One more theorem on norm equivalence in the Lebesgue space

TL;DR

This paper proves a new theorem establishing the equivalence of certain norms in Lebesgue spaces, using the infinitesimal generator of the shift semigroup, which has implications for fractional integro-differential operators.

Contribution

It introduces a novel norm based on the infinitesimal generator and proves its equivalence with existing norms in functional spaces.

Findings

Established norm equivalence in Lebesgue spaces

Linked fractional integro-differential operators to shift semigroup generators

Provided a new perspective on functional space analysis

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 fractional integro-differential 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.