A Carleson measure theorem for Bergman spaces

Abstract

Let μ \mu be a finite, positive measure on U n {U^n} , the unit polydisc in C n {{\mathbf {C}}^n} , and let σ n {\sigma _n} be 2 n 2n -dimensional Lebesgue volume measure on U n {U^n} . For 1 ⩽ p ⩽ q > ∞ 1 \leqslant p \leqslant q > \infty a necessary and sufficient condition on μ \mu is given in order that { ∫ U n f q ( z ) d μ ( z ) } 1 / q ⩽ C { ∫ U n f p ( z ) d σ n ( z ) } 1 / p \{ \int {_{{U^n}}{f^q}(z)d\mu (z){\} ^{1/q}} \leqslant } C\{ \int {_{{U^n}}{f^p}(z)d{\sigma _n}(z){\} ^{1/p}}} for every positive n n -subharmonic function f f on U n {U^n} .