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

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.