Riesz means

Ferenc Weisz 1

1 
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Ferenc Weisz. Riesz means of Fourier transforms and Fourier series on Hardy spaces. Studia Mathematica, Tome 131 (1998) no. 3, pp. 253-270. doi: 10.4064/sm-131-3-253-270

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