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

Abstract

Let QNS(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)=sup_{𝑧∈𝐷𝑐(𝜉)}|𝑢(𝑧)|,𝐷_𝑐(𝜉)={𝑧∈𝐁∶|𝜉−𝑧|<𝑐(1−|𝑧|)}, and 𝜉∈𝐒,𝑐>1. Also, we prove a maximal theorem for Bergman type spaces.

1. Introduction and Preliminaries

Let ℝ^𝑛 (𝑛≥2) denote the 𝑛-dimensional Euclidean space. Let 𝐁=𝐁_𝑛 be the unit ball centered at the origin. The boundary of 𝐁 will be denoted by 𝐒.

The Hardy space 𝐻^𝑝(𝐁)  (0<𝑝<∞) consists of functions 𝑢 harmonic in 𝐁 for which ‖𝑢‖_𝐻^𝑝∶=𝑢⁺_𝐿𝑝=_𝐒𝑢⁺𝑦^𝑝𝑑𝜎𝑦^1/𝑝<∞, (1) where 𝑑𝜎 denotes the normalized surface measure on 𝐒 and 𝑢⁺ is the radial maximal function 𝑢⁺𝑦=sup_0≤𝑟<1𝑢𝑟𝑦,𝑦=1. (2) Also, define a function 𝑢^∗ by 𝑢^∗(𝜉)=sup_{𝑧∈𝐷𝑐(𝜉)}|𝑢(𝑧)|, (3) where 𝐷_𝑐(𝜉)=𝑧∈𝐁∶𝜉−𝑧<𝑐(1−|𝑧|),𝜉∈𝐒,𝑐>1. (4)

Throughout the paper, we write 𝐶 (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 𝑢(𝑎)≤𝑄𝜀^−𝑁_{𝐁()𝑎,𝜀}𝑢𝑑𝑣 (5) whenever 𝐁(𝑎,𝜀)∶={𝑥∶|𝑥−𝑎|<𝜀}⊂𝐁. (6) Here 𝑑𝑣 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.

Theorem A (see [2, 3]). Let 𝑝>0. If 𝑢∈𝑄𝑁𝑆(𝐁), then 𝑢^𝑝∈𝑄𝑁𝑆(𝐁), and 𝑄(𝑢^𝑝)≤𝐶_𝑝,𝑁𝑄(𝑢).

For a function 𝜑∶𝐒→ℝ, 𝜑∈𝐿¹(𝐒), let 𝑀𝜑(𝜉)=sup_𝑟>01𝜎𝐒_𝑟(𝜉)_{𝐒𝑟(𝜉)}𝜑𝜂𝑑𝜎𝜂, (7) where 𝐒_𝑟(𝜉)=𝜂∈𝐒∶𝜂−𝜉<𝑟. (8)

Theorem B (Hardy-Littlewood maximal theorem). If 1<𝑝<∞, then _𝐒𝑀𝜑^𝑝𝑑𝜎≤𝐶_𝑝𝐒𝜑^𝑝𝑑𝜎. (9)

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 0<𝑝<∞. If 𝑢∈𝑄𝑁𝑆(𝐁), then _𝐒𝑢^∗𝑝𝑑𝜎≤𝐶_𝐒𝑢^+𝑝𝑑𝜎, (10) where 𝐶 depends only on 𝑝, 𝑁, and 𝑄(𝑢).

Proof. In view of Theorem A, we can assume that 𝑝>1. Let 𝑢 be a QNS function on 𝐁. Then sup_{𝑧∈𝐷𝑐(𝜉)}𝑢(𝑧)≤𝐾sup_{𝑧∈𝐷𝑐(𝜉)}1(1−|𝑧|)^𝑛_{𝐁𝜀(𝑧)}𝑢(𝑤)𝑑𝑣(𝑤), (11) where 𝐁_𝜀(𝑧)={𝑤∶|𝑤−𝑧|<𝜀(1−|𝑧|)}, 0<𝜀<1.
To continue the proof, we need the following lemma.
Lemma  2. If 𝑧∈𝐷_𝑐(𝜉) and 𝑤∈𝐁_𝜀(𝑧), 0<𝜀<1, then𝜌−𝑟<𝜀(1−𝑟),𝜂−𝜉<𝑘|1−𝑟|, (12)where  𝜌=|𝑤|, 𝑟=|𝑧|.
Proof. Let 𝑧−𝜉<𝑐(1−|𝑧|),𝑤∈𝐵_𝜀(𝑧). (13) That is, |𝑤−𝑧|<𝜀(1−|𝑧|). (14) Hence, ||𝑤|−|𝑧||<𝜀(1−|𝑧|). (15) That is, 𝜌−𝑟<𝜀(1−𝑟). (16) On the other hand, ≤+≤().𝜂−𝜉𝜂−𝑧𝑧−𝜉𝜂−𝑧+𝑐1−𝑟 (17) Since =≤+=+||≤()()()=(),𝜂−𝑧𝜂−𝑟𝑦𝜂−𝜌𝜂𝜌𝜂−𝑟𝑦1−𝜌𝑤−𝑧1−𝜌+𝜀1−𝑟≤1−𝑟+𝜀1−𝑟+𝜀1−𝑟1+2𝜀)(1−𝑟 (18) from this and (17), we get 𝜂−𝜉≤(𝑐+1+2𝜀)(1−𝑟). (19) The proof of the lemma is complete.
We continue the proof of the theorem.
From (11) and Lemma 2, we get sup_{𝑧∈𝐷𝑐(𝜉)}𝑢(𝑧)≤𝐾sup_{𝑧∈𝐷𝑐(𝜉)}1(1−𝑟)^𝑛_{()𝜂−𝜉<𝑘1−𝑟}𝜂×𝑑𝜎_{()𝜌−𝑟≤𝜀1−𝑟}𝑢𝜌𝜂𝑛𝜌^𝑛−1𝑑𝜌≤𝐾sup_{𝑧∈𝐷𝑐(𝜉)}𝑛(1−𝑟)^𝑛×_{()𝜂−𝜉<𝑘1−𝑟}𝑑𝜎𝜂2𝜀(1−𝑟)𝑢⁺𝜂=𝐶sup_{𝑧∈𝐷𝑐(𝜉)}1(1−𝑟)^{𝑛−1()𝜂−𝜉<𝑘1−𝑟}𝑢⁺𝜂𝑑𝜎𝜂=𝐶sup_0<𝑟<11(1−𝑟)^{𝑛−1()𝜂−𝜉<𝑘1−𝑟}𝑢⁺𝜂𝑑𝜎𝜂. (20)
Hence, if 𝑝>1, then, according to Theorem B, _𝐒𝑢^∗(𝜉)^𝑝𝑑𝜎(𝜉)≤𝐶_𝐁𝑢⁺𝜂^𝑝𝑑𝜎𝜂. (21)
The proof of the theorem is complete.

