By Otto Liess
The most subject of the publication is better analytic microlocalization and its software to difficulties ofpropagation of singularities. The half on highermicrolocalization may well function an advent to thesubject. the implications on propagation check with recommendations oflinear partial differentialoperators with characteristicsof variable multiplicity and are of conical refraction type.The relation and interaction among those effects and resultsor structures from geometrical optics in crystal theoryis mentioned with many info. The notes are writtenforemost for researchers operating in microlocal analysis,but it's was hoping that they could even be of curiosity formathematicians and physicists who paintings in propagationphenomena from a extra classical standpoint.
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Extra info for Conical Refraction and Higher Microlocalization
7) since it is more natural in propagation phenomena. 7) reduce to In~ ~1 _> e', In, &+'/Ino & . 1 : Higher order wave front sets 41 This has a tendency to localize estimates to the set (I - II2)~ = 0. 7 are considered void. In, particular, semi-isotropic microlocalization differs from the microlocalization considered above for the first time at the level of tri-microlocalization whereas in Lebeaus 's paper the difference already appears for second microlocalization. , ~k-1 ) and of the structure of the associated multineighborhoods, and is not visible at the level of inequalities.
6, or Liess ). The conclusion of the lemma is then satisfied for hi = e~h - f , h2 = e~h + f. 2. conditions of type c), (which shall play an important role in what follows) are in a deeper way related to decompositions of entire functions of the type above. For a discussion of this, cf. Liess [~]. 2. 1 the following two results are often useful. 3. Let ~a : R" --~ R+ satisfy I~(~ 1) - ~(~2)1 < Lt~ 1 - ~21, V~I,~ 2 E R n. 1]. The proof is once more rather short, so we present it also. 4. We denote the variables in C" x R+ by (z,t).
In fact, geometrically, there are two distinct points of intersection, but the point ~o is a double point since L was tangent to "~io. This gives at least m + 1 points of intersection between L and S, which is a contradiction since S is given by an rn-th order equation which does not contain any linear factors. Chapter 2 H i g h e r o r d e r w a v e front s e t s In this chapter we introduce the higher order wave front sets on which our arguments are based. Generally speaking these wave front sets are modelled on the original definition of H5rmander of the first analytic wave front set, but when the order of localization is greater than two, then we shall consider two variants of such wave front sets which are not completely equivalent one to the other.
Conical Refraction and Higher Microlocalization by Otto Liess