{"paper":{"title":"Remarks on functions with bounded Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Tarek M. Elgindi","submitted_at":"2016-05-17T17:57:58Z","abstract_excerpt":"$\\Delta \\psi:=\\frac{\\partial^2 \\psi}{\\partial x_1^2}+\\frac{\\partial^2 \\psi}{\\partial x_2^2}$ being locally bounded does not imply that $D^2\\psi$ is locally bounded. However, we prove that if $\\psi$ is invariant under rotation by $\\frac{2\\pi}{m}$, for some $m\\geq 3$, and $\\Delta \\psi$ is locally bounded, then $$\\sup_{x\\in B_1(0)}\\frac{|\\nabla \\psi(x)|}{|x|}<\\infty.$$ This is sharp in that there are examples of functions $\\psi$ for which $\\Delta \\psi$ is locally bounded, which are invariant under rotation by $\\pi$ with $|\\psi(x)-\\psi(0)|\\approx |x|^2 |\\log|x||$ as $|x|\\rightarrow 0$. This bound "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.05266","kind":"arxiv","version":1},"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"}