Riesz means of Fourier transforms and Fourier series on Hardy spaces

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 ${\displaystyle H_p(ℝ)}$ to ${\displaystyle L_p(ℝ)}$ (1/(α+1) < p < ∞) and is of weak type (1,1), where ${\displaystyle H_p(ℝ)}$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function ${\displaystyle ⨍\in L_1(ℝ)}$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on ${\displaystyle H_p(ℝ)}$ whenever 1/(α+1) < p < ∞. Thus, in case ${\displaystyle ⨍\in H_p(ℝ)}$, the Riesz means converge to ⨍ in ${\displaystyle H_p(ℝ)}$ norm (1/(α+1) < p < ∞). The same results are proved for the conjugate Riesz means and for Fourier series of distributions.