Boundedness for fractional Hardy-type operator on Herz-Morrey spaces with variable exponent

Original · EN

In this paper, the fractional Hardy-type operator of variable order β(x) is shown to be bounded from the Herz-Morrey spaces MKₚ₁,q₁₍·₎α,λ(Rⁿ) with variable exponent q₁(x) into the weighted space MKₚ₂,q₂₍·₎α,λ(Rⁿ,ω), where ω=(1+|x|)⁻γ⁽ˣ⁾ with some γ(x)>0 and 1/q₁(x)-1/q₂(x)=β(x)/n when q₁(x) is not necessarily constant at infinity. It is assumed that the exponent q₁(x) satisfies the logarithmic continuity condition both locally and at infinity that 1< q₁(∞)≤ q₁(x)≤(q₁)₊<∞ (x∈ Rⁿ).