Weak inequalities for maximal functions in Orlicz–Lorentz spaces and applications

Abstract

Let 00, denote a net of intervals of the form (x-epsilon,x+epsilon) subset [0,alpha). Let f^{epsilon}(x) be any best constant approximation of f in Lambda_{w,phi´} on B(x,epsilon). Weak inequalities for maximal functions associated with {f^{epsilon}(x)}_epsilon, in Orlicz-Lorentz spaces, are proved. As a consequence of these inequalities we obtain a generalization of Lebesgue´s Differentiation Theorem and the pointwise convergence of f^{epsilon}(x) to f(x), as epsilon tends to 0.