Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 116, pp. 1-43.

Title: Pseudodifferential operators with generalized symbols 
       and regularity theory
       
Authors: Claudia Garetto (Univ. Innsbruck, Austria)
         Todor Gramchev (Univ. di Cagliari, Italy)
	 Michael Oberguggenberger (Univ. Innsbruck, Austria)

Abstract: 
 We study pseudodifferential operators with amplitudes
 $a_\varepsilon (x,\xi)$ depending on a singular parameter
 $\varepsilon \to 0$ with asymptotic properties measured by different
 scales. We  prove, taking into account the asymptotic behavior
 for $\varepsilon \to 0$, 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.
  
Submitted June 13, 2005. Published October 21, 2005.
Math Subject Classifications: 35S50, 35S30, 46F10, 46F30, 35D10.
Key Words: Pseudodifferential operators; small parameter; 
           slow scale net; algebras of generalized functions.