Note on the Equivalence of Special Norms on the Lebesgue Space

Abstract

In this paper, we consider a norm based on the infinitesimal generator of the shift semigroup in a direction. The relevance of such a focus is guaranteed by an abstract representation of a uniformly elliptic operator by means of a composition of the corresponding infinitesimal generator. The main result of the paper is a theorem establishing equivalence of norms in functional spaces. Even without mentioning the relevance of this result for the constructed theory, we claim it deserves to be considered itself.

Keywords

• equivalence of norms
• compact embedding of spaces
• infinitesimal generator
• m-accretive operator
• uniformly elliptic operator