Multipliers in Holomorphic Mean Lipschitz Spaces on the Unit Ball

and Applied Analysis 3 We denote by Rf the radial derivative of f inH B defined by Rf n ∑


Introduction
Let X and Y be two function spaces. We call ϕ a pointwise multiplier from X to Y if ϕf ∈ Y for every f ∈ X. The collection of all pointwise multipliers from X to Y is denoted by M X → Y . When X Y , we let M X M X → X . Multipliers arise in the theory of differential equations. Coefficients of differential operators can be naturally considered as multipliers. The same is true for symbols of more general pseudodifferential operators.
To give some motivations for our study, we recall studies on multipliers of Sobolev spaces. Strichartz 1 was the first who studied on multipliers of Sobolev spaces. Let Ω be a bounded domain in R n with Lipschitz boundary. Let 1 < p < ∞. For s > 0, let W s,p Ω be the Sobolev spaces over Ω. Given f and g in W s,p Ω , one cannot in general expect that their product fg will belong to W s,p Ω . However, if s > n/p, then there exists a constant K depending on s, p, n, and Ω, such that 1-3 In the complex case, multipliers on Hardy-Sobolev spaces on the unit ball in C n were studied by 4, 5 for n 1 and 6 for n ≥ 1. Let B n {z ∈ C n : |z| < 1} denote the open unit ball in C n . Let  For points z z 1 , . . . , z n and w w 1 , . . . , w n in C n , we write z, w z 1 w 1 · · · z n w n , |z| |z 1 | 2 · · · |z n | 2 .

1.4
Let S n {ζ ∈ C n : |ζ| 1} denote the unit sphere in C n . The normalized Lebesgue measure on S n will be denoted by dσ. Let H B n denote the space of all holomorphic functions in B n . Given 0 < r < 1, 0 < p < ∞, and f ∈ H B n , we define When p ∞, we write for m > s. We define the norm of Λ p s as follows: It can be shown that the norm is independent of the choice of m; see 20 . When p ∞, this is exactly the classical holomorphic Lipschitz space Λ s ; see 19 .
We adapt the first order H p mean variation defined as follows: where U denotes the group of all unitary operators on C n , I denotes the identity of U, and U − I : sup ζ∈S |Uζ − ζ|. Then, we have for 0 < s < 1 see 20 . This justifies our usage of the term holomorphic mean Lipschitz space for Λ p s with 0 < s < 1. Now, for s ≥ 1, we consider the second-order H p mean variation defined as follows:

1.13
It was shown in 21 that, if 0 < s < 2, 1 ≤ p < ∞, and f ∈ H p , then ii If s ≤ n/p (nonregular), then Λ p s is not a multiplicative algebra.
By ii of Theorem 1.1, the space Λ p n/p is not a multiplicative algebra. We give some sufficient conditions for a holomorphic function to be a pointwise multiplier of Λ p n/p as follows. We do not know if our sufficient condition is also necessary.
Throughout the paper, we write X Y or Y X for nonnegative quantities X and Y whenever there is a constant C > 0 independent of the parameters in X and Y such that X ≤ CY . Similarly, we write X ≈ Y if X Y and Y X.

Auxiliary Embedding Results
The ball algebra A B n is the class of all functions f : B n → C that are continuous on the closed ball B n and that are holomorphic in its interior B n . This function was constructed in 7 . In fact, it was shown in 7 that this function is not contained in any Hardy-Sobolev space. Thus, the result of i of Proposition 2.1 follows, since every mean Lipschitz space is contained in a Hardy-Sobolev space.
Since the series converges uniformly on B n , its sum f is therefore in the ball algebra. It is enough to prove that f / ∈ Λ 1 s for 0 < s < 1. If f ∈ Λ 1 s , then

2.3
However, since the polynomials p k are orthogonal, for any 0 < r < 1,

2.4
Take Then, Making a change of variables and replacing ρz by z, we get By 2.10 and Hölder's inequality, we have

2.12
Therefore, f ∈ Λ s−n/p ⊂ A B n if s > n/p.
Proof. i Let 1/2 < r < 1 and α s − m, where m is the greatest integer less than s.

2.17
By 2.14 and 2.17 , we have

Regular Cases
We need an elementary variant of Hölder's inequality.
Lemma 3.1. Let 0 < p, q, δ < ∞ with 1/p 1/q 1/δ. Then, for f ∈ L q and g ∈ L δ , the product fg is in L p and fg L p ≤ f L q g L δ .
3.1 Proof. Let m be the greatest integer less than s and α s − m. Let f, g ∈ Λ p s . We will prove that M p r, R m 1 fg 1