By Marco Bramanti
Hörmander's operators are a huge type of linear elliptic-parabolic degenerate partial differential operators with soft coefficients, that have been intensively studied because the past due Nineteen Sixties and are nonetheless an lively box of analysis. this article offers the reader with a normal evaluate of the sector, with its motivations and difficulties, a few of its primary effects, and a few contemporary strains of development.
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Additional info for An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields
N, these operators have been studied more generally for any α ∗ C, encountering interesting phenomena: Theorem 26 There exists a sequence of forbidden values α = ±n, ± (n + 2) , ± (n + 4) , . . such that for any admissible (=not forbidden) α the operator Lα is both locally solvable and hypoelliptic. Remark 27 The comparison of different operators Lα having admissible or forbidden values exhibits interesting examples of operators which have a similar structure but are or are not hypoelliptic and solvable.
A detailed proof of this fact, firstly contained in , can be found in [20, Chap. 13, Sect. 3]. Let us summarize. Around 1970 the theory of Calderón-Zygmund singular integrals was extended at two different levels of generality. The proof of L 2 continuity was achieved in homogeneous groups (translations + dilations), assuming a quite rich underlying structure; the proof of L p continuity, 1 < p < ∗ (for an operator already known to be L 2 continuous) was established in a much greater generality, namely those of spaces of homogeneous type (quasidistance + doubling measure).
Inverting b is closely related to solving the ξ-Neumann problem, but is not identical to it. Note that, for b , there are no “boundary conditions”; however the nonellipticity of the ξ-Neumann problem is reflected here in that the second order operator b is not elliptic (. ): there is always a missing direction. The operator b is usually called the Kohn Laplacian. Again, 1/2-subelliptic estimates have been proved for b on ξ D when D is a strongly pseudoconvex domain. This implies that the Kohn Laplacian b on the boundary of a strongly pseudoconvex domain is hypoelliptic.
An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields by Marco Bramanti