Boundedness and compactness of Hardy operators on Lorentz-type spaces

Abstract

Given non-negative measurable functions ϕ,ψ on Rn we study the high dimensional Hardy operator Hf(x)=ψ(x)∫B(0,|x|)f(y)ϕ(y)dy between Orlicz–Lorentz spaces ΛXG(w), where f is a measurable function of x∈Rn and B(0,t) is the ball of radius t in Rn. We give sufficient conditions of boundedness of H:Λu0G0(w0)→Λu1G1(w1) and H:Λu0G0(w0)→Λu1G1,∞(w1). We investigate also boundedness and compactness of H:Λu0p0(w0)→Λu1p1,q1(w1) between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.