{"paper":{"title":"Mixed-norm of orthogonal projections and analytic interpolation on dimensions of measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bochen Liu","submitted_at":"2022-03-06T14:34:39Z","abstract_excerpt":"Suppose $\\mu, \\nu$ are compactly supported Radon measures on $\\mathbb{R}^d$ and $V\\in G(d,n)$ is an $n$-dimensional subspace. In this paper we systematically study the mixed-norm $$\\int\\|\\pi^y\\mu\\|_{L^p(G(d,n))}^q\\,d\\nu(y),\\ \\forall\\,p,q\\in[1,\\infty),$$ where $\\pi_V:\\mathbb{R}^d\\rightarrow V$ denotes the orthogonal projection and $$\\pi^y\\mu(V)=\\int_{y+V^\\perp}\\mu\\,d\\mathcal{H}^{d-n}=\\pi_V\\mu(\\pi_Vy),\\ \\text{if $\\mu$ has continuous density}.$$ When $n=d-1$ and $p=q$, our result significantly improves a previous result of Orponen.\n  In the proof we consider integer exponents first, then interpol"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2203.02973","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2203.02973/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}