Abstract
We prove that the maximal Riesz operator $\sigma^{\alpha,\gamma}_*$ is of strong type from $L^1(\R) \cap H^p$ $ (\R)$ to $L^p (\R)$ for $\alpha, \gamma>0$ and $1/(1+\alpha) < p \le 1$, it is of weak type for $\alpha,\gamma>0$ and $1/(1+\alpha) = p$, and these results are best possible. The proofs are based on sharp estimates of the derivatives of the Riesz kernel. We characterize the real Hardy space $H^p(\R)$ in terms of $\sigma^{\alpha,1}_*$ for $1/(1+ \alpha) < p \le 1$, and draw consequences for real Hardy spaces on $\R^2$, as well. For example, an integrable function $f$ belongs to $H^1(\R)$ if and only if the maximal Fej\’er operator $\sigma^{1,1}_*$ applied to $f$ belongs to $L^1(\R)$. We also establish analogous results for real Hardy spaces on $\T$ and $\T^2$.
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Brown, G., Dai, F. & Móricz, F. The Maximal Riesz, Fejér, and Cesàro Operators on Real Hardy Spaces. J. Fourier Anal. Appl. 10, 27–50 (2004). https://doi.org/10.1007/s00041-004-8002-6
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DOI: https://doi.org/10.1007/s00041-004-8002-6