Integral inequalities for conjugate harmonic functions of several variables

We say that a harmonic vector $F(x,y)=(u,v_1,\ldots,v_n)$ belongs to the class $S_p$ $(p>0)$ in the half-space $R^n\times(0,+\infty)$ if for any $y_0>0$ there exists a constant $C(y_0,F)$ depending only on $F$ and $y_0$ such that
$$ \int_{R^n}|F(x,y)|^p\,dx\leqslant C(y_0,F),\quad y\geqslant y_0. $$
Let $F\in S^p$ in $R^n\times(0,+\infty)$, $p>\frac{n-1}n$, $a>0$ and $\bigl\{\int_{R^n}|u(x,y)|^p\,dx\bigr\}^{1/p}\leqslant Cy^{-a}$ where $C=\mathrm{const}$. Then for $q\geqslant p$ we have
$$ \biggl\{\int_{R^n}|F(x,y)|^p\,dx\biggr\}^{1/p}\leqslant BCy^{-a-n/p+n/q}, $$
where $B$ depends only on $n$, $p$ and $a$.
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