On Iwaniec-Sbordone Spaces on Sets Which May Have Infinite Measure (pp. 67-84)

Abstract

We introduce grand Lebesgue spaces on open sets Ω of infinite measure in R n , controlling the integrability of |f(x)| p - e at infinity by means of a weight (depending also on e); in general, such spaces are different for different ways to introduce dependence of the weight on e. We prove some properties of these spaces. We introduce also a version of weighted grand Lebesgue spaces, different from the usual ones, for bounded sets, in which together with the passage from p to p−e we introduce a weight also depending on e. In both versions we show that every linear operator bounded in a Lebesgue space with Muckenhoupt weights is also bounded in the corresponding grand Lebesgue space a Muckenhoupt weight (for bounded or unbounded Ω).