Weighted Norm Inequality for the Poisson Integral on the Sphere

Benjamin Bordin, Iara A.A. Fernandes and Sergio A. Tozoni

Abstract: We obtain, for each $p$, $1<p<\infty$, a necessary and sufficient condition for the Poisson integral of functions defined on the sphere $S^n$, to be bounded from a weighted space $L^p(S^n,Wd\sigma)$ into a space $L^p(\B,\nu)$, where $\sigma$ is the Lebesgue measure on $S^n$ and $\nu$ is a positive measure on the unit ball $\B$ of $\R^{n+1}$.

Keywords: Poisson integral; Carleson condition; homogeneous space; weight; sphere.

Classification (MSC2000): 42B25, 43A85.

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