Hardy-Sobolev Type Integrability for Generalized Riesz Potentials in Weighted Morrey-Orlicz Spaces

As an extension of Hardy inequalities, we study Hardy-Sobolev integrabilities for generalized Riesz potentials Rα,mf(x)=∫B| x−y |α−nf(y)dy+∫Rn\B(|x−y|α−n−∑|ℓ|≤m−1xℓℓ!(DℓRα)(−y))f(y)dy of functions f in weighted Morrey-Orlicz spaces, where B denotes the unit ball B(0,1) in Rn and Rα(x)=|x|α−n. As an application, we prove the Hardy-Sobolev integrabilities for generalized Riesz potentials in the framework of general double phase functionals given by ϕ(x,t)=ϕ1(t)+ϕ2(b(x)t), x∈Rn,t≥0 , where ϕ1,ϕ2 are positive convex functions on (0,∞) and b is a non-negative function on [0,∞) which is Hölder continuous of order θ∈(0,1].