Hypo-analytic structures.

*(English)*Zbl 0565.35079
Microlocal analysis, Proc. Conf., Boulder/Colo. 1983, Contemp. Math. 27, 23-44 (1984).

[For the entire collection see Zbl 0527.00007.]

This article is a survey of recent results on the local solvability of overdetermined systems of first-order linear partial differential equations with non-constant coefficients. Important applications to CR structures suggest the vitality of the developed theory.

The author restricts his attention mainly to the case of systems of homogeneous equations \(L_ ju=0\), \(L_ j\) being complex vector fields on a real manifold \(\Omega\). The basic viewpoint in the paper is highly conceptual. The study of the above mentioned system is given in terms of the subbundle \(V\) of the complexified tangent bundle \({\mathbb{C}}T\Omega\) spanned by \(L_ j\) where by definition \(V\) verifies the formal integrability condition, i.e. \([V,V]\subset V\). Thus the object of the study become functions and distributions which are annihilated by any smooth section of \(V\). This generalizes the notion of solution. Having in mind the subbundle \(T'\) of \({\mathbb{C}}T^*\Omega\) orthogonal to V with respect to the natural pairing of tangent and cotangent vectors we can express the local integrability of \(V\) by the condition that \(T'\) is locally generated by exact differentials. The problem of local integrability is a deep problem of the local theory (see for instance Newlander-Nirenberg theorem).

The topics concerned in the paper are the following: 1. Existence of sufficiently many independent local solutions, e.g. extension of the Newlander-Nirenberg theorem for elliptic structures, generalization of Nirenberg’s examples of a formally integrable structure which is not locally integrable.

2. Approximation formula and its applications concerning polynomial approximation, local constancy on fibres, uniqueness in the Cauchy problem and local representation of distribution solutions.

3. The culminating point of the author’s ideas in the notion of a hypo- analytic structure, defined by equivalence classes of maps modulo holomorphisms. A criterion provided by the Fourier-Bros-Jagolitzer (FBI) transform in order to study the local or microlocal hypo-analyticity of solutions is given and discussed. Another application of the FBI transform is the study of the tube structure. The obtained results are in fact a microlocal version of the classical Bochner tube theorem. A result about the propagation of hypo-analyticity is obtained by means of the FBI transform as well. The connection with elliptic submanifolds is discussed.

4.The problem of the solvability of \(L_ ju=f_ j\) for any smooth right-side \(f=(f_ j)\) satisfying the compatibility condition \(L_ jf_ k=L_ kf_ j\) is briefly discussed.

I can recommend this rich survey to everybody who wants to learn this beautiful area of analysis.

Reviewer’s remark. Every non-integrable almost complex manifold \((M,J)\) of constant local type \(q\) gives an example of hypo-analytic structure. Constant local type \(q\) means that every point \(m\) of \(M\) has a neighborhood on which there exists a maximal system of \(q\) \(J\)-almost-holomorphic functions with \(\mathbb C\)-linear differentials. See S. Dimiev [Complex analysis, Banach Cent. Publ. 11, 61–75 (1983; Zbl 0581.32006); Analytic functions, Proc. Conf., Błazejewko/Pol. 1982, Lect. Notes Math. 1039, 102–117 (1983; Zbl 0553.35002)] and L. Apostolova [C. R. Acad. Bulg. Sci. 38, 977–979 (1985; Zbl 0587.53027)].

This article is a survey of recent results on the local solvability of overdetermined systems of first-order linear partial differential equations with non-constant coefficients. Important applications to CR structures suggest the vitality of the developed theory.

The author restricts his attention mainly to the case of systems of homogeneous equations \(L_ ju=0\), \(L_ j\) being complex vector fields on a real manifold \(\Omega\). The basic viewpoint in the paper is highly conceptual. The study of the above mentioned system is given in terms of the subbundle \(V\) of the complexified tangent bundle \({\mathbb{C}}T\Omega\) spanned by \(L_ j\) where by definition \(V\) verifies the formal integrability condition, i.e. \([V,V]\subset V\). Thus the object of the study become functions and distributions which are annihilated by any smooth section of \(V\). This generalizes the notion of solution. Having in mind the subbundle \(T'\) of \({\mathbb{C}}T^*\Omega\) orthogonal to V with respect to the natural pairing of tangent and cotangent vectors we can express the local integrability of \(V\) by the condition that \(T'\) is locally generated by exact differentials. The problem of local integrability is a deep problem of the local theory (see for instance Newlander-Nirenberg theorem).

The topics concerned in the paper are the following: 1. Existence of sufficiently many independent local solutions, e.g. extension of the Newlander-Nirenberg theorem for elliptic structures, generalization of Nirenberg’s examples of a formally integrable structure which is not locally integrable.

2. Approximation formula and its applications concerning polynomial approximation, local constancy on fibres, uniqueness in the Cauchy problem and local representation of distribution solutions.

3. The culminating point of the author’s ideas in the notion of a hypo- analytic structure, defined by equivalence classes of maps modulo holomorphisms. A criterion provided by the Fourier-Bros-Jagolitzer (FBI) transform in order to study the local or microlocal hypo-analyticity of solutions is given and discussed. Another application of the FBI transform is the study of the tube structure. The obtained results are in fact a microlocal version of the classical Bochner tube theorem. A result about the propagation of hypo-analyticity is obtained by means of the FBI transform as well. The connection with elliptic submanifolds is discussed.

4.The problem of the solvability of \(L_ ju=f_ j\) for any smooth right-side \(f=(f_ j)\) satisfying the compatibility condition \(L_ jf_ k=L_ kf_ j\) is briefly discussed.

I can recommend this rich survey to everybody who wants to learn this beautiful area of analysis.

Reviewer’s remark. Every non-integrable almost complex manifold \((M,J)\) of constant local type \(q\) gives an example of hypo-analytic structure. Constant local type \(q\) means that every point \(m\) of \(M\) has a neighborhood on which there exists a maximal system of \(q\) \(J\)-almost-holomorphic functions with \(\mathbb C\)-linear differentials. See S. Dimiev [Complex analysis, Banach Cent. Publ. 11, 61–75 (1983; Zbl 0581.32006); Analytic functions, Proc. Conf., Błazejewko/Pol. 1982, Lect. Notes Math. 1039, 102–117 (1983; Zbl 0553.35002)] and L. Apostolova [C. R. Acad. Bulg. Sci. 38, 977–979 (1985; Zbl 0587.53027)].

Reviewer: Stanco Dimiev (Sofia)

##### MSC:

35N10 | Overdetermined systems of PDEs with variable coefficients |

32D15 | Continuation of analytic objects in several complex variables |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

58J47 | Propagation of singularities; initial value problems on manifolds |