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

Annales de l'institut Fourier (1982)

- Volume: 32, Issue: 3, page 183-213
- ISSN: 0373-0956

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topDe Gosson, Maurice. "Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)." Annales de l'institut Fourier 32.3 (1982): 183-213. <http://eudml.org/doc/74545>.

@article{DeGosson1982,

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 $\partial WF_\omega (u)$ for distributions $u \in D^\{\prime \}(\{\bf R\}^n_+)$, regular in the normal variable $x_n$ (thus, $\partial WF_\omega (u)= \emptyset$ means that $u\in \cap _\{s+t=1/2\} H^\{s+t\}$ near the boundary), and it is shown that $\partial WF_\{\omega -m\} [P(u^0)_\{x_n>0\}]$ is a subset of $\partial WF(u)$ 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.},

author = {De Gosson, Maurice},

journal = {Annales de l'institut Fourier},

keywords = {boundary singular spectrum; differential Cauchy problems; bicharacteristic transversal},

language = {eng},

number = {3},

pages = {183-213},

publisher = {Association des Annales de l'Institut Fourier},

title = {Microlocal regularity at the boundary for pseudo-differential operators with the transmission property (I)},

url = {http://eudml.org/doc/74545},

volume = {32},

year = {1982},

}

TY - JOUR

AU - De Gosson, Maurice

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

JO - Annales de l'institut Fourier

PY - 1982

PB - Association des Annales de l'Institut Fourier

VL - 32

IS - 3

SP - 183

EP - 213

AB - 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 $\partial WF_\omega (u)$ for distributions $u \in D^{\prime }({\bf R}^n_+)$, regular in the normal variable $x_n$ (thus, $\partial WF_\omega (u)= \emptyset$ means that $u\in \cap _{s+t=1/2} H^{s+t}$ near the boundary), and it is shown that $\partial WF_{\omega -m} [P(u^0)_{x_n>0}]$ is a subset of $\partial WF(u)$ 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.

LA - eng

KW - boundary singular spectrum; differential Cauchy problems; bicharacteristic transversal

UR - http://eudml.org/doc/74545

ER -

## References

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- [8] L. HÖRMANDER, Pseudo-differential operators and non-elliptic boundary problems. Ann. Math., 83 (1966). Zbl0132.07402MR38 #1387
- [9] L. HÖRMANDER, On the existence and the regularity of solutions of linear pseudo-differential equations, L'Ens. Math., t. XVII, fasc. 2 (1972).
- [10] LIONS, MAGENES, Problèmes aux limites non homogènes et applications, vol. I, Dunod, 1968. Zbl0165.10801
- [11] R.B. MELROSE, Transformations of boundary problems, Preprint, Acta Math. (1980). Zbl0436.58024
- [12] R.B. MELROSE, J. SJÖSTRAND, Singularities of boundary problems, I, Comm. on Pure and Appl. Math., XXXI (1978). Zbl0368.35020
- [13] F. TREVES, Linear partial differential equations with constant coefficients, Gordon and breach, 1963.
- [14] J. SJÖSTRAND, Operators of principal type with interior boundary conditions, Acta Math., 130 (1973). Zbl0253.35076MR55 #9174

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