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

Abstract

Let 0 < α ≤ ∞ and let { B ( x , ϵ ) } ϵ , ϵ > 0 , denote a net of intervals of the form ( x − ϵ , x + ϵ ) ⊂ [ 0 , α ) . Let f ϵ ( x ) be any best constant approximation of f ∈ Λ w , ϕ ′ on B ( x , ϵ ) . Weak inequalities for maximal functions associated with { f ϵ ( x ) } ϵ , 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 ϵ ( x ) to f ( x ) , as ϵ → 0 .