Hardy-Steklov operators and embedding inequalities of Sobolev type

TL;DR

This paper characterizes weighted norm inequalities related to the embedding of absolutely continuous functions into fractional Sobolev spaces, providing necessary and sufficient conditions for the Hardy--Steklov operator's boundedness in weighted Lebesgue spaces.

Contribution

It introduces new necessary and sufficient conditions for the Hardy--Steklov operator's boundedness, advancing understanding of Sobolev space embeddings and integral operator inequalities.

Findings

Derived conditions for Hardy--Steklov operator boundedness

Established embedding inequalities for fractional Sobolev spaces

Provided auxiliary results of independent mathematical interest

Abstract

We characterize a weighted norm inequality which corresponds to the embedding of a class of absolutely continuous functions into the fractional order Sobolev space. The auxiliary result of the paper is of independent interest. It comprises of several types of necessary and sufficient conditions for the boundedness of the Hardy--Steklov operator (an integral operator with two variable boundaries of integration) in weighted Lebesgue spaces.