Pseudodifferential operators with generalized symbols and regularity theory
journal contributionposted on 17.04.2015 by Claudia Garetto, Todor Gramchev, Michael Oberguggenberger
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We study pseudodifferential operators with amplitudes aε(x,ξ) depending on a singular parameter ε → 0 with asymptotic properties measured by different scales. We prove, taking into account the asymptotic behavior for ε → 0, refined versions of estimates for classical pseudodifferential operators. We apply these estimates to nets of regularizations of exotic operators as well as operators with amplitudes of low regularity, providing a unified method for treating both classes. Further, we develop a full symbolic calculus for pseudodifferential operators acting on algebras of Colombeau generalized functions. As an application, we formulate a sufficient condition of hypoellipticity in this setting, which leads to regularity results for generalized pseudodifferential equations. © 2005 Texas State University - San Marcos.
C. Garetto was supported by INDAM–GNAMPA, Italy. T. Gramchev was supported by INDAM–GNAMPA, Italy and by grant PST.CLG.979347 from NATO. M. Oberguggenberger was supported by project P14576-MAT from FWF, Austria.
- Mathematical Sciences