View source for Wave front

''wave front set, of a [[Generalized function|generalized function]] (distribution) or hyperfunction''

A conical set in the cotangent bundle to the manifold on which the generalized function or hyperfunction in question is given, which characterizes its singularities. A hyperfunction is a sum of formal boundary values of holomorphic functions. Two such sums are identified if they are equivalent in the sense of equivalence given by an analogue of Bogolyubov's "edge-of-the-wedge" theorem (cf. [[Bogolyubov theorem|Bogolyubov theorem]]), in which, however, one in no sense assumes that the holomorphic functions in question have limits.

The wave front set of a hyperfunction is also often called the analytic wave front set or the singular support (the last term is more often used in a completely-different sense, when it denotes the complement to the set of some sort of regularity of the generalized function on the manifold itself, and not in the cotangent bundle). The concept of the wave front set lies behind micro-local analysis, which is a complex of ideas and methods using wave front sets and other related concepts and techniques (in particular, pseudo-differential operators and Fourier integral operators) for studying partial differential equations (mainly linear equations).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971401.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971402.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971403.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971404.png" /> is a generalized function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971405.png" />. Then the wave front set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971406.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971407.png" /> is the closed conical subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971408.png" /> defined as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w0971409.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714010.png" /> means that there is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714011.png" />, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714012.png" /> in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714013.png" />, and a conical neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714016.png" />, such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714017.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714018.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714019.png" /></td> </tr></table>

that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714020.png" /> is the Fourier transform of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714021.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714022.png" /> is a manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714023.png" /> is a generalized function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714024.png" /> (or, more generally, a generalized section of a smooth vector bundle), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714025.png" /> is defined in the same way as above (after transition to local coordinates). In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714026.png" /> turns out to be a well-defined conical subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714027.png" /> (the cotangent bundle without the zero section).

One introduces the canonical projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714028.png" />. Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714029.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714030.png" /> is the complement of the largest open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714031.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714032.png" /> coincides with an infinitely-differentiable function. This relationship shows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714033.png" /> is actually a finer characteristic of the singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714034.png" /> than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714035.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714036.png" /> be a [[Pseudo-differential operator|pseudo-differential operator]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714037.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714038.png" /> with principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714039.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714040.png" /> be the set of its characteristic directions, that is,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714041.png" /></td> </tr></table>

Then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714042.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Here the first inclusion characterizes the pseudo-locality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714043.png" />, and the second is a far-reaching generalization of the theorem on the smoothness of solutions of elliptic equations with smooth coefficients.

If the principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714045.png" /> is real-valued, then the following theorem on the propagation of singularities holds: If one is given a connected piece <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714046.png" /> of a bicharacteristic (that is, a trajectory of the Hamiltonian vector field on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714047.png" /> with Hamiltonian <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714048.png" />) that does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714049.png" />, then either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714050.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714051.png" />.

This theorem shows that the singularities of the solutions (that is, their wave front sets) of an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714052.png" /> with a smooth right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714053.png" /> propagate along the bicharacteristics of the principal symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714055.png" /> (see [[#References|[3]]], [[#References|[4]]], [[#References|[8]]], [[#References|[11]]], [[#References|[12]]], [[#References|[16]]]).

The analytic wave front set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714056.png" /> for a generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714057.png" /> can be defined in one of the following three equivalent (see [[#References|[13]]]) ways (here, for simplicity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714058.png" /> is a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714059.png" />):

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714060.png" /> if there are a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714062.png" />, open proper convex cones <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714063.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714064.png" /> and functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714065.png" />, holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714066.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714068.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714069.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714070.png" /> is the cone dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714072.png" /> is the boundary value of the holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714073.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714075.png" />, understood in the sense of weak convergence of generalized functions. This definition is also applicable to hyperfunctions if the boundary value is interpreted differently.

2) Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714076.png" /></td> </tr></table>

(a generalized Fourier transform); then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714077.png" /> if and only if for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714078.png" /> that is analytic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714079.png" /> there are a conical neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714080.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714081.png" /> and positive constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714082.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714083.png" /></td> </tr></table>

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714084.png" /> if and only if there are a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714085.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714086.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714087.png" />, a bounded sequence of generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714089.png" /> with compact support, and a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714090.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714091.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714092.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714093.png" /></td> </tr></table>

