Mean Lipschitz Spaces Research Articles

Mean Lipschitz conditions on Bergman space

For f analytic on the unit disc let rt(f)(z)=f(eitz) and fr(z)=f(rz), rotations and dilations respectively. We show that for f in the Bergman space Ap and 0<α≤1 the following are equivalent.(i)‖rt(f)−f‖Ap=O(|t|α), t→0,(ii)‖(f′)r‖Ap=O((1−r)α−1), r→1−,(iii)‖fr−f‖Ap=O((1−r)α), r→1−.The Hardy space analogues of these conditions are known to be equivalent by results of Hardy and Littlewood and of E. Storozhenko, and in that setting they describe the mean Lipschitz spaces Λ(p,α).On the way, we provide an elementary proof of the equivalence of (ii) and (iii) in Hardy spaces, and show that similar assertions are valid for certain weighted mean Lipschitz spaces.

Holomorphic mean Lipschitz spaces and Hardy–Sobolev spaces on the unit ball

We study two classes of holomorphic functions in the unit ball 𝔹 n of ℂ n : mean Lipschitz spaces and Hardy–Sobolev spaces. Main results include new characterizations in terms of fractional radial differential operators and various comparisons between these two classes.

CHARACTERIZATION OF LIPSCHITZ-TYPE FUNCTIONS BY GARSIA-TYPE NORMS ON THE UPPER HALF SPACE

It is well-known that the BMO norm is equivalent to the Garsia norm. In this paper, we characterize mean-Lipschitz spaces by using Garsia-type norms on the upper half space <TEX>$\mathbb{R}_+^{n+1}$</TEX>.

Multipliers in Holomorphic Mean Lipschitz Spaces on the Unit Ball

For 1 ≤ p ≤ ∞ and s &gt; 0, let be holomorphic mean Lipschitz spaces on the unit ball in ℂn. It is shown that, if s &gt; n/p, the space is a multiplicative algebra. If s &gt; n/p, then the space is not a multiplicative algebra. We give some sufficient conditions for a holomorphic function to be a pointwise multiplier of .

Mean Lipschitz Spaces Characterization via Mean Oscillation

In this paper we characterize mean Lipschitz spaces in terms of some $L^p$-mean oscillation on the unit disc.

FATOU THEOREM AND EMBEDDING THEOREMS FOR THE MEAN LIPSCHITZ FUNCTIONS ON THE UNIT BALL

We investigate the boundary values of the holomorphic mean Lipschitz function. In fact, we prove that the admissible limit exists at every boundary point of the unit ball for the holomorphic mean Lipschitz functions under some assumptions on the Lipschitz order. Moreover, we get embedding theorems of holomorphic mean Lipschitz spaces into Hardy spaces or into the Bloch space on the unit ball in <TEX>$\mathbb{C}_n$</TEX>.

Criteria for Membership of the Mean Lipschitz Spaces

Our aim is to characterize the elements in certain function spaces by means of the Cesáro means and/or partial sums of their Fourier series. Firstly, we seek to extend known results for the Besov spaces B^s_{pq} (1 \le p,q &lt; \infty) to the case where q = \infty . Secondly, we consider the Mean Lipschitz spaces \Lambda(p,s) . We confine attention to the values 1 \le p &lt; \infty and 0 &lt; s \le 1 for the parameters. For s &lt; 1 , the spaces B^s _{p\infty} and \Lambda(p,s) coincide. For the case p = 1 certain counter-examples are provided; some positive results are also given. We then treat the case s = 1 and consider the spaces B^1_{p\infty} and \Lambda(p,1) separately. Analogues of some known results for the spaces \Lambda_s are given.

Some Results on Mean Lipschitz Spaces of Analytic Functions

Some Results on Mean Lipschitz Spaces of Analytic Functions

Mean Lipschitz spaces and bounded mean oscillation

Mean Lipschitz spaces and bounded mean oscillation