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Hörmander’s Operators: Why they are Studied

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An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

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Abstract

The chapter describes some examples of PDEs written in the form of Hörmander’s operators which arise both from physical applications and from other fields of mathematics, to give some more motivations for the study of these equations. We deal with two different areas, which represent the main “historical” motivations for this field of research, namely Kolmogorov-Fokker-Planck equations arising in the study of stochastic systems, and some relevant PDEs which arise in the theory of several complex variables, in particular the Kohn-Laplacian on the Heisenberg group.

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Notes

  1. 1.

    In the quotation I have used the symbol \(w^{\prime }\) instead of \(n\) used by Mumford, and replaced the term “Brownian motion” with “Wiener process”, to keep the terminology we are using. This does not change the meaning of the text.

  2. 2.

    This is not the standard definition, but is shown to be equivalent to it. See [7, p. 52–53].

  3. 3.

    We suggest the short survey paper [24] as an introduction to this idea. Here we just say that pseudoconvexity can be defined via the Levi form, which is a kind of “complex hessian” of the defining function of the domain.

  4. 4.

    See [7, p. 85] for precise references.

  5. 5.

    See also [27, Chap. 13, Sect. 1] for this point.

  6. 6.

    See [15, p. 98] for the meaning of this condition.

  7. 7.

    This is a stronger notion than pseudoconvexity. Let us state a theorem which characterizes it (See [7, Corollary 3.4.5]):

    Let \(D\) be a bounded pseudoconvex domain with \(C^{2}\) boundary in \(\mathbb {C}^{n}\), \(n\ge 2\). Then \(D\) is strongly pseudoconvex if and only if \(D\) is locally biholomorphically equivalent to a strictly convex domain near every boundary point.

  8. 8.

    Here we skip the indices \(\left( p,q\right) \) of the operators \(\overline{\partial }_{b}\) and \(\overline{\partial }_{b}^{*}\), to keep more readable the expression. The right indices are the same as in (2.15).

  9. 9.

    via the following mapping:

    $$\begin{aligned} w_{n+1}&=\frac{i-z_{n+1}}{i+z_{n+1}};w_{k}=\frac{2iz_{k}}{i+z_{n+1}},k=1,2,\ldots ,n.\\ z_{n+1}&=i\left( \frac{1-w_{n+1}}{1+w_{n+1}}\right) ;z_{k}=\frac{w_{k}}{1+w_{n+1}},k=1,2,\ldots ,n. \end{aligned}$$

    Note that for \(n=1\) this map realizes the biholomorphic equivalence between the halfplane Im \(z>0\) and the circle \(\left| w\right| <1\) in the complex plane.

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  1. Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

    Marco Bramanti

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Bramanti, M. (2014). Hörmander’s Operators: Why they are Studied. 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_2

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