Globally Bisingular Elliptic Operators

Sharp Weyl estimates for tensor products of pseudodifferential operators

“…This clear spectral meaning of the Dirichlet divisor problem was one of the main motivation of the papers [BGRP13,GPRVar]. For the connection between Dirichlet divisor problem and standard bisingular operators on the product of closed manifolds see [Bat12].…”

“…In 1849, Dirichlet proved that In order to have a spectral interpretation of the Dirichlet divisor problem, a global bisingular calculus based on Shubin calculus has been introduced in [BGRP13]. Then, the following Hermite-type operator…”

“…A version of bisingular operators, based on Shubin pseudodifferential calculus on R n , was introduced in [BGRP13]. The counting function was studied also in this setting, obtaining results analogous to those which hold for the "standard" bisingular calculus.…”

“…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.…”

Gelfand–Shilov Type Spaces Through Hermite Expansions

“…In the last part of the paper we introduce one more class of Gelfand-Shilov type spaces S ⊗,σ σ (R n ), σ ≥ 1/2, and Σ ⊗,σ σ (R n ), σ > 1/2. These spaces were obtained through the iteration of Harmonic oscilators and are related to our study of Weyl formula for tensorised products of elliptic Shubin type operators (see [1]). We compare all the considered spaces through the estimates of Hermite coefficients.…”