Quasiconformal composition operators on harmonic Bergman spaces on the real unit ball
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Suppose $n\geq 2$ and $\mathbb{B}_n$ is the unit ball in $\mathbb{R}^n$. Let $\Phi:\mathbb{B}_n\to\mathbb{B}_n$ be a quasiconformal mapping. In this paper, we give a necessary and sufficient condition such that the composition operator $C_\Phi$ is bounded between the harmonic Bergman space $b^p(\mathbb{B}_n)$ and $b^q(\mathbb{B}_n)$ for $1<p,q<\infty$. Our results generalize Fang-Guo-Wang's recent work to harmonic settings.
Abstract
Suppose $n\geq 2$ and $\mathbb{B}_n$ is the unit ball in $\mathbb{R}^n$. Let $\Phi:\mathbb{B}_n\to\mathbb{B}_n$ be a quasiconformal mapping. In this paper, we give a necessary and sufficient condition such that the composition operator $C_\Phi$ is bounded between the harmonic Bergman space $b^p(\mathbb{B}_n)$ and $b^q(\mathbb{B}_n)$ for $1<p,q<\infty$. Our results generalize Fang-Guo-Wang's recent work to harmonic settings.