Weighted Hardy operators in complementary Morrey spaces
Summary: We study the weighted \(p \to q\)-boundedness of the multidimensional weighted Hardy-type operators \(H^\alpha_w\) and \(\mathcal H^\alpha_w\) with radial type weight \(w = w(|x|)\), in the generalized complementary Morrey spaces \(^\complement \mathcal L^{p,\psi}_{\{0\}}(\mathbb R^n)\) 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 \(w\), 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 \(w\) are power functions. We also prove that the spaces \(^\complement \mathcal L^{p,\psi}_{\{0\}}(\Omega)\) 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.
scientific article; zbMATH DE number 6134538
scientific article