Some particular norm in the Sobolev space $H^{1}[a,b].$

TL;DR

This paper explores a special norm in the Sobolev space H^1[a,b], providing an alternative characterization linked to a generalized isoperimetric inequality, extending previous work on a related reproducing kernel Hilbert space.

Contribution

It offers a new description of a Sobolev space with a norm connected to geometric inequalities, building on prior constructions of a related Hilbert space.

Findings

Identifies a Sobolev space with a specialized norm.

Connects the norm to a generalized isoperimetric inequality.

Provides an alternative characterization of the space.

Abstract

This paper is a continuation of the recent paper of the author, where a certain reproducing kernel Hilbert space $X_{\mathcal{S}}$ was constructed. The norm in $X_{\mathcal{S}}$ is related to a certain generalized isoperimetric inequality in ${\mathbb R^2}$. In the present paper we give an alternative description of the space $X_{\mathcal{S}}$, which appears to be a Sobolev space $H^{1}[a,b]$ with some special norm.