Mapping properties of operator-valued pseudo-differential operators

Abstract

In this paper, we investigate the mapping properties of pseudo-differential operators with operator-valued symbols. Thanks to the smooth atomic decomposition of the operatorvalued Triebel-Lizorkin spaces F α,c 1 (R d , M) obtained in our previous paper, we establish the F α,c 1 -regularity of regular symbols for every α ∈ R, and the F α,c 1 -regularity of forbidden symbols for α > 0. As applications, we obtain the same results on the usual and quantum tori.

Quantum Differentiability on Quantum Tori

“…The theory of pseudodifferential operators goes back to Kohn-Nirenberg [31] and Hörmander [28]. It has been extended to the noncommutative setting, especially the quantum torus case, by many authors; see for instance [39,32,34,23,57,59]. Our main references of this part are [6,1] and [16], while the details can be found in [26,27].…”

“…Also note that, by the noncommutativity, if we change the order of ρ(ξ) and α s (a) in (5.2) (or ρ(n) and U n in (5.3)), we get another pseudodifferential operator with the same symbol. In [59], these two operators are distinguished as column and row operators. But in this paper, we will not need to consider both kinds of operators, and so we focus only on those with the form (5.2) or (5.3).…”

“…where the first derivative D α ξ is the derivative of ρ 1 with respect to the variable ξ ∈ R d , and the second derivative D α is the derivation on C ∞ (T d θ ) described in Section 2.2.2. Many authors have considered the question of the mapping properties of pseudodifferential operators on functions spaces on quantum tori [59,23,27]. In this paper we are concerned solely with the boundedness of a pseudodifferential operator on L 2 (T d θ ).…”

Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry

“…Harmonic analysis on T d θ has been investigated by many authors. In particular, pseudodifferential operator theory has been developed in some detail, beginning with the early work of Baaj [1] and Connes [15], and has now reached a state of maturity, see [32,33,38,34,50,51,73,71,40,47,48]. For the theory of function spaces on T d θ , we refer the reader to [74,66,71].…”