Geodesic
Integral inequalities for conjugate harmonic functions of several variables
Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 191-208 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $a$. Bibliography: 14 titles. 
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     author = {A. A. Bonami},
     title = {Integral inequalities for conjugate harmonic functions of several variables},
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     volume = {16},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1972_16_2_a3/}
}
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A. A. Bonami. Integral inequalities for conjugate harmonic functions of several variables. Sbornik. Mathematics, Tome 16 (1972) no. 2, pp. 191-208. http://geodesic.mathdoc.fr/item/SM_1972_16_2_a3/

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