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

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.