Formulae for the distance in some quasi-Banach spaces

Abstract

Let (A_{0, A1}) be a compatible pair of quasi-Banach spaces and 1etA be a corresponding space of real interpolation type such that A₀∩A₁ is not dense in A. Upper and lower estimates are obtained for the distance of any element f of A from A₀∩A₁. These lead to formulae for the distance in a large number of concrete situations, such as when A₀∩A₁=L^∞ and A is either weak-L^q, a ‘grand’ Lebesgue space or an Orlicz space of exponential type.