A unified approach to self-improving property via K-functionals
摘要
In this paper we obtain new quantitative estimates that improve the classical inequalities: Poincaré–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.