Geodesic
Riesz means of Fourier transforms and Fourier series on Hardy spaces
Studia Mathematica, Tome 131 (1998) no. 3, pp. 253-270
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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 $H_p(ℝ)$ 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 
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     title = {Riesz means of {Fourier} transforms and {Fourier} series on {Hardy} spaces},
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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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