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''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 [[singularity|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  $  X $ be a domain in  $  \mathbf R  ^ {n} $ and let  $  u \in \mathcal D  ^  \prime  ( X) $, 
that is,  $  u $ is a generalized function on  $  X $. 
Then the wave front set  $   \mathop{\rm WF} ( u) $ of  $  u $
is the closed conical subset of  $  T  ^ {*} X \setminus  0 = X \times ( \mathbf R  ^ {n} \setminus  0) $
defined as follows: If  $  ( x _ {0} , \xi _ {0} ) \in X \times ( \mathbf R  ^ {n} \setminus  0) $, 
then  $  ( x _ {0} , \xi _ {0} ) \notin  \mathop{\rm WF} ( u) $
means that there is a function  $  \phi \in C _ {0}  ^  \infty  ( X) $, 
equal to  $  1 $
in a neighbourhood of  $  x _ {0} $, 
and a conical neighbourhood  $  \Gamma $
of  $  \xi _ {0} $
in  $  \mathbf R  ^ {n} \setminus  0 $, 
such that for every  $  N > 0 $,

$$ 
| \widehat{ {\phi u }}  ( \xi ) |  \leq   C _ {N} ( 1 + | \xi | )  ^ {-N} ,\ \ 
\xi \in \Gamma ,
$$

where

$$ 
C _ {N}  >  0,\ \ 
\widehat{ {\phi u }}  ( \xi )  =  \langle  u ( x), \phi ( x) e ^ {- ix \cdot \xi } \rangle ,
$$

that is,  $  \widehat{ {\phi u }}  $
is the [[Fourier transform]] of  $  \phi u $.

If  $  X $
is a manifold and  $  u $
is a generalized function on  $  X $(
or, more generally, a generalized section of a smooth vector bundle), then  $   \mathop{\rm WF} ( u) $
is defined in the same way as above (after transition to local coordinates). In this case  $   \mathop{\rm WF} ( u) $
turns out to be a well-defined conical subset of  $  T  ^ {*} X \setminus  0 $(
the cotangent bundle without the zero section).

One introduces the canonical projection  $  \pi :  T  ^ {*} X \setminus  0 \rightarrow X $. 
Then

$$ \tag{1 }
\pi (  \mathop{\rm WF} ( u))  =  \singsupp  u ,
$$

where  $  \singsupp  u $
is the complement of the largest open subset of  $  X $
on which  $  u $
coincides with an infinitely-differentiable function. This relationship shows that  $   \mathop{\rm WF} ( u) $
is actually a finer characteristic of the singularities of  $  u $
than  $  \singsupp  u $.

Let  $  A $
be a [[Pseudo-differential operator|pseudo-differential operator]] of order  $  m $
on  $  X $
with principal symbol  $  a _ {m} ( x, \xi ) $, 
and let  $   \mathop{\rm char}  A $
be the set of its characteristic directions, that is,

$$ 
 \mathop{\rm char}  A  = \ 
\{ {( x, \xi ) \in T  ^ {*} X \setminus  0 } : {a _ {m} ( x, \xi ) = 0 } \}
.
$$

Then

$$ \tag{2 }
 \mathop{\rm WF} ( Au)  \subset    \mathop{\rm WF} ( u)  \subset  \ 
 \mathop{\rm WF} ( Au) \cup  \mathop{\rm char}  A.
$$

Here the first inclusion characterizes the pseudo-locality of  $  A $, 
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  $  a _ {m} ( x, \xi ) $
of  $  A $
is real-valued, then the following theorem on the propagation of singularities holds: If one is given a connected piece  $  \gamma $
of a bicharacteristic (that is, a trajectory of the Hamiltonian vector field on  $  T  ^ {*} X \setminus  0 $
with Hamiltonian  $  a _ {m} $) 
that does not intersect  $   \mathop{\rm WF} ( Au) $, 
then either  $  \gamma \subset   \mathop{\rm WF} ( u) $
or  $  \gamma \cap  \mathop{\rm WF} ( u) = \emptyset $.

