On Ren-Kähler's Paper “Hardy-Littlewood Inequalities and $Q_p$-Spaces” Z. Anal. Anwendungen 24 (2005), 375 – 388

Abstract

In this note we prove that a harmonic function $u$ on the unit ball $B\subset {\mathbb{R}}^n$ belongs to the harmonic mixed norm space ${\mathcal{A}}_s^{p,q}(B)$, when $p,q\in (0,\infty ]$ and $s>0$, if and only if all weighted tangential derivatives of order $k$ (with positive orders of derivatives) belong to the related weighted Lebesgue mixed norm space ${\mathcal{L}}_s^{p,q}(B).$ Our proof of the result for the case $q\in (0,1)$ and $k$ is odd, corrects the corresponding one in the paper: G. Ren and U. Kähler, Hardy–Littlewood inequalities and $Q_p$-spaces, Z. Anal. Anwendungen 24 (2005), 375–388.