A Carleson measure theorem for Bergman spaces

Hastings, William W.

doi:10.1090/S0002-9939-1975-0374886-9

A Carleson measure theorem for Bergman spaces
HTML articles powered by AMS MathViewer

by William W. Hastings PDF

Proc. Amer. Math. Soc. 52 (1975), 237-241 Request permission

Abstract:

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

References

Lennart Carleson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921–930. MR 117349, DOI 10.2307/2372840
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
Peter L. Duren, Extension of a theorem of Carleson, Bull. Amer. Math. Soc. 75 (1969), 143–146. MR 241650, DOI 10.1090/S0002-9904-1969-12181-6

Theory of

spaces

42

Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841