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 for distributions , regular in the normal variable (thus, means that near the boundary), and it is shown that is a subset of if has degree 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 pour les distributions régulières en la variable normale signifiant que près du bord), et montrons que est inclus dans si est le degré 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}, pages = {183--213}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {32}, number = {3}, year = {1982}, doi = {10.5802/aif.884}, zbl = {0488.35080}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.884/} }
TY - JOUR AU - Gosson, Maurice De TI - Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I) JO - Annales de l'Institut Fourier PY - 1982 SP - 183 EP - 213 VL - 32 IS - 3 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.884/ DO - 10.5802/aif.884 LA - en ID - AIF_1982__32_3_183_0 ER -
%0 Journal Article %A Gosson, Maurice De %T Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I) %J Annales de l'Institut Fourier %D 1982 %P 183-213 %V 32 %N 3 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.884/ %R 10.5802/aif.884 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.884/
[1] The propagation of singularities along gliding rays, Invent. Math., 41 (1977). | MR | Zbl
, ,[2] Boundary problems for pseudo differential operators, Acta. Math., 126 (1971). | MR | Zbl
,[3] Reflection of C∞ singularities for a class of operators with multiple characteristics, Rims - Kyoto University, vol. 12, supp. 1977. | MR | Zbl
,[4] Hypoellipticité partielle à la frontière pour les opérateurs pseudo-différentiels de transmission, Annali di Mat. Pura ed Appl. serie IV, t. cxxiii (1980). | MR | Zbl
,[5] Parametrix de transmission pour des opérateurs de type parabolique etc, C.R. Acad. Sc., Paris, t. 292. | Zbl
,[6] Résultats microlocaux en hypoellipticité partielle à la frontière pour les O.P.D. de transmission, C.R. Acad Sc., Paris, t. 292. | Zbl
,[7] Linear partial differential operators. Springer Verlag, 1964.
,[8] Pseudo-differential operators and non-elliptic boundary problems. Ann. Math., 83 (1966). | MR | Zbl
,[9] On the existence and the regularity of solutions of linear pseudo-differential equations, L'Ens. Math., t. XVII, fasc. 2 (1972).
,[10] Problèmes aux limites non homogènes et applications, vol. I, Dunod, 1968. | Zbl
, ,[11] Transformations of boundary problems, Preprint, Acta Math. (1980). | Zbl
,[12] Singularities of boundary problems, I, Comm. on Pure and Appl. Math., XXXI (1978). | Zbl
, ,[13] Linear partial differential equations with constant coefficients, Gordon and breach, 1963.
,[14] Operators of principal type with interior boundary conditions, Acta Math., 130 (1973). | MR | Zbl
,Cited by Sources: