{"paper":{"title":"Bounded compositions on scaling invariant Besov spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.CA","authors_text":"Eero Saksman, Herbert Koch, Pekka Koskela, Tom\\'as Soto","submitted_at":"2012-09-28T10:45:00Z","abstract_excerpt":"For $0 < s < 1 < q < \\infty$, we characterize the homeomorphisms $\\varphi : \\real^n \\to \\real^n$ for which the composition operator $f \\mapsto f \\circ \\varphi$ is bounded on the homogeneous, scaling invariant Besov space $\\dot{B}^s_{n/s,q}(\\real^n)$, where the emphasis is on the case $q\\not=n/s$, left open in the previous literature. We also establish an analogous result for Besov-type function spaces on a wide class of metric measure spaces as well, and make some new remarks considering the scaling invariant Triebel-Lizorkin spaces $\\dot{F}^s_{n/s,q}(\\real^n)$ with $0 < s < 1$ and $0 < q \\leq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.6477","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"}