Operators on weighted Bergman spaces

Mathematische Nachrichten, 2003

Duke Mathematical Journal, 1992

We describe the boundedness of a linear operator from B p (ρ) = {f : D → C analytic : D ρ(1 − |z|) (1 − |z|) |f (z)| p dA(z) 1/p < ∞} , for 0 < p ≤ 1 under some conditions on the weight function ρ, into a general Banach space X by means of the growth conditions at the boundary of certain fractional derivatives of a single X-valued analytic function. This, in particular, allows us to characterize the dual of B p (ρ) for 0 < p < 1 and to give a formulation of generalized Carleson measures in terms of the inclusion B 1 (ρ) ⊂ L 1 (D, µ). We then apply the result to the study of multipliers, Hankel operators and composition operators acting on B p (ρ) spaces.

emis.ams.org

MARGARITA MATHEMATICA EN MEMORIA DE JOSÉ JAVIER (CHICHO) GUADALUPE HERNÁNDEZ (Luis Espa˜nol y Juan L. Varona, editores), Servicio de Publicaciones, Universidad de La Rioja, Logro˜no, Spain, 2001. ... A NOTE ON THE BOUNDEDNESS OF OPERATORS ON ...

Proceedings of the American Mathematical Society

Let D denote the unit disk in the complex plane. We consider a class of superbiharmonic weight functions w : D → R + whose growth are subject to the condition 0 ≤ w(z) ≤ C(1 − |z|) for some constant C. We first establish a Reisz-type representation formula for w, and then use this formula to prove that the polynomials are dense in the weighted Bergman space with weight w.

JOURNAL OF OPERATOR THEORY

We obtain a formula for the essential norm of any operator be-tween weighted Bergman spaces of infinite order. Then we apply it to obtain or estimate essential norms of operators acting on Bloch type spaces and to differences of composition operators or Toeplitz operators on some weighted Bergman spaces. KEYWORDS: Weighted Bergman spaces of infinite order, essential norm, composition operator, Toeplitz operator. MSC (2000): 47B25.

International Journal of Mathematics and Mathematical Sciences, 1990

The radial limits of the weighted derivative of an bounded analytic function is considered.

Arkiv for Matematik, 2004

Using a geometric method, we characterize all entire functions that transform the Bloch space into a Bergman space by superposition in terms of their order and type. We also prove that all superposition operators induced by such entire functions act boundedly. Similar results hold for superpositions from BMOA into Bergman spaces and from the Bloch space into certain weighted Hardy spaces.

Turkish Journal of Mathematics, 2021

We study some classical operators defined on the weighted Bergman Fréchet space $A^p_{\alpha+}$ (resp. weighted Bergman (LB)-space $A^p_{\alpha-}$) arising as the projective limit (resp. inductive limit) of the standard weighted Bergman spaces into the growth Fréchet space $H^\infty_{\alpha+}$ (resp. growth (LB)-space $H^\infty_{\alpha-}$) which is the projective limit (resp. inductive limit) of the growth Banach spaces. We show that the continuity of the Volterra integral operator $T_g$ and the pointwise multiplication operator $M_g$ defined via the identical symbol function is characterized by the same condition determined by the symbol's state of belonging to a Bloch-type space. Under a certain additional limitation on the composition symbol function $\varphi$, we also show that this condition is also equivalent to the continuity of the weighted composition operator $W_{g,\varphi}$. These results have consequences related to the invertibility of $W_{g,\varphi}$ acting on a weighted Bergman Fréchet or (LB)space. Some results concerning eigenvalues of such composition operators $C_\varphi$ are presented.

Nagoya Mathematical Journal, 2005

LetH(Dn)be the space of holomorphic functions on the unit polydisk Dn, and let, wherep, q&gt; 0,α = (α1,…,αn) with αj&gt; -1,j =1,...,n, be the class of all measurable functions f defined on Dnsuch thatwhereMp(f,r)denote thep-integral means of the functionf. Denote the weighted Bergman space on. We provide a characterization for a functionfbeing in. Using the characterization we prove the following result: Letp&gt; 1, then the Cesàro operator is bounded on the space.

Journal of Inequalities and Applications, 2021

For 1 ≤ p < ∞, let A p ω be the weighted Bergman space associated with an exponential type weight ω satisfying D K z (ξ) ω(ξ) 1/2 dA(ξ) ≤ Cω(z)-1/2 , z ∈ D, where K z is the reproducing kernel of A 2 ω. 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 ω *. As an application, we prove the boundedness of the Bergman projection on L p ω , identify the dual space of A p ω , 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 ω into A q ω , 1 ≤ p, q < ∞, such as Toeplitz and (big) Hankel operators.