Spectral Invariance of Non-Smooth Pseudodifferential Operators

In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in H\"older spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results for certain subsets of non-smooth pseudodifferential operators with coefficients in H\"older spaces, we improve the characterization of non-smooth pseudodifferential operators in a previous work by the authors.Comment: 43 page

LpL^p and Sobolev boundedness of pseudodifferential operators with non-regular symbol: a regularisation approach

In this paper we investigate $L^p$ and Sobolev boundedness of a certain class of pseudodifferential operators with non-regular symbols. We employ regularisation methods, namely convolution with a net of mollifiers (\rho_\eps)_\eps, and we study the corresponding net of pseudodifferential operators providing $L^p$ and Sobolev estimates which relate the parameter \eps with the non-regularity of the symbol