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This article is a survey of Louis Nirenberg's contributions to linear partial differential equations, focusing on his groundbreaking work on pseudo-differential operators and solvability.
One cannot overestimate Louis Nirenberg's impact on twentieth century mathematics, especially on the analysis of both linear and nonlinear partial differential equations. In this article, we shall concentrate on Nirenberg's achievements in linear PDEs, in particular his development (with Kohn) of the calculus of pseudo-differential operators and microlocal analysis. These $\Psi$DOs were developed as tools for the analysis of PDEs, but they have now become indispensible both for analysis and for other areas of mathematics. In connection with this, we shall also treat Nirenberg's work (with Treves) on the solvability of partial and pseudo-differential operators.