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.


Introduction and Preliminaries
Let R  ( ≥ 2) denote the -dimensional Euclidean space.Let B = B  be the unit ball centered at the origin.The boundary of B will be denoted by S.
The Hardy space   (B) (0 <  < ∞) consists of functions  harmonic in B for which ‖‖   :=      +     = (∫ S      + ()       ()) where  denotes the normalized surface measure on S 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
Here V denotes the normalized Lebesgue measure on B. Members of QNS(B) are called quasinearly subharmonic functions (see [1,2]).The class QNS contains nonnegative subharmonic functions.
We need the following results.
For a function  : S → R,  ∈  1 (S), let where The following theorem is well known in the case of nonnegative subharmonic functions and is due to Fefferman and Stein (see [4]).
where  depends only on , , and ().
Proof.In view of Theorem A, we can assume that  > 1.Let  be a QNS function on B. Then sup To continue the proof, we need the following lemma.
That is, Hence, The proof of the lemma is complete.We continue the proof of the theorem.From (11) and Lemma 2, we get sup Hence, if  > 1, then, according to Theorem B, The proof of the theorem is complete.

A Maximal Theorem for Bergman Type Spaces
The harmonic Bergman space A   = A   (B) consists of functions  harmonic in B for which Define now the maximal function  ⋆ by where  0 = /||,  ̸ = 0.
Since || ∈ QNS, this theorem is a special case of the following.
Further, we have (28) This completes the proof of the theorem.