{"paper":{"title":"Littlewood--Paley--Stein Estimates for Non-local Dirichlet Forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Huaiqian Li, Jian Wang","submitted_at":"2017-04-10T03:00:23Z","abstract_excerpt":"We obtain the boundedness in $L^p$ spaces for all $1<p<\\infty$ of the so-called vertical Littlewood--Paley functions for non-local Dirichlet forms in the metric measure space under some mild assumptions. For $1<p\\le 2$, the pseudo-gradient is introduced to overcome the difficulty that chain rules are not available for non-local operators, and then the Mosco convergence is used to pave the way from the finite jumping kernel case to the general case, while for $2\\le p<\\infty$, the Burkholder--Davis--Gundy inequality is effectively applied. The former method is analytic and the latter one is prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.02690","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"}