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 $\nabla K\in L^p_{\mathrm{loc}}$ if and only if $p 1$, 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}.