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

Authors: Olivera R. Mihić
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

LetQNS(B) denote the class of quasinearly subharmonic functions in unit ball B. We provide, following result: if 0<p<∞ and if u∈QNS(B), then ∫S‍(u*)pdσ≤C∫S‍(u+)pdσ, where u+ is the radial maximal function and u*(ξ)=supz∈Dc(ξ)|u(z)|,Dc(ξ)={z∈B:|ξ−z|<c(1-|z|)}, and ξ∈S,c>1. Also, we prove a maximal theorem for Bergman type spaces.


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## Body

## 1. Introduction and Preliminaries

Letℝn (n≥2) denote the n-dimensional Euclidean space. Let B=Bn be the unit ball centered at the origin. The boundary of  B will be denoted by S.The Hardy spaceHp(B)  (0<p<∞) consists of functions u harmonic in B for which
(1)∥u∥Hp:=∥u+∥Lp=(∫S|u+(y)|pdσ(y))1/p<∞,
where dσ denotes the normalized surface measure on S and u+ is the radial maximal function
(2)u+(y)=sup0≤r<1|u(ry)|,|y|=1.
Also, define a function u* by
(3)u*(ξ)=supz∈Dc(ξ)|u(z)|,
where
(4)Dc(ξ)={z∈B:|ξ-z|<c(1-|z|)},ξ∈S,c>1.Throughout the paper, we writeC (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   QNS(B). Let QNS(B) denote the class of nonnegative measurable functions u on B for which a constant Q=Q(u) exists such that
(5)u(a)≤Qε-N∫B(a,ε)udv
whenever
(6)B(a,ε):={x:|x-a|<ε}⊂B.
Here dv denotes the normalized Lebesgue measure on B.Members ofQNS(B) are called quasinearly subharmonic functions (see [1, 2]). The class QNS contains nonnegative subharmonic functions.We need the following results.Theorem A (see [2, 3]).
Letp>0. If u∈QNS(B), then up∈QNS(B), and Q(up)≤Cp,NQ(u).For a functionφ:S→ℝ, φ∈L1(S), let
(7)(Mφ)(ξ)=supr>01σ(Sr(ξ))∫Sr(ξ)|φ(η)|dσ(η),
where
(8)Sr(ξ)={η∈S:|η-ξ|<r}.Theorem B (Hardy-Littlewood maximal theorem).
If1<p<∞, then
(9)∫S|Mφ|pdσ≤Cp∫S|φ|pdσ.The following theorem is well known in the case of nonnegative subharmonic functions and is due to Fefferman and Stein (see [4]).Theorem 1.
Let0<p<∞. If u∈QNS(B), then
(10)∫S(u*)pdσ≤C∫S(u+)pdσ,
where C depends only on p, N, and Q(u).Proof.
In view of Theorem A, we can assume thatp>1. Let u be a QNS function on B. Then
(11)supz∈Dc(ξ)u(z)≤Ksupz∈Dc(ξ)1(1-|z|)n∫Bε(z)u(w)dv(w),
where  Bε(z)={w:|w-z|<ε(1-|z|)},  0<ε<1.
To continue the proof, we need the following lemma.
Lemma  2.Ifz∈Dc(ξ)andw∈Bε(z),  0<ε<1,then(12)|ρ-r|<ε(1-r),|η-ξ|<k|1-r|,where  ρ=|w|, r=|z|.
Proof. Let
(13)|z-ξ|<c(1-|z|),w∈Bε(z).
That is,
(14)|w-z|<ε(1-|z|).
Hence,
(15)||w|-|z||<ε(1-|z|).
That is,
(16)|ρ-r|<ε(1-r).
On the other hand,
(17)|η-ξ|≤|η-z|+|z-ξ|≤|η-z|+c(1-r).
Since
(18)|η-z|=|η-ry|≤|η-ρη|+|ρη-ry|=(1-ρ)+|w-z|≤(1-ρ)+ε(1-r)≤1-r+ε(1-r)+ε(1-r)=(1+2ε)(1-r),
from this and (17), we get
(19)|η-ξ|≤(c+1+2ε)(1-r).
The proof of the lemma is complete.
We continue the proof of the theorem.
From (11) and Lemma 2, we get
(20)supz∈Dc(ξ)u(z)≤Ksupz∈Dc(ξ)1(1-r)n∫|η-ξ|<k(1-r)dσ(η)supz∈Dc(ξ)u(z)≤×∫|ρ-r|≤ε(1-r)u(ρη)nρn-1dρsupz∈Dc(ξ)u(z)≤Ksupz∈Dc(ξ)n(1-r)nsupz∈Dc(ξ)u(z)≤×∫|η-ξ|<k(1-r)dσ(η)2ε(1-r)u+(η)supz∈Dc(ξ)u(z)=Csupz∈Dc(ξ)1(1-r)n-1∫|η-ξ|<k(1-r)u+(η)dσ(η)supz∈Dc(ξ)u(z)=Csup0<r<11(1-r)n-1∫|η-ξ|<k(1-r)u+(η)dσ(η).
Hence, ifp>1, then, according to Theorem B,
(21)∫S‍u*(ξ)pdσ(ξ)≤C∫B‍u+(η)pdσ(η).
The proof of the theorem is complete.


