Some summability methods for power series of functions in $H^p(D^n)$, $0<p<\infty$

Let $H^p(D^n)$ be a Hardy space in the unit polydisc
$$ D^n=\{z\in\mathbb C^n:|z_j|<1,\,j=1,\dots,n\} $$
and let
$$ R^{l,\alpha}_\varepsilon(f,e^{i\theta})=\sum_k(1-(\varepsilon|k|)^l)^\alpha_+\widehat f_ke^{ik\theta}, \qquad l>0, \quad \alpha>0, $$
be the generalized Riesz means of a function $f\in H^p(D^n)$. For certain standard relations between $p$, $l$, $n$ and $\alpha$ the following estimate is established:
$$ C_1(\alpha,l,p)\widetilde{\omega}_l(\varepsilon,f)_p \le\bigl\|f(e^{i\theta})-R_\varepsilon^{l,\alpha}(f,e^{i\theta})\bigr\|_p \le C_2(\alpha,l,p)\omega_l(\varepsilon,f)_p, $$
where $\widetilde\omega_l(\varepsilon,f)_p$ and $\omega_l(\varepsilon,f)_p$ are integral moduli of continuity of order $l$.
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