{"paper":{"title":"Boundedness of non-homogeneous square functions and $L^q$ type testing conditions with $q \\in (1,2)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Henri Martikainen, Mihalis Mourgoglou","submitted_at":"2014-01-21T20:44:56Z","abstract_excerpt":"We continue the study of local $Tb$ theorems for square functions defined in the upper half-space $(\\mathbb{R}^{n+1}_+, \\mu \\times dt/t)$. Here $\\mu$ is allowed to be a non-homogeneous measure in $\\mathbb{R}^n$. In this paper we prove a boundedness result assuming local $L^q$ type testing conditions in the difficult range $q \\in (1,2)$. Our theorem is a non-homogeneous version of a result of S. Hofmann valid for the Lebesgue measure. It is also an extension of the recent results of M. Lacey and the first named author where non-homogeneous local $L^2$ testing conditions have been considered."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5457","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"}