On Ren-Kähler's Paper &#34;Hardy-Littlewood Inequalities and <em>Q_p</em>-Spaces&#34;, <em>Z. Anal. Anwendungen</em> 24 (2005), 375 – 388

Abstract

In this note we prove that a harmonic function $u$ on the unit ball $B\subset\RR^n$ belongs to the harmonic mixed norm space ${\cal A}^{p,q}_s(B),$ 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 ${\cal L}^{p,q}_s(B).$ 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.~K\"ahler, Hardy-Littlewood inequalities and $Q_p$-spaces, {\it Z. Anal. Anwendungen} {24} (2005), 375 -- 388.