Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)
Annales de l'Institut Fourier, Volume 32 (1982) no. 3, p. 183-213

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 uD (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.

Nous nous livrons dans cet article à une étude systématique des propriétés de régularité microlocale des opérateurs pseudo-différentiels possédant la propriété de transmission. Nous définissons à cet effet une notion de spectre singulier sur le bord, noté WF ω (u) pour les distributions uD (R + n ) régulières en la variable normale x n (WF ω (u))= signifiant que u s+t=1/2 H s+t près du bord), et montrons que WF ω-m P(u 0 ) x n >0 est inclus dans WF(u) si P est le degré m et possède la propriété de transmission. Nous montrons finalement que ces résultats permettent d’obtenir un théorème de régularité (microlocale) pour les solutions des problèmes de Cauchy associés à des opérateurs différentiels à bicaractéristiques transverses à l’hyperplan des données initiales.

@article{AIF_1982__32_3_183_0,
     author = {Gosson, Maurice De},
     title = {Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Louis-Jean},
     address = {Gap},
     volume = {32},
     number = {3},
     year = {1982},
     pages = {183-213},
     doi = {10.5802/aif.884},
     zbl = {0488.35080},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1982__32_3_183_0}
}
Gosson, Maurice De. Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I). Annales de l'Institut Fourier, Volume 32 (1982) no. 3, pp. 183-213. doi : 10.5802/aif.884. http://www.numdam.org/item/AIF_1982__32_3_183_0/

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