Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)

Abstract

This work is devoted to a systematic study of the microlocal regularity properties of pseudo-differential operators with the transmission property. We introduce a “boundary singular spectrum”, denoted ∂WF ω (u) for distributions u∈D ′ (R + n ), regular in the normal variable x n (thus, ∂WF ω (u)=∅ means that u∈∩ s+t=1/2 H s+t near the boundary), and it is shown that ∂WF ω-m [P(u 0 ) x n >0 ] is a subset of ∂WF(u) if P has degree m and the transmission property. We finally prove that these results can bef used to examinate the (microlocal) regularity of the solutions of differential Cauchy problems, with bicharacteristics transversal to the hyperplane supporting the Cauchy data.