Pseudodifferential operators with generalized symbols and regularity theory

Garetto, Claudia
Gramchev, Todor
Oberguggenberger, Michael
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Texas State University-San Marcos, Department of Mathematics
We study pseudodifferential operators with amplitudes ɑε(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 pseudo-differential 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.
Pseudodifferential operators, Small parameter, Slow scale net, Algebras of generalized functions
Garetto, C., Gramchev, T., & Oberguggenberger, M. (2005). Pseudodifferential operators with generalized symbols and regularity theory. <i>Electronic Journal of Differential Equations, 2005</i>(116), pp. 1-43.