Generalized fractional integral operators on generalized local Morrey spaces (Q492563)

Summary: We study the continuity properties of the generalized fractional integral operator \(I_\rho\) on the generalized local Morrey spaces \(LM_{p,\varphi}^{\{x_0\}}\) and generalized Morrey spaces \(M_{p,\varphi}\). We find conditions on the triple \((\varphi_1,\varphi_2,\rho)\) which ensure the Spanne-type boundedness of \(I_\rho\) from one generalized local Morrey space \(LM_{p,\varphi_1}^{\{x_0\}}\) to another \(LM_{q,\varphi_2}^{\{x_0\}}\), \(1<p<q<\infty\), and from \(LM_{1,\varphi_1}^{\{x_0\}}\) to the weak space \(WLM_{q,\varphi_2}^{\{x_0\}}\), \(1<q<\infty\). We also find conditions on the pair \((\varphi,\rho)\) which ensure the Adams-type boundedness of \(I_\rho\) from \(M_{p,\varphi^{1/p}}\) to \(M_{q,\varphi^{1/q}}\) for \(1<p<q<\infty\) and from \(M_{1,\varphi}\) to \(WM_{q,\varphi^{1/q}}\) for \(1<q<\infty\). In all cases the conditions for the boundedness of \(I_\rho\) are given in terms of Zygmund-type integral inequalities on \((\varphi_1,\varphi_2,\rho)\) and \((\varphi,\rho)\), which do not assume any assumption on monotonicity of \(\varphi_1(x,r)\), \(\varphi_2(x,r)\), and \(\varphi(x,r)\) in \(r\).