A double integral characterization of a Bergman type space and its Mobius invariant subspace
Bull. Korean Math. Soc.
Published online June 10, 2019
Cheng Yuan and Hong-Gang Zeng
School of Applied Mathematics, Guangdong University of Technology, School of Mathematics, Tianjin University
Abstract : This paper shows that if $1<p<\infty$, $\alpha\ge-n-2$, $\alpha>-1-\frac{p}{2} $ and $f$ is holomorphic on the unit ball $\mathbb{B}_n$, then
$$\int_{\mathbb{B}_n} |Rf(z)|^p(1-|z|^2)^{p+\alpha} \mathrm{d} v_\alpha(z)<\infty$$
if and only if
$$\int_{\mathbb{B}_n}\int_{\mathbb{B}_n}\frac{|f(z)-f(w)|^p}{|1-\langle z,w\rangle|^{n+1+s+t-\alpha}} (1-|w|^2)^s(1-|z|^2)^t \mathrm{d} v(z)\mathrm{d} v(w)<\infty,$$
where $s,t>-1$ with $\min(s,t)>\alpha $.
Keywords : Bergman space, $Q_p$ spaces
MSC numbers : 30H25, 32A36
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