Weighted hardy operators in complementary morrey spaces - Lukkassen, Dag (2018)
We study the weighted p -> q-boundedness of the multidimensional weighted Hardy-type operators H-omega(alpha) and H-omega(alpha) with radial type weight omega = omega(vertical bar x vertical bar), in the generalized complementary Morrey spaces (C) L-(0)(p,psi) (R-n) defined by an almost increasing function psi = psi(r). We prove a theorem which provides conditions, in terms of some integral inequalities imposed on psi and omega, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the function psi and the weight omega are power functions. We also prove that the spaces (C) L-(0)(p,psi) (Omega) over bounded domains Omega are embedded between weighted Lebesgue space L-p with the weight psi and such a space with the weight psi, perturbed by a logarithmic factor. Both the embeddings are sharp.