{"paper":{"title":"Two-weight $L^p\\to L^q$ bounds for positive dyadic operators in the case $0<q< 1 \\le p<\\infty$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Igor E. Verbitsky, Timo S. H\\\"anninen","submitted_at":"2017-06-27T03:18:55Z","abstract_excerpt":"Let $\\sigma$, $\\omega$ be measures on $\\mathbb{R}^d$, and let $\\{\\lambda_Q\\}_{Q\\in\\mathcal{D}}$ be a family of non-negative reals indexed by the collection $\\mathcal{D}$ of dyadic cubes in $\\mathbb{R}^d$. We characterize the two-weight norm inequality, \\begin{equation*} \\lVert T_\\lambda(f\\sigma)\\rVert_{L^q(\\omega)}\\le C \\, \\lVert f \\rVert_{L^p(\\sigma)}\\quad \\text{for every $f\\in L^p(\\sigma)$,} \\end{equation*} for the positive dyadic operator \\begin{equation*} T_\\lambda(f\\sigma):= \\sum_{Q\\in \\mathcal{D}} \\lambda_Q \\, \\Big(\\frac{1}{\\sigma(Q)} \\int_Q f\\mathrm{d}\\sigma\\Big) \\, 1_Q \\end{equation*} "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.08657","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"}