## 2. A Maximal Theorem for Bergman Type Spaces

The harmonic Bergman space𝒜αp=𝒜αp(B) consists of functions u harmonic in B for which
(22)∥u∥p,αp:=∫B‍|u(z)|p(1-|z|)αdv(z)<∞.
Define now the maximal function u⋆ by
(23)u⋆(z)=sup{|u(w)|:|w-z0|<2(1-|w|),|w|<1+|z|2},
where z0=z/|z|, z≠0.Theorem 3.
Ifu∈𝒜αp, then u⋆∈𝒜αp, and ∥u⋆∥p,α≤C∥u∥p,α, where C is independent of u.Since|u|∈QNS, this theorem is a special case of the following.Theorem 4.
Ifu∈QNS  and
(24)Kp,α(f):=∫B‍u(z)p(1-|z|2)αdv(z)<∞,
then Kp,α(u⋆)<∞, and there is a constant C depending only on Q(u), p, and α such that K(u⋆)≤CK(u).Proof.
Forρ=(3+r)/4, r=|z|, we have
(25)|w-z0ρ|≤|w-z0|+1-ρ≤2(1-|w|)+1-ρ≤2(ρ-|w|)+3(1-ρ)≤5(ρ-|w|)
because 1-ρ≤ρ-|w|, and we have
(26)|w-z0|<2(1-|w|),|w|<1+|z|2⟹|w-z03+|z|4|<5(3+|z|4-|w|).
Further, we have(27)∫Bu⋆(z)(1-|z|2)αdv(z)=n∫01(1-r2)αrn-1dr∫Su⋆(ry)dσ(y),u⋆(ry)≤sup{|u(w)|:|w-3+r4y|u⋆(ry)≤supb<5(3+r4-|w|)}=sup{|uρ(z)|:|z-ρy|<5(1-|z|)},
where ρ=(3+r)/4,  z=w/ρ, |(w/ρ)-y|<5|1-(|w|/ρ)|, and  |w-ρ|<5(ρ-|w|).
Hence, by Theorem1,
(28)∫B‍(u⋆(z))p(1-|z|2)αdv(z)≤n∫01‍(1-r2)αrn-1dr∫S‍(u⋆(ry))pdσ(y)≤n∫01‍(1-r2)αrn-1dr∫S‍(uρ⋆(y))pdσ(y)≤n∫01‍(1-r2)αrn-1dr∫S‍(uρ+(y))pdσ(y)≤C∫B‍u(z)p(1-|z|2)αdv(z).
This completes the proof of the theorem.



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