This theorem shows that the singularities of the solutions (that is, their wave front sets) of an equation  $  Au = f $
with a smooth right-hand side  $  f $
propagate along the bicharacteristics of the principal symbol  $  a _ {m} $
of  $  A $(
see [[#References|[3]]], [[#References|[4]]], [[#References|[8]]], [[#References|[11]]], [[#References|[12]]], [[#References|[16]]]).

The analytic wave front set  $   \mathop{\rm WF} _ {a} ( u) $
for a generalized function  $  u \in D  ^  \prime  ( X) $
can be defined in one of the following three equivalent (see [[#References|[13]]]) ways (here, for simplicity,  $  X $
is a domain in  $  \mathbf R  ^ {n} $):

1)  $  ( x _ {0} , \xi _ {0} ) \notin  \mathop{\rm WF} _ {a} ( u) $
if there are a neighbourhood  $  \omega $
of  $  x _ {0} $, 
open proper convex cones  $  \Gamma _ {1} \dots \Gamma _ {N} $
in  $  \mathbf R  ^ {n} $
and functions  $  f _ {j} $, 
holomorphic in  $  \omega + i \Gamma _ {j} $, 
such that  $  \xi _ {0} \notin \Gamma _ {j}  ^ {0} $, 
$  j = 1 \dots N $, 
and  $  u = \sum _ {j = 1 }   ^ {N} b ( f _ {j} ) $, 
where  $  \Gamma _ {j}  ^ {0} $
is the cone dual to  $  \Gamma _ {j} $
and  $  b ( f _ {j} ) $
is the boundary value of the holomorphic function  $  f _ {j} ( x + iy) $
for  $  y \rightarrow 0 $, 
$  y \in \Gamma _ {j} $, 
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

$$ 
F _ {u} ( \xi , \lambda ;  x)  = \ 
\int\limits  \mathop{\rm exp}  [- iy \cdot \xi - \lambda  | y - x |  ^ {2} ] u ( y)  dy
$$

(a generalized Fourier transform); then  $  ( x _ {0} , \xi _ {0} ) \notin  \mathop{\rm WF} ( u) $
if and only if for any function  $  \chi \in C _ {0}  ^  \infty  ( X) $
that is analytic in a neighbourhood of  $  x _ {0} $
there are a conical neighbourhood  $  \Gamma $
of  $  \xi _ {0} $
and positive constants  $  \alpha , \gamma , C _ {N} $
such that

$$ 
F _ {\chi u }  ( \xi , \lambda ;  x _ {0} )  \leq  \ 
C _ {N} ( 1 + | \xi | )  ^ {-N} e ^ {- \lambda \alpha } ,\ \ 
\xi \in \Gamma ,\  0 < \lambda < \gamma | \xi | .
$$

3)  $  ( x _ {0} , \xi _ {0} ) \notin  \mathop{\rm WF} _ {a} ( u) $
if and only if there are a neighbourhood  $  \omega $
of  $  x _ {0} $
in  $  X $, 
a bounded sequence of generalized functions  $  u _ {k} $, 
$  k = 1, 2 \dots $
with compact support, and a constant  $  C > 0 $, 
such that  $  u _ {k} = u $
in  $  \omega $
and

$$ 
| {\widehat{u}  _ {k} } ( \xi ) |  \leq  \ 
C ^ {k + 1 } k!  | \xi |  ^ {-k} ,\ \ 
\xi \in \Gamma .
$$

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

$$ 
\pi (  \mathop{\rm WF} _ {a} ( u))  =  \singsupp _ {a}  u ,
$$

where  $  \singsupp _ {a}  u $
is the complement of the largest set on which  $  u $
is real-analytic. There is an analogue of the property (2), where one can take for  $  A $
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  $  A $
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>