Generalized Fractional Maximal Functions In Lorentz Spaces Λ

Abstract

In this paper we give the complete characterization of the boundedness of generalized fractional maximal operator M-phi,Lambda(alpha)(b)f(x):=Sup(Q(sic)x) parallel to f chi Q parallel to(Lambda alpha(b))/phi(vertical bar Q vertical bar) (x is an element of R-n), between the classical Lorentz spaces Lambda(P)(v) and Lambda(q)(w), as well as between Lambda(P)(v) and weaktype Lorentz spaces Lambda(q)(,infinity)(w), and between Lambda(P,infinity)(v) and Lambda(q,infinity)(w), and between Lambda(P,infinity)(v) and Lambda(q)(w), for appropriate functions phi, where 0 < p, q,alpha < infinity, v,w, b are weights on (0, infinity) such that 0 < B(t) := integral(t)(0) b infinity, t 0, B is an element of Delta(2) and B(t)/t(r) is quasi-increasing for some 0 < r <= 1.