On sharp constants in one-dimensional embedding theorems of arbitrary order

TL;DR

This paper investigates conditions under which the extremal function in one-dimensional embedding theorems of arbitrary order is constant, providing a complete characterization of such cases.

Contribution

It establishes necessary and sufficient conditions for the extremal function to be constant in embedding theorems of arbitrary (including non-integer) order.

Findings

Characterization of when the extremal function is constant

Necessary and sufficient conditions derived

Applicable to non-integer order embedding theorems

Abstract

We discuss the problem where the extremal function in embedding theorem of some (generally speaking, non-integer) order is the constant function. We obtain the necessary and sufficient conditions of this.