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Zeitschrift für Analysis und ihre Anwendungen


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Volume 26, Issue 4, 2007, pp. 473–480
DOI: 10.4171/ZAA/1337

Published online: 2007-12-31

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

Stevo Stevic[1]

(1) Serbian Academy of Science, Beograd, Serbia

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.

Keywords: Harmonic function, mixed norm space, unit ball, HL-property

Stevic Stevo: On Ren-Kähler's Paper "Hardy-Littlewood Inequalities and Qp-Spaces", Z. Anal. Anwendungen 24 (2005), 375 – 388. Z. Anal. Anwend. 26 (2007), 473-480. doi: 10.4171/ZAA/1337