Sharp reverse fractional Hausdorff inequality on power-weighted Lebesgue spaces

Abstract

Our main focus in this paper is to explore the mapping properties for the n-dimensional fractional Hausdorff operator \(H_{\Phi ,\beta }\) from \(L^{p}(\mathbb {R}^{n},|x|^{\alpha })\) to \(L^{q}(\mathbb {R}^{n},|x|^{\gamma })\), where \(p,q<1~(p,q\ne 0)\), \(\alpha ,\gamma \in \mathbb {R}\), \(0\le \beta <n\) and \(\Phi \) is a nonnegative measurable function on \(\mathbb {R}^n\). For \(p,q<1~(p,q\ne 0)\) satisfying some additional assumptions, we give sufficient conditions for the validity of the reverse fractional Hausdorff inequality \(\left\| H_{\Phi ,\beta }f\right\| _{L^{q}(\mathbb {R}^{n},|x|^{\gamma })}\ge C\Vert f\Vert _{L^{p}(\mathbb {R}^{n},|x|^{\alpha })}\) for some positive constant C and all nonnegative functions \(f\in L^{p}(\mathbb {R}^{n},|x|^{\alpha })\). For the particular case \(0<p=q<1\), we obtain the sharp reverse fractional Hausdorff inequality. As applications, we establish the sharp reverse inequalities for the n-dimensional fractional Hardy operator and its adjoint operator, and also the n-dimensional fractional Hardy–Littlewood–Pólya operator on power-weighted Lebesgue spaces.