Anisotropic pseudodifferential operators of type \(1,1\) (Q1568763)

This paper deals with several properties as boundedness and microlocal inclusion of a class of pseudodifferential operators (\(\psi\).d.o) of Hörmander's type 1,1 in the frame of anisotropic Sobolev spaces. The author introduces the anisotropic Littlewood-Paley decomposition and imposes some conditions on the symbols guaranteeing the continuity of the corresponding \(\psi\).d.o. in the anisotropic Sobolev spaces. The symbolic calculus developed in \S 3 enables the author to prove a ``sharp Gårding'' type inequality which includes the well-known Hörmander-Gårding inequality as well as its extension to anisotropic \(\psi\).d.o. proposed by Segala. Moreover, a microlocal inclusion for the wave-front sets of \(a(x,{\mathcal D})u\) and \(u\) of Sato type in the case of anisotropic Sobolev singularities is also shown.