On the range space of Yano's extrapolation theorem and new extrapolation estimates at infinity

Abstract

Given a sublinear operator T satisfying that T f Lp (ν) ≤ C p-1 f Lp (µ), for every 1 < p ≤ p0, with C independent of f and p, it was proved in [C] that ∞ λν f (y) dy T 1/r sup |f (x)|(1 + log+ |f (x)|) dµ(x). r>0 1 + log+ r M This estimate implies that T : L log L → B, where B is a re- arrangement invariant space. The purpose of this note is to give several characterizations of the space B and study its associate space. This last information allows us to formulate an extrap- olation result of Zygmund type for linear operators satisfying T f Lp (ν) ≤q Cp f Lp (µ), for every p ≥ p0.