(A) characterization of carleson measures for the bergman spaces on the ball

Abstract

A characterization of Carleson measures for the Bergman spaces on the ball Let E($w,r$) denote the pseudo-hyperbolic disc of the unit disc D of the complex plane C. It is known that if $0 < r <1,\; 1 \le p< \infty$ and $\mu$ is a positive finite Borel measure on D, then the following two quantities are equivalent: \begin{eqnarray*}(i)& & \sup \{\int_D \mid f \mid^p d\mu/ \parallel f \parallel^p_{A^p} : f \in A^p(D),\; f \not\equiv 0 \}\\ (ii)& & \sup \{\mu(E(w,r))/m(E(w,r)) : w \in D \} \end{eqnarray*}\\ Where $A^p$(D) denotes the Bergman space on D and m denotes the area measure on D. In this thesis, we extend this result to the unit ball of C$^n$.