A double integral characterization of a Bergman type space and its M\"obius invariant subspace
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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 $\bbn$, then $$\ibn |Rf(z)|^p(1-|z|^2)^{p+\alpha} \rd v_\alpha(z)<\infty$$ if and only if $$\ibn\ibn\frac{|f(z)-f(w)|^p}{|1-\langle z,w\rangle|^{n+1+s+t-\alpha}} (1-|w|^2)^s(1-|z|^2)^t \rd v(z)\rd v(w)<\infty,$$ where $s,t>-1$ with $\min(s,t)>\alpha $.
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 $\bbn$, then $$\ibn |Rf(z)|^p(1-|z|^2)^{p+\alpha} \rd v_\alpha(z)<\infty$$ if and only if $$\ibn\ibn\frac{|f(z)-f(w)|^p}{|1-\langle z,w\rangle|^{n+1+s+t-\alpha}} (1-|w|^2)^s(1-|z|^2)^t \rd v(z)\rd v(w)<\infty,$$ where $s,t>-1$ with $\min(s,t)>\alpha $.
참고문헌 (10)
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