Endpoint Lorentz Inequality for Riesz Potentials with Respect to General Measures

Let $n\ge 2$, $\alpha \in (0,n)$, and $\mu$ be a locally finite nonnegative Borel measure on ${\mathbb{R}}^n$. We prove that, for any $f\in C_{\mathrm{c}}^{\mathrm{\infty }}({\mathbb{R}}^n)$, \begin{align*} |I_αf|{L^{\frac{n}{n-α},1}(μ)} \le C{n,α} \int_{$\mathbb{R}$^n} |Rf(x)|[Mμ(x)]^{1-\fracα{n}},dx, \end{align*} where $I_{\alpha }$ is the Riesz potential, $R$ is the Riesz transform, $M\mu$ is the Hardy–Littlewood maximal function of $\mu$, and $C_{n,\alpha }$ is a positive constant depending only on $n$ and $\alpha$. This answers a question posed by Bortz, Moen, Olivo, Pérez, and Rela, with the case $\alpha =1$ already known, and extends Spector’s optimal unweighted Lorentz inequality to general measures. As applications, we obtain a Lorentz estimate for curl-free vector fields, Sobolev–Lorentz and Hardy inequalities with respect to general measures, and an $A_1$-weighted Lorentz estimate with a sharp exponent of the $A_1$ characteristic.