An inhomogeneous radial operator and an improvement of the Stein-Weiss inequality

Abstract

We give the necessary and sufficient conditions for the weighted inequality \begin{document}$ \begin{equation*} \||x|^{-b} T_{{\alpha}_{1}, \dots, {\alpha}_{m}, {\beta}_{1}, \dots, {\beta}_{m}, {\gamma}_{1}, \dots, {\gamma}_{m}}f \|_{L^q(\mathbb{R}^{n})} \leq C \||x|^{a} f\|_{L^p(\mathbb{R}^{n})} \end{equation*} $\end{document} both inside and outside the scaling regime of the singular operator$ \begin{equation*} T_{{\alpha}_{1}, \dots, {\alpha}_{m}, {\beta}_{1}, \dots, {\beta}_{m}, {\gamma}_{1}, \dots, {\gamma}_{m}}f(x): = \int_{\mathbb{R}^n} \prod\limits_{i = 1}^{m}||x|^{\alpha_i}-|y|^{\beta_i}|^{ -\gamma_i} f(y) dy, \end{equation*} $with $ a, b, \alpha_i, \beta_i>0 $, $ 0<\gamma_i<1 $, $ i = 1, \cdots, m $. As a consequence, we characterize an improvement of the Stein-Weiss inequality.