NOTES ON CARLESON TYPE MEASURES ON BOUNDED SYMMETRIC DOMAIN

Title & Authors
NOTES ON CARLESON TYPE MEASURES ON BOUNDED SYMMETRIC DOMAIN
Choi, Ki-Seong;

Abstract
Suppose that $\small{\mu}$ is a finite positive Borel measure on bounded symmetric domain $\small{\Omega{\subset}\mathbb{C}^n\;and\;\nu}$ is the Euclidean volume measure such that $\small{\nu(\Omega)=1}$. Suppose 1 < p < $\small{\infty}$ and r > 0. In this paper, we will show that the norms $\small{sup\{\int_\Omega{\mid}k_z(w)\mid^2d\mu(w)\;:\;z\in\Omega\}}$, $\small{sup\{\int_\Omega{\mid}h(w)\mid^pd\mu(w)/\int_\Omega{\mid}h(w)^pd\nu(w)\;:\;h{\in}L_a^p(\Omega,d\nu),\;h\neq0\}}$ and $\small{sup\{\frac{\mu(E(z,r))}{\nu(E(z,r))}\;:\;z\in\Omega\}}$ are are all equivalent. We will also show that the inclusion mapping $\small{ip\;:\;L_a^p(\Omega,d\nu){\rightarrow}L^p(\Omega,d\mu)}$ is compact if and only if lim $\small{w\rightarrow\partial\Omega\frac{\mu(E(w,r))}{\nu(E(w,r))}=0}$.
Keywords
Bergman space;Bergman projection;
Language
English
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