The Harmonic Bloch and Besov Spaces on the Real Unit Ball by an Oscillation

LetB be the real unit ball inR andf ∈ C(B). Given amulti-indexm = (m 1 , . . . , m n ) of nonnegative integers with |m| = N, we set the quantity sup x∈B,y∈E(x,r),x ̸ =y(1− |x| 2)α(1− |y|2)β(|∂mf(x)−∂mf(y)|/|x−y|γ[x, y]1−γ), x ̸ = y,where 0 ≤ γ ≤ 1 and α+β = N+1. In terms of it, we characterize harmonic Bloch and Besov spaces on the real unit ball. This generalizes the main results of Yoneda, 2002, into real harmonic setting.


Introduction
Let B be the real unit ball in R  with  ≥ 2, where V is the normalized volume measure on B and  is the normalized surface measure on the unit sphere  = B.We denote the class of all harmonic functions on the unit ball by (B).For  ∈ (B), ∇() denotes the gradient of .Given a multiindex  = ( 1 , . . .,   ) of nonnegative integers, we use the notations || =  1 + ⋅ ⋅ ⋅ +   and For each  > 0, the harmonic -Bloch space B  consists of all functions  ∈ (B) such that          = sup and the little -Bloch space B  0 consists of the functions  ∈ B  such that lim ∇ ()     = 0.
The harmonic Besov space B  is the space of all functions in (B) for which where  > −1 and () = (1 − || Let D be the open unit disk in the complex plane C and let  be a continuous complex-value function in D. Denote by where (, ) is the hyperbolic disc with center  ∈ D and radius ,  ≥ 1 an integer,  +  = , and In [2], Yoneda characterized harmonic Bloch and Besov spaces in D in terms of  as follows.
Theorem C. Let  be a complex-value harmonic function in D. Fix an integer  ≥ 1 and a pair of real numbers ,  such that  +  = .Then  ∈ B 1 (D) if and only if  is bounded.
Theorem D. Let  be a complex-value harmonic function in D. Fix an integer  ≥ 2 and a pair of real numbers ,  such that  +  = .Then for each  ≥ 1,  ∈ B  (D) if and only if where See [3][4][5] for various characterizations of the Bloch, little Bloch, and Besov spaces in the unit ball of C  .
The main purpose of this paper is to give some characterizations for the spaces B  , B  0 , and B  in the real unit ball along Yoneda's direction.In Section 2, we collect some known results that will be needed in the proof of our results.Our main results and their proofs are presented in Sections 3 and 4.
Throughout this paper, constants are denoted by ; they are positive and may differ from one occurrence to the other.The notation  ≍  means there is a positive constant  such that / ≤  ≤ .

Preliminaries
We will be using the same notation in [1,6]: we write ,  ∈ R  in polar coordinates by  = ||  and  = ||  .For any ,  ∈ R  , let Then the symmetry lemma in [7] shows that For any  ∈ B, denote by   the Möbius transformation in B.
It is an involution of B such that   (0) =  and   () = 0, which is of the form By simple computations, we have For any  ∈ B and  ∈ (0, 1), we define the pseudohyperbolic ball with center  and radius  as Clearly, (, ) =   (B(0, )).
The following is a characterization of the space B 1 (resp., B 1 0 ) which is proved in [8].
Lemma 2. Let ℎ ∈ (B) and  be a positive integer.Then for all multi-index  with || = .
As an application of Lemma 2, we can obtain the following.
For the converse, we assume that ℎ ∈ B ( It follows from [9] that there exists  > 0 such that This implies that sup ,∈B, ̸ = So the result follows.
Combining Theorem A and Lemma 3, we extend [2, Corollary 2.4] into the real harmonic setting as follows.
In the following, we give an example which shows that Corollary 4 does not hold for  ≥ 3.
2 ); then  ∈ B 1 .By a simple computation we have

Results and Discussions
3.1.Harmonic Bloch Spaces.In this section, we give some characterizations of the spaces B  which can be viewed as the generalizations of Yoneda's results into the real unit ball B of R  .For a continuous function ℎ in B and 0 ≤  ≤ 1, we write By using the notation, we characterize B 1 , B 1 0 as follows.
In the following, by removing the restriction  ∈ (, ) in Theorem 6, we obtain the following.
Proof.We only need to prove the necessity since the proof of sufficiency is similar to that in proof of Theorem Since   () ≤   , we see that (54)