Non-commutative residues for anisotropic pseudo-differential operators in Rn

Toeplitz and Hankel operators and Dixmier traces on the unit ball of $\mathbb C^n$

“…Our result can thus be viewed as a generalization of the compactification to weighted Bergman spaces and an application of the ideas in [7] of computing Dixmier traces. In particular our Theorem 4.1 is closely related to the results in [4], where the residue trace of pseudo-differential operators of a certain class is computed; here we use the Weyl transforms and they differ from pseudo-differential operators of lower order, so that Theorem 4.1 can also be obtained from [4] provided one proves that the lower order terms are of trace class.…”

“…There has been an intensive study of Dixmier trace and residue trace of pseudodifferential operators, mostly on compact manifolds where the analysis is relatively easier; see e.g. [4] and [15] and the references therein. Thus the Toeplitz operators on Hardy spaces on the boundary of a bounded strictly pseudo-convex domain can be treated using the techniques developed there.…”

Anisotropic Operator Symbols Arising From Multivariate Jump Processes

“…Furthermore, for the closely related analysis of hypo-and multi-quasi-elliptic operators we refer to [1,3,4,6,22,25,42,41]. Even though the focus of this work lies on classical Sobolev-and hence L 2 -based results, note that a great number of L p -boundedness results for (different classes of) anisotropic integral operators can be found in [10,16,29,40,46] and the references there.…”

Sharp Weyl estimates for tensor products of pseudodifferential operators

“…Similar formulae can be obtained in many other different settings, see [SV97] and [ANPS09] for a detailed analysis and several developments. To mention a few specific situations, see [Shu87,HR81] for the case of the Shubin calculus on R n , [BN03] for the anisotropic Shubin calculus, [BC11,CM13,Nic03] for the SG-operators on R n and the manifolds with ends, [GL02] for operators on conic manifolds, [Mor08] for operators 1 on cusp manifolds, [DD13] for operators on asymptotic hyperbolic manifolds, [Bat12,BGRP13] for bisingular operators.…”