Fractional maximal operator in the local Morrey–Lorentz spaces and some applications

Abstract

In this study, we obtain the necessary and sufficient conditions for the boundedness of the fractional maximal operator \(M_{\alpha }\) in the local Morrey–Lorentz spaces \(M_{p,q;\lambda }^{loc}({\mathbb {R}}^n)\). We use sharp rearrangement inequalities while proving our result. We apply this result to the Schrödinger operator \(-\Delta + V\) on \({\mathbb {R}}^n\), where the nonnegative potential V belongs to the reverse Hölder class \(B_{\infty }({\mathbb {R}}^n)\). The local Morrey–Lorentz \(M_{p,r;\lambda }^{loc}({\mathbb {R}}^n) \rightarrow M_{q,s;\lambda }^{loc}({\mathbb {R}}^n)\) estimates for the Schrödinger type operators \(V^{\gamma } (-\Delta +V)^{-\beta }\) and \(V^{\gamma } \nabla (-\Delta +V)^{-\beta }\) are obtained.