Norms of inclusions between some spaces of analytic functions

The inclusions between the Besov spaces $$B^q$$ B q , the Bloch space $$\mathcal {B}$$ B and the standard weighted Bergman spaces $$A^p_{\alpha}$$ A α p are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.

Bergman spaces with exponential type weights

For $1\le p<\infty $ 1 ≤ p < ∞ , let $A^{p}_{\omega }$ A ω p be the weighted Bergman space associated with an exponential type weight ω satisfying $$ \int _{{\mathbb{D}}} \bigl\vert K_{z}(\xi ) \bigr\vert \omega (\xi )^{1/2} \,dA(\xi ) \le C \omega (z)^{-1/2}, \quad z\in {\mathbb{D}}, $$ ∫ D | K z ( ξ ) | ω ( ξ ) 1 / 2 d A ( ξ ) ≤ C ω ( z ) − 1 / 2 , z ∈ D , where $K_{z}$ K z is the reproducing kernel of $A^{2}_{\omega }$ A ω 2 . This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂̅-equation (Theorem 2.5) for more general weight $\omega _{*}$ ω ∗ . As an application, we prove the boundedness of the Bergman projection on $L^{p}_{\omega }$ L ω p , identify the dual space of $A^{p}_{\omega }$ A ω p , and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from $A^{p}_{\omega }$ A ω p into $A^{q}_{\omega }$ A ω q , $1\le p,q<\infty $ 1 ≤ p , q < ∞ , such as Toeplitz and (big) Hankel operators.

The wandering subspace property and Shimorin’s condition of shift operator on the weighted Bergman spaces

In the present paper, we first study the wandering subspace property of the shift operator on the $$I_{a}$$ I a type zero based invariant subspaces of the weighted Bergman spaces $$L_{a}^{2}(dA_{n})(n=0,2)$$ L a 2 ( d A n ) ( n = 0 , 2 ) via the spectrum of some Toeplitz operators on the Hardy space $$H^{2}$$ H 2 . Second, we give examples to show that Shimorin’s condition for the shift operator fails on the $$I_{a}$$ I a type zero based invariant subspaces of the weighted Bergman spaces $$L_{a}^{2}(dA_{\alpha })(\alpha >0)$$ L a 2 ( d A α ) ( α > 0 ) .

OPERATORS ON BERGMAN SPACES

Bergman space theory has been at the core of complex analysis research for many years. Indeed, Hardy spaces are related to Bergman spaces. The popularity of Bergman spaces increased when functional analysis emerged. Although many researchers investigated the Bergman space theory by mimicking the Hardy space theory, it appeared that, unlike their cousins, Bergman spaces were more complex in different aspects. The issue of invariant subspace constitutes one common problem in mathematics that is yet to be resolved. For Hardy spaces, each invariant subspace for shift operators features an elegant description. However, the method for formulating particular structures for the large invariant subspace of shift operators upon Bergman spaces is still unknown. This paper aims to characterize bounded Hankel operators involving a vector-valued Bergman space compared to other different vector value Bergman spaces.