Harmonic Bergman spaces of the half-space and their some operators (Q2770347)

The authors consider the Bergman space \(b^p\) of functions \(f(z)\) harmonic in the upper half-space \(y>0\), \(z=(x,y)\in H_n=\{(x,y): x\in R^{n-1}\), \(y>0\}\) with NEWLINE\[NEWLINE\int_{H_n}|f(x,y)|^p dx dy<\inftyNEWLINE\]NEWLINE Based on a preliminary consideration of properties of the reproducing kernel for the half-space, they give some conditions for the imbedding \(b^p\rightarrow L^p(H_n,d\mu)\) to be compact. They also study compactness of Toeplitz operator \(T_f: b^2\rightarrow b^2\), the main statement being the following: let \(f\in L^\infty\) and \(\lim_{z\to\infty}f(z)=0\), then \(T_f\) is compact if and only if \(f\in C_0(H_n)\).