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ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 515398, 3 pages
http://dx.doi.org/10.1155/2013/515398
Research Article

Some Properties of Quasinearly Subharmonic Functions and Maximal Theorem for Bergman Type Spaces

Faculty of Organizational Sciences, University of Belgrade, Jove Ilića 154, 11000 Belgrade, Serbia

Received 17 July 2013; Accepted 24 August 2013

Academic Editors: O. Miyagaki and A. M. Peralta

Copyright © 2013 Olivera R. Mihić. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let denote the class of quasinearly subharmonic functions in unit ball . We provide, following result: if and if , then , where is the radial maximal function and , and . Also, we prove a maximal theorem for Bergman type spaces.

1. Introduction and Preliminaries

Let () denote the -dimensional Euclidean space. Let be the unit ball centered at the origin. The boundary of will be denoted by .

The Hardy space    consists of functions harmonic in for which where denotes the normalized surface measure on and is the radial maximal function Also, define a function by where

Throughout the paper, we write (sometimes with indexes) to denote a positive constant which might be different at each occurrence (even in a chain of inequalities) but is independent of the functions or variables being discussed.

The Class. Let denote the class of nonnegative measurable functions on for which a constant exists such that whenever Here denotes the normalized Lebesgue measure on .

Members of are called quasinearly subharmonic functions (see [1, 2]). The class contains nonnegative subharmonic functions.

We need the following results.

Theorem A (see [2, 3]). Let . If , then , and .

For a function , , let where

Theorem B (Hardy-Littlewood maximal theorem). If , then

The following theorem is well known in the case of nonnegative subharmonic functions and is due to Fefferman and Stein (see [4]).

Theorem 1. Let . If , then where depends only on , , and .

Proof. In view of Theorem A, we can assume that . Let be a QNS function on . Then where , .
To continue the proof, we need the following lemma.
Lemma  2. If and , , thenwhere, .
Proof. Let That is, Hence, That is, On the other hand, Since from this and (17), we get The proof of the lemma is complete.
We continue the proof of the theorem.
From (11) and Lemma 2, we get
Hence, if , then, according to Theorem B,
The proof of the theorem is complete.

2. A Maximal Theorem for Bergman Type Spaces

The harmonic Bergman space consists of functions harmonic in for which Define now the maximal function by where , .

Theorem 3. If , then , and , where is independent of .

Since , this theorem is a special case of the following.

Theorem 4. If and then , and there is a constant depending only on , , and such that .

Proof. For , , we have because , and we have
Further, we have where , , , and .
Hence, by Theorem 1,
This completes the proof of the theorem.

References

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