Pseudodifferential operators with generalized symbols and regularity theory
Claudia Garetto
Todor Gramchev
Michael Oberguggenberger
2134/17305
https://repository.lboro.ac.uk/articles/journal_contribution/Pseudodifferential_operators_with_generalized_symbols_and_regularity_theory/9385358
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.
2015-04-17 15:10:27
Algebras of generalized functions
Pseudodifferential operators
Slow scale net
Small parameter
Mathematical Sciences not elsewhere classified