{"paper":{"title":"A decomposition technique for integrable functions with applications to the divergence problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fernando L\\'opez Garc\\'ia","submitted_at":"2013-08-20T16:45:30Z","abstract_excerpt":"Let $\\Omega\\subset \\mathbb{R}^n$ be a bounded domain that can be written as $\\Omega=\\bigcup_{t} \\Omega_t$, where $\\{\\Omega_t\\}_{t\\in\\Gamma}$ is a countable collection of domains with certain properties. In this work, we develop a technique to decompose a function $f\\in L^1(\\Omega)$, with vanishing mean value, into the sum of a collection of functions $\\{f_t-\\tilde{f}_t\\}_{t\\in\\Gamma}$ subordinated to $\\{\\Omega_t\\}_{t\\in\\Gamma}$ such that $Supp\\,(f_t-\\tilde{f}_t)\\subset\\Omega_t$ and $\\int f_t-\\tilde{f}_t=0$. As an application, we use this decomposition to prove the existence of a solution in we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.4346","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"}