imho, this topic will be incomplete without the mention of V. Maslov. There is an original presentation of a "symbolic calculus" which he coined algebra with µ-structure in his book "Operationnal methods". I the vol. 7 of Dieudonne "treatise on analysis", there is a chapter entitled "Operators of Lax-maslov" which is coined for Fourier intergal operators if I am not mistaken. I think that WKB-method (WKB stands for Wentzel, Brillouin and Kramers) of solution of PDE is also important to cite in this context and I found the introduction of Duistermaat book, "Fourier integral operators" very interesting.

imho, this topic will be incomplete without the mention of V. Maslov. There is an original presentation of a "symbolic calculus" which he coined algebra with $\mu$-structure in his book "Operationnal methods". In the vol. 7 of Dieudonne "treatise on analysis", there is a chapter entitled "Operators of Lax-Maslov" which is coined for Fourier integral operators if I am not mistaken. I think that WKB-method (WKB stands for Wentzel, Brillouin and Kramers) of solution of PDE is also important to cite in this context and I found the introduction of Duistermaat book, "Fourier integral operators" very interesting.