New Extrapolation Estimates

Given a sublinear operator T satisfying that ‖TχA‖Lp(ν)⩽Cp−1 ‖χA‖Lp(μ), for every measurable set A and every 1<p⩽p0, with C independent of A and p, we show that supr>0 ∫∞1/r λνTf(y) dy1+log+ r≲∫M |f(x)| (1+log+ |f(x)|) dμ(x). This estimate allows us to improve Yanoʹs extrapolation theorem and also to obtain that for every f∈L log L(μ), r→∞ ∫∞1/r λνTf(y) dylog r≲‖f‖1. Other types of extrapolation results are also given.