On geometric characterizations of mappings generating composition operators on Sobolev spaces

In this paper, we consider refined geometric characterizations of weak p-quasiconformal mappings φ:Ω→Ω~, where Ω and Ω~ are domains in Rn. We prove that mappings with bounded geometric p-dilatation on the set Ω\\S, where S is a set with σ-finite (n-1)-measure, are Sobolev Wp,loc1-mappings and generate bounded composition operators on Sobolev spaces.

We describe the $$(p,q)$$ Fock–Carleson measures for weighted Fock–Sobolev spaces in terms of the objects $$(s,t)$$ -Berezin transforms, averaging functions, and averaging sequences on the complex space $$\mathbb{C }^n$$ . The main results show that while these objects may have growth not faster than polynomials to induce the $$(p,q)$$ measures for $$q\ge p$$ , they should be of $$L^{p/(p-q)}$$ integrable against a weight of polynomial growth for $$q<p$$ . As an application, we characterize the bounded and compact weighted composition operators on the Fock–Sobolev spaces in terms of certain Berezin type integral transforms on $$\mathbb{C }^n$$ . We also obtained estimation results for the norms and essential norms of the operators in terms of the integral transforms. The results obtained unify and extend a number of other results in the area.

Composition operators on holomorphic Sobolev spaces in [formula omitted

Maps F: S × P ↦ S, where S is a Riemann surface and P is the set of positive reals, are studied under the assumption that f t : p ↦ F(p,t),p ∈ S, is holomorphic and f s+t = f s ○ f t , s, t ∈ P. Apart from cases where the fundamental group of S is abelian, the f t are conformal automorphisms of S and F is trivial if it is continuous. When S is the open unit disk and F is continuous, the latter admits a simple geometric characterization with the aid of the notion of a region convex in a given direction thanks to work of Grunsky (J.M.A.A. 34 (1971), 685–701). The case, S a plane region and F continuous, was studied independently by Berkson and Porta (Mich. Math. J. 25 (1978), 101–115) with the view to investigating semigroups of composition operators on Hardy classes. Their work does not include the trivialization of F when the fundamental group of S is non-abelian nor the connection with the theory of regions convex in a given direction.