On weighted spaces of holomorphic functions of several variables

Peer reviewe

ON SOLID CORES AND HULLS OF WEIGHTED BERGMAN SPACES Aμ1

Funding Information: Open Access funding provided by University of Helsinki including Helsinki University Central Hospital. The research of Bonet was partially supported by the project MCIN PID2020-119457GB-I00/AEI/10.13039/501100011033. Publisher Copyright: © 2022, The Author(s).We consider weighted Bergman spaces Aμ1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of Aμ1. Also, as a consequence of a characterization of solid Aμ1-spaces, we show that, in the case of entire functions, there indeed exist solid Aμ1-spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of Aμ1, when μ equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted A1- and H∞-spaces, the validity of which requires the assumption that - log v belongs to the class W, studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight v(z) = exp (- 1 / (1 - | z|) provides an example satisfying both assumptions.Peer reviewe

On the boundedness of Toeplitz operators with radial symbols over weighted sup-norm spaces of holomorphic functions

We prove sufficient conditions for the boundedness and compactness of Toeplitz operators T-a in weighted sup-normed Banach spaces H-v(infinity) of holomorphic functions defined on the open unit disc D of the complex plane; both the weights v and symbols a are assumed to be radial functions on D. In an earlier work by the authors was shown that there exists a bounded, harmonic (thus non-radial) symbol a such that T-a is not bounded in any space H-v(infinity) with an admissible weight v. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol a at the boundary of D guarantees the boundedness of T-a. The sufficient conditions for the boundedness and compactness of T-a, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition (B), which includes in addition to standard weight classes also many rapidly decreasing weights. (c) 2020 Elsevier Inc. All rights reserved.Peer reviewe

Solid hulls and cores of weighted H-infinity-spaces

We determine the solid hull and solid core of weighted Banach spaces H-upsilon(infinity) of analytic functions functions f such that upsilon vertical bar f vertical bar is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights upsilon. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if H-upsilon(infinity) is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in H-upsilon(infinity). As a consequence H-upsilon(infinity) does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions.Peer reviewe

On boundedness and compactness of Toeplitz operators in weighted H-infinity-spaces

Peer reviewe

SOLID CORES AND SOLID HULLS OF WEIGHTED BERGMAN SPACES

We determine the solid hull for 2 <p <infinity and the solid core for 1 <p <2 of weighted Bergman spaces A(mu)(p) , 1 <p <infinity, of analytic functions on the disk and on the whole complex plane, for a very general class of nonatomic positive bounded Borel measures mu . New examples are presented. Moreover, we show that the space A(mu)(p) , 1 <p <infinity, is solid if and only if the monomials are an unconditional basis of this space.Peer reviewe

Schauder bases and the decay rate of the heat equation

We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space $${\mathbb {R}}^N$$RN. We show that given a weighted $$L^p$$Lp-space $$L_w^p({\mathbb {R}}^N)$$Lwp(RN)with $$1 \le p 0such that, if the initial data f belong to the closed linear space of$$e_n$$enwith$$n \ge n_m$$n≥nm, then the decay rate of the solution of the heat equation is at least$$t^{-m}$$t-m. Such a basis can be constructed as a perturbation of any given Schauder basis. The proof is based on a construction of a basis of$$L_w^p( {\mathbb {R}}^N)$$Lwp(RN), which annihilates an infinite sequence of bounded functionals.Peer reviewe