The maximal Riesz, Fejér, and Cesàro operators on real Hardy spaces (Q1882638)

For any bounded distribution \(f\) in the real Hardy space \(H^p(\mathbb R)\) \((0<p<\infty)\) the Riesz means can be expressed as \[ \sigma_T^{\alpha,\gamma}(f)=f*K_T^{\alpha,\gamma}\quad (\alpha, \gamma >0),\;T>0 \] with the kernel \[ K_T^{\alpha,\gamma}(u)=\sqrt{2\over \pi}\int_0^T\Bigg(1-\Big({t\over T}\Big)^\gamma\Bigg)^\alpha\cos ut\,dt, \;T>0, u\in\mathbb R. \] The authors show that the maximal operator \(\sigma_*^{\alpha,\gamma}\) is of strong type from \(L^1({\mathbb R})\cap H^p({\mathbb R})\) to \(L^p(\mathbb R)\) for \(\alpha, \gamma>0,\;{1\over 1+\alpha}<p\leqslant 1\) (Theorem 1), and is of weak type for \(\alpha, \gamma>0,\;p={1\over 1+\alpha} \) (Theorem 2). The proofs are based on sharp estimates of the derivatives of the kernel \(K_T^{\alpha,\gamma} \). The results are best possible (Theorem 3). The space \(H^p(\mathbb R)\) is characterized in terms of \(\sigma_*^{\alpha, 1}\) for \({1\over 1+\alpha}<p\leqslant 1\;(\alpha>0)\) (Theorem 4 and Theorem 5). Some consequences for real Hardy spaces on \({\mathbb R}^2\) are obtained (Theorem 6 and Theorem 7). An almost everywhere convergence result of double Fejér means is derived. Analogous results for real Hardy spaces on \(\mathbb T\) and \({\mathbb T}^2\) are also established (Theorem 8 and Theorem 9). Valuable remarks are also given.