Geometric Criteria for Morrey Admissibility via the Real Log-Canonical Threshold

Abstract

We study local integrability of gradients of singular interaction kernels in aggregation equations. Suppose near the singularity that $|\nabla K|\asymp |\nabla f||f{|}^{-(\kappa +1)}$, where $\kappa >0$ and $f$ is real-analytic with $f(0)=0$. A real log-resolution of $f$ and its Jacobian ideal gives the exact criterion $\nabla K\in L_{\mathrm{loc}}^p$ if and only if $p<p^{\ast }(\kappa )$, where \[ p^*(κ)=\min_i\frac{a_i+1}{(κ+1)ν_i-M_i}. \] Here ${\nu }_i$, $M_i$, and $a_i$ are the vanishing orders of $f$, those of its Jacobian ideal, and the discrepancy exponents. Moreover, ${\mathrm{rlct}}_0(f)/(\kappa +1)\le p^{\ast }(\kappa )$, with equality exactly when an RLCT-computing divisor has $M_i=0$; in particular, the inequality is strict for isolated zeros in dimension at least two. For quasi-homogeneous singularities with isolated real zero, $p^{\ast }(\kappa )=|w|/(\kappa d+{\max }_i{w}_i)$. Finally, if $p^{\ast }(\kappa )>1$, a suitable global truncation gives $\nabla K\in L^1({\mathbb{R}}^n)$, yielding the kernel hypothesis in the Morrey-space well-posedness theorem of \cite{suleiman2023existence}.