{"paper":{"title":"A weak type $(p,a)$ criterion for operators, and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CA"],"primary_cat":"math.FA","authors_text":"Bernhard H. Haak, El-Maati Ouhabaz","submitted_at":"2025-09-10T07:16:28Z","abstract_excerpt":"Let $(X, d, \\mu)$ be a space of homogeneous type and $\\Omega$ an open subset of $X$. Given a bounded operator $T: L^p(\\Omega) \\to L^q(\\Omega)$ for some $1 \\le p \\le q < \\infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for $p_0$ and $a$ such that $\\frac{1}{p_0} - \\frac{1}{a} = \\frac{1}{p}-\\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0, a)$ for Riesz potentials $L^{-\\frac{\\alpha}{2}}$ or for Riesz transform type operators $\\nabla \\Delta^{-\\frac{\\alpha}{2}}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.08334","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"}