Pointwise estimates of pseudo-differential operators

The subject of this project is simply to estimate the action of an general pseudo-differential operator at an arbitrary point in Euclidean space. This is done in terms of the maximal function of Peetre-Fefferman-Stein type and a symbol factor carrying the entire information on the symbol; they both enter the so-called factorisation inequality.

The symbol factor is easily controlled via symbolic estimates and classical techniques for the Fourier transformation; hence fits well into the Littlewood-Paley analysis of symbols and operators, which yields the paradifferential 3-term decomposition.

In this way, a rather direct proof of continuity in the standard Sobolev scale and Hölder-Zygmund scale is obtained; with immediate extensions to the Besov spaces. Secondly the fact that the space of slowly growing smooth functions is invariant under pseudo-differential operators follows with these techniques. Thirdly, as a surprise, every operator of type 1,1 is bounded on the functions in Lp with compact spectra.

The project has been crucial for the recent progress in the theory of pseudo-differential operators of type 1,1 (2010-2011). An article has appeared in J. Pseudo-diff. Ops. Appl. (2011).