Abstract
With the creation of distribution theory, around 1950, a systematic study of properties of linear PDEs with smooth coefficients started. In this context, two basic concepts are those of solvability and hypoellipticity of an operator. Looking for a characterization of linear second order hypoelliptic operators with real coefficients, Hörmander’s proved in 1967 a fundamental result which opened the way to the study of what are now called “Hörmander’s operators”. The chapter describes this circle of ideas.
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Notes
- 1.
An interesting account of the early impact of the theory of distributions on the mathematical environment is given by Lars Gårding in [6, Chap. 12].
- 2.
Lewy’s paper appeared on “Annals of Math.”. In its review on the Math. Rev. we read: “Experience with linear partial differential equations has shown that they generally possess smooth local solutions provided the equations are sufficiently smooth. This paper produces the first example of a system with coefficients in \(C^{\infty }\) having no smooth solutions in any domain”.
- 3.
Analogously, one can introduce the notion of Gevray-hypoellipticity, replacing \(C^{\omega }\) with Gevray classes. See [5, pp. 91–92].
- 4.
Think to the fundamental solution of the heat equation, which vanishes identically for \(t<0\) but not for \(t>0\), hence is not analytic.
- 5.
To be more precise, the statement “\(L\) cannot be hypoelliptic” is correct if the operator, after the change of variables, does not possess a zero order term, for in that case any rough function of the variables which are not involved in the derivatives is a solution to the equation. In the general case one can say that if the equation \(Lu=0\) admits a nontrivial solution \(u\), then one can modify \(u\) in a suitable halfspace (normal to one of the direction of the variables not involved in the equation), getting a discontinuous function which is still a distributional solution to the equation \(Lu=0\). Hence \(L\) is not hypoelliptic.
- 6.
We will explain what is the Heisenberg group in Sect. 2.2.4. For the moment, just consider this operator as defined in \(\mathbb {R}^{3}\).
- 7.
In Sect. 2.2.4 we will say more about this.
- 8.
In particular, the first half of that book is devoted to boundary value problems for these elliptic-parabolic degenerate operators, a topic we will not touch here.
- 9.
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Bramanti, M. (2014). Hörmander’s Operators: What they are. In: An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-02087-7_1
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