MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).
show/hide this revision's text 2 added 199 characters in body; deleted 8 characters in body

Let us follow the history of classical PDE with a few (hopefully) well-chosen examples illustrating the rôle of pseudodifferential operators in classical analysis.

(1) Before 1950. Prehistory. A long tradition of the russian school of Mikhlin introduced singular integrals, to be developed considerably by Calder\'on Calderon and Zygmund. In the late fifties and early sixties, it is quite clear that approximate inverse inverses of elliptic operators are pseudodifferential operators and that it is a good way to prove that elliptic operators are hypoelliptic in the primitive sense that singsupp$u=$singsupp$Pu$ for $P$ elliptic.

(2) 1959. The true beginning of pseudodifferential methods in PDE: Calderon's proof in 1959 of Cauchy uniqueness for a large class of principal type operators, using a pseudodifferential factorization to prove a Carleman estimate. The first resolution of a classical analysis problem by a microlocal method.

(3) 1968. After R.T. Seeley proved the invariance of classical pseudodifferential operators by diffeomorphism, M. Atiyah and I. Singer prove the index theorem for elliptic operators.

(4) 1971. Microellipticity: introduction in 1971 by Sato and then Hormander of the wave-front-set, proof that $WF u= WF Pu$ for $P$ elliptic and more generally $WF u\subset WF Pu\cup char P$ (elliptic microlocal regularity).

(4) 5) 1971, the apex. Proof by Sato and Hormander of the Huygens principlein 1971, formulated in the seventieth century, by the same authors. Although the final proof will involve Fourier integral operators, it is possible to prove the propagation of singularity theorem by a multiplier method, and for a real principal type operator $P$ and a distribution $u$ such that $Pu\in C^\infty$, $WF u$ is invariant by the flow of $H_p$, exactly as predicted by Huygens who lacked correct definitions.

(5) 6) 1973: Proof by Richard R. Beals and Charles C. Fefferman of local solvability of principal type differential operators satisfying Nirenberg-Treves condition (P). A problem of local analysis solved by the introduction of nonhomegeneous class of pseudodifferential operators.

(6) 7) 1978: subelliptic estimates. Characterization by Egorov, Hormander of operators $P$ of order $m$ such that $Pu\in H^s$ implies $u\in H^{s+m-\frac{k}{k+1}}$. The case $k=0$ is the elliptic case and the cases $k\ge 1$ involve iterated Poisson brackets of the real and imaginary part of the principal symbol of $P$.

(7) 8) 1981: paradifferential calculus. Introduction by J.-M. Bony of pseudodifferential operators with limited regularity, proof of propagation of weak singularities for classes of nonlinear PDE.
show/hide this revision's text 1

Let us follow the history of classical PDE with a few (hopefully) well-chosen examples illustrating the rôle of pseudodifferential operators in classical analysis.

(1) Before 1950. Prehistory. A long tradition of the russian school of Mikhlin introduced singular integrals, to be developed considerably by Calder\'on and Zygmund. In the late fifties and early sixties, it is quite clear that approximate inverse of elliptic operators are pseudodifferential operators and that it is a good way to prove that elliptic operators are hypoelliptic in the primitive sense that singsupp$u=$singsupp$Pu$ for $P$ elliptic.

(2) 1959. The true beginning of pseudodifferential methods in PDE: Calderon's proof in 1959 of Cauchy uniqueness for a large class of principal type operators, using a pseudodifferential factorization to prove a Carleman estimate. The first resolution of a classical analysis problem by a microlocal method.

(3) 1971. Microellipticity: introduction in 1971 by Sato and then Hormander of the wave-front-set, proof that $WF u= WF Pu$ for $P$ elliptic and more generally $WF u\subset WF Pu\cup char P$ (elliptic microlocal regularity).

(4) 1971, the apex. Proof of the Huygens principle in 1971, formulated in the seventieth century, by the same authors. Although the final proof will involve Fourier integral operators, it is possible to prove the propagation of singularity theorem by a multiplier method, and for a real principal type operator $P$ and a distribution $u$ such that $Pu\in C^\infty$, $WF u$ is invariant by the flow of $H_p$, exactly as predicted by Huygens who lacked correct definitions.

(5) 1973: Proof by Richard Beals and Charles Fefferman of local solvability of principal type differential operators satisfying condition (P). A problem of local analysis solved by the introduction of nonhomegeneous class of pseudodifferential operators.

(6) 1978: subelliptic estimates. Characterization by Egorov, Hormander of operators $P$ of order $m$ such that $Pu\in H^s$ implies $u\in H^{s+m-\frac{k}{k+1}}$. The case $k=0$ is the elliptic case and the cases $k\ge 1$ involve iterated Poisson brackets of the real and imaginary part of the principal symbol of $P$.

(7) 1981: paradifferential calculus. Introduction by Bony of pseudodifferential operators with limited regularity, proof of propagation of weak singularities for classes of nonlinear PDE.