{"paper":{"title":"Failure of $L^2$ boundedness of gradients of single layer potentials for measures with zero low density","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Jos\\'e M. Conde-Alonso, Mihalis Mourgoglou, Xavier Tolsa","submitted_at":"2018-01-25T15:35:23Z","abstract_excerpt":"Consider a totally irregular measure $\\mu$ in $\\mathbb{R}^{n+1}$, that is, the upper density $\\limsup_{r\\to0}\\frac{\\mu(B(x,r))}{(2r)^n}$ is positive $\\mu$-a.e.\\ in $\\mathbb{R}^{n+1}$, and the lower density $\\liminf_{r\\to0}\\frac{\\mu(B(x,r))}{(2r)^n}$ vanishes $\\mu$-a.e. in $\\mathbb{R}^{n+1}$. We show that if $T_\\mu f(x)=\\int K(x,y)\\,d\\mu(y)$ is an operator whose kernel $K(\\cdot,\\cdot)$ is the gradient of the fundamental solution for a uniformly elliptic operator in divergence form associated with a matrix with H\\\"older continuous coefficients, then $T_\\mu$ is not bounded in $L^2(\\mu)$. This ext"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.08453","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}