A Carleson measure theorem for Bergman spaces
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- by William W. Hastings
- Proc. Amer. Math. Soc. 52 (1975), 237-241
- DOI: https://doi.org/10.1090/S0002-9939-1975-0374886-9
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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 ${H^p}$ spaces, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR 42 #3552.
- Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
Bibliographic Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 52 (1975), 237-241
- MSC: Primary 46E15; Secondary 30A78, 32A30
- DOI: https://doi.org/10.1090/S0002-9939-1975-0374886-9
- MathSciNet review: 0374886