Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 807-821

A HARDY–LITTLEWOOD THEOREM FOR BERGMAN SPACES

Guanlong Bao, Hasi Wulan and Kehe Zhu

Shantou University, Department of Mathematics
Shantou, Guangdong Province, P.R. China; glbao 'at' stu.edu.cn

Shantou University, Department of Mathematics
Shantou, Guangdong Province, P.R. China; wulan 'at' stu.edu.cn

State University of New York, Department of Mathematics and Statistics
Albany, NY 12222, U.S.A.; kzhu 'at' math.albany.edu

Abstract. We study positive weight functions ω(z) on the unit disk D such that

∫_D|f(z)|^pω(z) dA(z) < ∞

if and only if

∫_D(1 – |z|²)^p|f'(z)|^pω(z) dA(z) < ∞,

where f is analytic on D and dA is area measure on D. We obtain some conditions on ω that imply the equivalence above, and we apply our conditions to several important classes of weights that have appeared in the literature before.

2010 Mathematics Subject Classification: Primary 30H20.

Key words: Bergman spaces, Hardy–Littlewood theorem, subharmonic functions, superharmonic functions.

Reference to this article: G. Bao, H. Wulan and K. Zhu: A Hardy–Littlewood theorem for Bergman spaces. Ann. Acad. Sci. Fenn. Math. 43 (2018), 807-821.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4345