{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:2HQ6EGZTKQCRF7YNQLZA64Q2VI","short_pith_number":"pith:2HQ6EGZT","schema_version":"1.0","canonical_sha256":"d1e1e21b33540512ff0d82f20f721aaa08c52b6d30660227a16ec733760e1540","source":{"kind":"arxiv","id":"1603.04247","version":2},"attestation_state":"computed","paper":{"title":"Noncommutative harmonic analysis on semigroup and ultracontractivity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OA","authors_text":"Xiao Xiong","submitted_at":"2016-03-14T13:04:09Z","abstract_excerpt":"We extend some classical results of Cowling and Meda to the noncommutative setting. Let $(T_t)_{t>0}$ be a symmetric contraction semigroup on a noncommutative space $L_p(\\mathcal{M}),$ and let the functions $\\phi$ and $\\psi$ be regularly related. We prove that the semigroup $(T_t)_{t>0}$ is $\\phi$-ultracontractive, i.e. $\\|T_t x\\|_\\infty \\leq C \\phi(t)^{-1} \\|x\\|_1$ for all $x\\in L_1(\\mathcal{M})$ and $ t>0$ if and only if its infinitesimal generator $L$ has the Sobolev embedding properties: $\\|\\psi(L)^{-\\alpha} x\\|_q \\leq C'\\|x\\|_p$ for all $x\\in L_p(\\mathcal{M}),$ where $10}$ be a symmetric contraction semigroup on a noncommutative space $L_p(\\mathcal{M}),$ and let the functions $\\phi$ and $\\psi$ be regularly related. We prove that the semigroup $(T_t)_{t>0}$ is $\\phi$-ultracontractive, i.e. $\\|T_t x\\|_\\infty \\leq C \\phi(t)^{-1} \\|x\\|_1$ for all $x\\in L_1(\\mathcal{M})$ and $ t>0$ if and only if its infinitesimal generator $L$ has the Sobolev embedding properties: $\\|\\psi(L)^{-\\alpha} x\\|_q \\leq C'\\|x\\|_p$ for all $x\\in L_p(\\mathcal{M}),$ where $1