Characterization of weighted analytic Besov spaces in terms of operators of fractional differentiation

Abstract

Let $\mathbb{D}$ stand for the unit disc in the complex plane ℂ. Given 0 < p < ∞, −1 < λ < ∞, the analytic weighted Besov space $B_p^\lambda \left( \mathbb{D} \right)$ is defined to consist of analytic in $\mathbb{D}$ functions such that

$$\int\limits_\mathbb{D} {\left( {1 - \left| z \right|^2 } \right)^{Np - 2} \left| {f^{\left( N \right)} \left( z \right)} \right|^p d\mu _\lambda \left( z \right) < \infty ,}$$

where dμ _λ (z) = (λ + 1)(1 − |z|²)^λ dμ(z), $d\mu (z) = \tfrac{1} {\pi }dxdy$, and N is an arbitrary fixed natural number, satisfying N _p > 1 − λ.

We provide a characterization of weighted analytic Besov spaces $B_p^\lambda \left( \mathbb{D} \right)$, 0 < p < ∞, in terms of certain operators of fractional differentiation R ^α,t _z of order t. These operators are defined in terms of construction known as Hadamard product composition with the function b. The function b is calculated from the condition that R ^α,t _z (uniquely) maps the weighted Bergman kernel function $\left( {1 - z\bar w} \right)^{ - 2 - \alpha }$ to the similar (weight parameter shifted) kernel function $\left( {1 - z\bar w} \right)^{ - 2 - \alpha - t}$, t > 0. We also show that $B_p^\lambda \left( \mathbb{D} \right)$ can be thought as the image of certain weighted Lebesgue space $L^p \left( {\mathbb{D},d\nu _\lambda } \right)$ under the action of the weighted Bergman projection $P_\mathbb{D}^\alpha$.