Notes on Carleson Type Measures On bounded symmetric domain
Commun. Korean Math. Soc. 2007 Vol. 22, No. 1, 65-74
Printed March 1, 2007
Ki Seong Choi
Konyang University
Abstract : Suppose that $\mu$ is a finite positive Borel measure on bounded symmetric domain $\Omega \subset \Bbb C^n$ and $\nu$ is the Euclidean volume measure such that $\nu (\Omega) =1$. Suppose $1 < p < \infty $ and $r> 0 $. In this paper, we will show that the norms $\sup \{ \int_\Omega \vert k_z (w) \vert^2 d \mu(w): z \in \Omega \} $, $\sup \{ {\int_{\Omega} \vert h(w) \vert^p d \mu(w)} /$ ${\int_{\Omega} \vert h(w) \vert^p d \nu(w)} : h \in L^p_a (\Omega, d \nu), h \neq 0 \}$ and $$\sup \{ \frac{\mu(E(z,r))}{\nu (E(z,r))} : z \in \Omega \}$$ are all equivalent. We will also show that the inclusion mapping $i_p : L^p_a (\Omega, d \nu) \rightarrow L^p (\Omega, d \mu)$ is compact if and only if $ \lim_{w \rightarrow \partial \Omega} \frac{\mu(E(w,r))}{\nu( E(w,r))} = 0 $.
Keywords : Bergman space, Bergman projection
MSC numbers : 32H25, 32E25, 30C40
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