Let P, a partial differential operator with real principal part and constant multiplicity characteristics. We have shown in some previous works [1], [2] that several results about operators with simple characteristics can be generalized to these operators with multiple characterics if we add an hypothesis, named Levi's condition, on the lower order terms. For instance, there is still propagation along bicharacteristics for the singularities of distributions u such that Pu∈C^∞(Ω); if moreover the principal part of P is hyperbolic, the Cauchy problem is well posed in the C^∞ setting.
In this paper, we shall give a generalization to such operators of the theorem of Lax-Nirenberg [6] concerning the reflection at the boundary ∂Ω of the singularities of distributions u satisfying Pu∈C^∞(\barΩ). In order to get a more precise result of regularity up to the boundary, we shall differ from Nirenberg's proof in the details, but the principle will be the same. That is, we factorise micro locally P in an elliptic and two hyperbolic factors satisfying the Levi condition; then the proof of the theorem will be reduce to the study of micro local regularity results for elliptic and hyperbolic boundary problems.

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The S-matrix discontinuity formulas are used to show that in neighborhoods of many physical points the S-matrix is a holonomic microfunction.

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The Wightman axioms are extended to the quantum field theory in terms of Fourier hyperfunctions. The support concept of hyperfunctions is crucial for the formulation of locality and spectral condition. The complete equivalence is proved between modified Wightman axioms for relativistic theory and modified Osterwalder-Schrader axioms for Euclidean theory.

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The notion of a complex-dimensional integral is introduced in the complex n-dimensional Minkowski space. Its basic properties, such as Lorentz invariance, are investigated. Complex-dimensional invariant delta functions Δ_n(x; m²), Δ⁽¹⁾_n(x; m²), etc. are explicitly calculated in position space. It is proposed to define products of singular functions in the ordinary Minkowski space by analytically continuing the corresponding n-dimensional ones to n=4. The light-cone singularities of [Δ(x; m²)]², Δ(x; m²)×Δ⁽¹⁾(x; m²) and [Δ⁽¹⁾(x; m²)]² are shown to be unambiguously determined in this way.

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Necessary conditions and sufficient conditions are given for the existence and the continuation of holomorphic solutions of partial differential equations near the characteristic boundary of an open subset of Cⁿ.

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We study, microlocally, the non elliptic boundary value problems, with the help of the sheaf \mathscrC_N/X of Kashiwara and Kawai. A hypothesis of “N-regularity” verified by several classes of preudo-differential operators allows us to obtain a theorem of reflection of analytic singularities analogous to that of Lax and Nirenberg.

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