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Media type:
E-Article
Title:
Noncommutative Harmonic Analysis on Semigroups and Ultracontractivity
Contributor:
Xiong, Xiao
imprint:
Department of Mathematics of Indiana University, 2017
Published in:Indiana University Mathematics Journal
Language:
English
ISSN:
0022-2518;
1943-5258
Origination:
Footnote:
Description:
<p>We extend some classical results of Cowling and Meda to the noncommutative setting. Let (Tt)t>0 be a symmetric contractive semigroup on a noncommutative space Lp(M), and let the functions φ and ψ be regularly related. We prove that the semigroup (Tt)t>0 is φ-ultracontractive if and only if its infinitesimal generator L has the Sobolev embedding properties; that is, ∥Ttχ∥∞ ≤ Cφ(t)⁻¹ ∥χ∥₁ for all χ ϵ L₁(M) and t > 0 if and only if ∥ψ(L)-αχ∥q ≤ C' ∥χ∥p for all x ϵ Lp(M), where 1 < p < q < ∞ and α = 1/p – 1/q. We establish some noncommutative spectral multiplier theorems and maximal function estimates for the generator of a φ-ultracontractive semigroup. We also show the equivalence between φ-ultracontractivity and logarithmic Sobolev inequality for some special φ. Finally, we give some results on local ultracontractivity.</p>