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

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 , , then where ？ , . 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.