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

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 zα,t 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 zα,t (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 $$P_\mathbb{D}^\alpha$$.

!ENTROPY NUMBERS OF EMBEDDINGS OF WEIGHTED BESOV SPACES. II

We investigate the asymptotic behaviour of the entropy numbers of the compact embedding $B^{s_1}_{p_1,q_1}(\mathbb{R}^d,w_1)\hookrightarrow B^{s_2}_{p_2,q_2}(\mathbb{R}^d,w_2)$. Here $B^s_{p,q}(\mathbb{R}^d,w)$ denotes a weighted Besov space. We present a general approach which allows us to work with a large class of weights.