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

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