On the Continuity of the Generalized Nemytskii Operator on Spaces of Differentiable Functions

We obtain sufficient conditions for the continuity of the general nonlinear superposition operator (generalized Nemytskii operator) acting from the space of differentiable functions on a bounded domain to the Lebesgue space . The values of operators on a function are locally determined by the values of both the function itself and all of its partial derivatives up to order inclusive. In certain particular cases, the sufficient conditions obtained are proved to be necessary as well. The results are illustrated by several examples, and an application to the theory of Sobolev spaces is also given.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ? ?n for functions defined onG to be extendable by zero to the entire space ?n with smoothness preserved in an integral norm.     

Embeddings of Spaces of Functions of Positive Smoothness on Irregular Domains

Doklady Mathematics - For spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space, an embedding theorem into spaces of the same type is proved and...     

Interpolation of Spaces of Functions of Positive Smoothness on a Domain

Doklady Mathematics - An interpolation theorem for spaces of functions of positive smoothness on a domain with flexible cone condition is established.     

Spaces of functions of fractional smoothness on an irregular domain

On an irregular domain G ⊂ ℝ ⁿ of a certain type, we introduce spaces of functions of fractional smoothness s > 0. We prove embedding theorems relating these spaces to the Sobolev spaces W _p ^m (G) and Lebesgue spaces L _p (G).     

The boundary behavior of components of polyharmonic functions

We consider the following representation of polyharmonic functions on the unit ballD ^m : 450-1 where the _j are harmonic onD ^m . We study the relation between uniform boundary properties ofƒ (its smoothness and growth while approaching the boundary) and the same properties of the terms in this representation. The theorems proved in this paper generalize some results obtained by Dolzhenko in the theory of polyanalytic functions.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 518–530, October, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01366.     

Function spaces of Lizorkin-Triebel type on an irregular domain

On an irregular domain G ⊂ ℝ ⁿ of a certain type, we introduce function spaces of fractional smoothness s > 0 that are similar to the Lizorkin-Triebel spaces. We prove embedding theorems that show how these spaces are related to the Sobolev and Lebesgue spaces W _p ^m (G) and L _p (G). Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 32–43.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.