Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 43, 2018, 807-821

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*)|^{p}ω(*z*) *dA*(*z*) < ∞

if and only if

∫_{D}(1 – |*z*|^{2})^{p}|*f*'(*z*)|^{p}ω(*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.

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

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