Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces

Abstract

In this paper, we give a boundedness criterion for the potential operator ℐα{ {\mathcal I} }^{\alpha } in the local generalized Morrey space LMp,φ{t0}(Γ)L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the generalized Morrey space Mp,φ(Γ){M}_{p,\varphi }(\text{Γ}) defined on Carleson curves Γ\text{Γ}, respectively. For the operator ℐα{ {\mathcal I} }^{\alpha }, we establish necessary and sufficient conditions for the strong and weak Spanne-type boundedness on LMp,φ{t0}(Γ)L{M}_{p,\varphi }^{\{{t}_{0}\}}(\text{Γ}) and the strong and weak Adams-type boundedness on Mp,φ(Γ){M}_{p,\varphi }(\text{Γ}).

Keywords

• carleson curve
• local generalized morrey space
• potential operator
• adams-type inequalities
• 26a33
• 42b25
• 42b35
• 47b38