2. A Maximal Theorem for Bergman Type Spaces

The harmonic Bergman space 𝒜^𝑝_𝛼=𝒜^𝑝_𝛼(𝐁) consists of functions 𝑢 harmonic in 𝐁 for which ‖𝑢‖^𝑝_𝑝,𝛼∶=_𝐁|𝑢(𝑧)|^𝑝(1−|𝑧|)^𝛼𝑑𝑣(𝑧)<∞. (22) Define now the maximal function 𝑢^⋆ by 𝑢^⋆(𝑧)=sup|𝑢(𝑤)|∶𝑤−𝑧₀(|𝑤,|𝑤|<|𝑧|<21−|)1+2, (23) where 𝑧₀=𝑧/|𝑧|, 𝑧≠0.

Theorem 3. If 𝑢∈𝒜^𝑝_𝛼, then 𝑢^⋆∈𝒜^𝑝_𝛼, 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. If u∈𝑄𝑁𝑆 and 𝐾_𝑝,𝛼𝑓∶=_𝐁𝑢(𝑧)^𝑝1−|𝑧|^2𝛼𝑑𝑣(𝑧)<∞, (24) then 𝐾_𝑝,𝛼(𝑢^⋆)<∞, and there is a constant 𝐶 depending only on 𝑄(𝑢), 𝑝, and 𝛼 such that 𝐾(𝑢^⋆)≤𝐶𝐾(𝑢).

Proof. For 𝜌=(3+𝑟)/4, 𝑟=|𝑧|, we have 𝑤−𝑧₀𝜌≤𝑤−𝑧₀(|𝑤|𝑤||𝑤|+1−𝜌≤21−|)+1−𝜌≤2𝜌−+31−𝜌≤5𝜌− (25) because 1−𝜌≤𝜌−|𝑤|, and we have 𝑤−𝑧₀(|𝑤,|𝑤|<|𝑧|<21−|)1+2⟹𝑤−𝑧₀3+|𝑧|4|𝑧|<53+4−|𝑤|. (26)
Further, we have _𝐁𝑢^⋆(𝑧)1−|𝑧|^2𝛼𝑑𝑣(𝑧)=𝑛¹₀1−𝑟^2𝛼𝑟^𝑛−1𝑑𝑟_𝐒𝑢^⋆𝑦,𝑢𝑟𝑦𝑑𝜎_⋆𝑟𝑦≤sup|𝑢(𝑤)|∶𝑤−3+𝑟4𝑦<53+𝑟4−|𝑤|=sup𝑢_𝜌(𝑧)∶𝑧−𝜌𝑦<5(1−|𝑧|), (27) where 𝜌=(3+𝑟)/4, 𝑧=𝑤/𝜌, |(𝑤/𝜌)−𝑦|<5|1−(|𝑤|/𝜌)|, and |𝑤−𝜌|<5(𝜌−|𝑤|).
Hence, by Theorem 1, _𝐁𝑢^⋆(𝑧)^𝑝1−|𝑧|^2𝛼𝑑𝑣(𝑧)≤𝑛¹₀1−𝑟^2𝛼𝑟^𝑛−1𝑑𝑟_𝐒𝑢^⋆𝑟𝑦^𝑝𝑑𝜎𝑦≤𝑛¹₀1−𝑟^2𝛼𝑟^𝑛−1𝑑𝑟_𝐒𝑢^⋆_𝜌𝑦^𝑝𝑑𝜎𝑦≤𝑛¹₀1−𝑟^2𝛼𝑟^𝑛−1𝑑𝑟_𝐒𝑢⁺_𝜌𝑦^𝑝𝑑𝜎𝑦≤𝐶_𝐁𝑢(𝑧)^𝑝1−|𝑧|^2𝛼𝑑𝑣(𝑧). (28)
This completes the proof of the theorem.