{"paper":{"title":"The $L^p$ primitive integral","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Erik Talvila","submitted_at":"2012-08-17T21:21:53Z","abstract_excerpt":"For each $1\\leq p<\\infty$ a space of integrable Schwartz distributions, $L^'^{\\,p}$, is defined by taking the distributional derivative of all functions in $L^p$. Here, $L^p$ is with respect to Lebesgue measure on the real line. If $f\\in L^'^{\\,p}$ such that $f$ is the distributional derivative of $F\\in L^p$ then the integral is defined as $\\int^\\infty_{-\\infty} fG=-\\int^\\infty_{-\\infty} F(x)g(x)\\,dx$, where $g\\in L^q$, $G(x)= \\int_0^x g(t)\\,dt$ and $1/p+1/q=1$. A norm is $\\lVert f\\rVert'_p=\\lVert F\\rVert_p$. The spaces $L^'^{\\,p}$ and $L^p$ are isometrically isomorphic. Distributions in $L^'^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.3694","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"}