{"paper":{"title":"Gaussian Integral Means of Entire Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.CV","authors_text":"Chunjie Wang, Jie Xiao","submitted_at":"2013-01-02T22:55:39Z","abstract_excerpt":"For an entire mapping $f:\\mathbb C\\mapsto\\mathbb C$ and a triple $(p,\\alpha, r)\\in (0,\\infty)\\times(-\\infty,\\infty)\\times(0,\\infty]$, the Gaussian integral means of $f$ (with respect to the area measure $dA$) is defined by $$ {\\mathsf M}_{p,\\alpha}(f,r)=\\Big({\\int_{|z|<r}e^{-\\alpha|z|^2}dA(z)}\\Big)^{-1}{\\int_{|z|<r}|f(z)|^p{e^{-\\alpha|z|^2}}dA(z)}. $$ Via deriving a maximum principle for ${\\mathsf M}_{p,\\alpha}(f,r)$, we establish not only Fock-Sobolev trace inequalities associated with ${\\mathsf M}_{p,p/2}(z^m f(z),\\infty)$ (as $m=0,1,2,...$), but also convexities of $r\\mapsto\\ln {\\mathsf M}_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0349","kind":"arxiv","version":3},"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"}