{"paper":{"title":"An lp-boundedness of stochastic singular integral operators and its application to spdes","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Ildoo Kim, Kyeonghun Kim","submitted_at":"2016-08-31T05:09:04Z","abstract_excerpt":"In this article we introduce a stochastic counterpart of the H\\\"ormander condtion on the kernel $K(r,t,x,y)$: there exists a pseudo-metric $\\rho$ on $(0,\\infty)\\times R^d$ and a positive constant $C_0$ such that for $X=(t,x), Y=(s,y), Z=(r,z) \\in (0,\\infty) \\times R^d$, $$ \\sup_{X,Y}\\int_{0}^\\infty \\left[ \\int_{\\rho(X,Z) \\geq C_0 \\rho(X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\\right]^2 dr <\\infty. $$\n  We prove that the stochastic singular integral of the type $$ \\mathbb{T} g(t,x) :=\\int_0^{t} \\int_{R^d} K(t,s,x,y) g(s,y)dy dW_s $$ is a bounded operator on $\\mathbb{L}_p=L_p(\\Omega \\times (0,\\infty"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08728","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"}