Garello, Gianluca

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Institutional Research Information System University of Turin

This is an author version of the contribution published on:Gianluca Garello.Review to the paper: Commutator criteria for magnetic pseudodifferen-tial operators. .AMS MathSciNet Mathematical Reviews, Vol. MR2753628 , .The definitive version is available at:http : //www.ams.org/mathscinet/search/publdoc.html?pg1 =RV RI&pg3 = authreviews&s1 = 343862&vfpref = html&r =8&mx− pid = 2753628The authors study some commutator techniques in the frame of thetwisted pseudodifferential calculus associated to Hamiltonian systems in abounded, smooth magnetic field. The main goal of the paper is to prove re-sults, analogous to the classic ones, for a generalization of the Weyl calculusadapted to the situation in which a variable magnetic field is also present.In order to take advantage of the gauge invariance the authors formulate thestatements about commutators in algebraic frame, using the twisted Moyalalgebra.The main result consists in a generalization of the Beals Criterion[MR0435933 (55 #8884)], in the algebraic setting, which is proved in thecase of symbols in the Ho¨rmander class S00(R2n) and Smρ (R2n). A Bony typecriterion, see [MR1482829 (98m:35233)], is also proved. Both the criteria areapplied to inverses and fractional powers of some twisted pseudo-differentialoperators. At the end, following the idea of Bony, the authors introduce aclass of Twisted Fourier Integral Operators (TFIO) and prove that, for alarge class of first order elliptic symbols, the unitary groups they generate(in Hilbert representation) are such TFIO.2

Review of the paper: Commutator criteria for magnetic pseudodifferential operators

https://iris.unito.it/bitstream/2318/93904/1/MATHREVCommutator_crit_Iftimie_Mantoiu_Purice2011.pdf

Review of the paper: Commutator criteria for magnetic pseudodifferential operators

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