A unified approach to self-improving property via K-functionals

TL;DR

This paper introduces a unified method using K-functionals to derive self-improving inequalities, improving classical results and establishing new limiting formulas for fractional operators on Banach spaces.

Contribution

It presents a novel approach leveraging K-functionals to obtain sharper inequalities and new limiting formulas, enhancing the understanding of self-improving properties in analysis.

Findings

Improved Poincaré-Ponce, Gaussian Sobolev, and John-Nirenberg inequalities.

Derived new Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova formulas.

Extended formulas to fractional powers of operators on Banach spaces.

Abstract

In this paper we obtain new quantitative estimates that improve the classical inequalities: Poincar\'e-Ponce, Gaussian Sobolev, and John-Nirenberg. Our method is based on the K-functionals and allows one to derive self-improving type inequalities. We show the optimality of the method by obtaining new Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova limiting formulas. In particular, we derive these formulas for fractional powers of infinitesimal generators of operator semigroups on Banach spaces.