{"paper":{"title":"Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Anna Kairema, Ji Li, Lesley Ward, M. Cristina Pereyra","submitted_at":"2015-09-12T17:12:28Z","abstract_excerpt":"We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for $L^p$. Next we focus on spaces $X$ of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces $H^1(X)$ and ${\\rm BMO}(X)$. In the setting of product spaces $\\widetilde{X} = X_1 \\times \\cdots \\times X_n$ of homogeneous type, we show that the spa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.03761","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"}