New extrapolation estimates

Given a sublinear operator T satisfying that \\T chi(A)\\(Lp(v)) less than or equal to C/p - 1 \\chi(A)\\(Lp(mu)), foe every measurable set A and every 1 < p less than or equal to p(0), with C independent of A and p, we show that r>0sup integral(1/r)((x)) lambda(Tj)(v) (y) dy/1 + log(+) r less than or si milar to integral(M) \f(X)\ (1 + log(+) \f(x)\) d mu(x). This estimate allows us to improve Yano's extrapolation theorem and also to obtain that for every f epsilon L log L(mu), r -->infinity lim integral(1/r)(infinity) lambda(Tf)(v)(y) dy/log r less th an or similar to \\f\\(I). Other types of extrapolation results are also given. (C) 2000 Academic Pres s.