There is an analogue of the property (1) for the analytic wave front:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714094.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714095.png" /> is the complement of the largest set on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714096.png" /> is real-analytic. There is an analogue of the property (2), where one can take for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714097.png" /> a differential operator with real-analytic coefficients or an analytic pseudo-differential operator (see [[#References|[6]]], [[#References|[9]]], [[#References|[11]]], [[#References|[15]]], [[#References|[16]]]). For such an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097140/w09714098.png" /> with a real principal symbol, a theorem on the propagation of the analytic wave front set holds, analogous to the theorem stated above for the ordinary wave front set (see [[#References|[11]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M. Sato, "Hyperfunctions and partial differential equations" , ''Proc. 2nd Conf. Functional Anal. Related Topics'' , Tokyo Univ. Press (1969) pp. 91–94 {{MR|0650826}} {{ZBL|0208.35801}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L. Hörmander, "Fourier integral operators I" ''Acta Math.'' , '''127''' (1971) pp. 79–183 {{MR|0388463}} {{ZBL|0212.46601}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.J. Duistermaat, L. Hörmander, "Fourier integral operators II" ''Acta Math.'' , '''128''' (1972) pp. 183–269 {{MR|0388464}} {{ZBL|0232.47055}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.J. Duistermaat, "Fourier integral operators" , Courant Inst. Math. (1973) {{MR|0451313}} {{ZBL|0272.47028}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M.A. Shubin, "Pseudo-differential operators and spectral theory" , Springer (1983) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> F. Trèves, "Introduction to pseudo-differential and Fourier integral operators" , '''1–2''' , Plenum (1980)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> M.E. Taylor, "Pseudo-differential operators" , Princeton Univ. Press (1981) {{MR|1567325}} {{ZBL|0289.35001}} {{ZBL|0207.45402}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> L. Nirenberg, "Lectures on linear partial differential equations" , Amer. Math. Soc. (1972) {{MR|0450756}} {{MR|0450755}} {{ZBL|0267.35001}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> M. Kashiwara, "Microfunctions and pseudo-differential equations" H. Komatsu (ed.) , ''Hyperfunctions and pseudodifferential equations. Proc. Conf. Katata, 1971'' , ''Lect. notes in math.'' , '''287''' , Springer (1973) pp. 265–529 {{MR|0420735}} {{ZBL|0277.46039}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> P. Schapira, "Théorie des hyperfonctions" , ''Lect. notes in math.'' , '''126''' , Springer (1970) {{MR|0631543}} {{MR|0270151}} {{ZBL|0201.44805}} {{ZBL|0192.47305}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> J. Sjöstrand, "Singularités analytiques microlocales" , Univ. Paris-Sud (1982) ((Prepublication.)) {{MR|0699623}} {{ZBL|0524.35007}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> R. Lascar, "Propagation des singularités des solutions d'Aeequations pseudo-differentielles à caractéristiques de multiplicités variables" , Springer (1981)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> J. Bony, "Equivalence des diverses notions de spectre singulier analytique" ''Sém. Goulaouic–Schwartz'' , '''III''' (1976–1977) {{MR|0650834}} {{ZBL|0367.46036}} </TD></TR><TR><TD valign="top">[14a]</TD> <TD valign="top"> J. Bros, D. Iagolnitzer, "Tuboides et structure analytique des distributions I. Tuboides et généralisation d'un théorème de Grauert" ''Sém. Goulaouic–Lions–Schwartz'' , '''16''' (1974) {{MR|0399493}} {{ZBL|}} </TD></TR><TR><TD valign="top">[14b]</TD> <TD valign="top"> J. Bros, D. Iagolnitzer, "Tuboides et structure analytique des distributions II. Support essential et structure analytique des distributions" ''Sém. Goulaouic–Lions–Schwartz'' , '''18''' (1975) {{MR|0399494}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> L. Hörmander, "On the singularities of solutions of partial differential equations" ''Comm. Pure Appl. Math.'' , '''23''' (1970) pp. 329–358 {{MR|0262646}} {{ZBL|0193.06603}} {{ZBL|0191.10901}} {{ZBL|0188.40901}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> L.V. Hörmander, "The analysis of linear partial differential operators" , '''1–4''' , Springer (1983–1985) {{MR|2512677}} {{MR|2304165}} {{MR|2108588}} {{MR|1996773}} {{MR|1481433}} {{MR|1313500}} {{MR|1065993}} {{MR|1065136}} {{MR|0961959}} {{MR|0925821}} {{MR|0881605}} {{MR|0862624}} {{MR|1540773}} {{MR|0781537}} {{MR|0781536}} {{MR|0717035}} {{MR|0705278}} {{ZBL|1178.35003}} {{ZBL|1115.35005}} {{ZBL|1062.35004}} {{ZBL|1028.35001}} {{ZBL|0712.35001}} {{ZBL|0687.35002}} {{ZBL|0619.35002}} {{ZBL|0619.35001}} {{ZBL|0612.35001}} {{ZBL|0601.35001}} {{ZBL|0521.35002}} {{ZBL|0521.35001}} </TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V. Guillemin, S. Sternberg, "Geometric asymptotics" , Amer. Math. Soc. (1977) {{MR|0516965}} {{ZBL|0364.53011}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) {{MR|}} {{ZBL|0734.53001}} </TD></TR></table>