Wave front

wave front set, of a 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), 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 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 [3], [4], [8], [11], [12], [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 [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 [6], [9], [11], [15], [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 [11]).

References

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[3]	J.J. Duistermaat, L. Hörmander, "Fourier integral operators II" Acta Math. , 128 (1972) pp. 183–269 MR0388464 Zbl 0232.47055
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[12]	R. Lascar, "Propagation des singularités des solutions d'Aeequations pseudo-differentielles à caractéristiques de multiplicités variables" , Springer (1981)
[13]	J. Bony, "Equivalence des diverses notions de spectre singulier analytique" Sém. Goulaouic–Schwartz , III (1976–1977) MR0650834 Zbl 0367.46036
[14a]	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) MR0399493
[14b]	J. Bros, D. Iagolnitzer, "Tuboides et structure analytique des distributions II. Support essential et structure analytique des distributions" Sém. Goulaouic–Lions–Schwartz , 18 (1975) MR0399494
[15]	L. Hörmander, "On the singularities of solutions of partial differential equations" Comm. Pure Appl. Math. , 23 (1970) pp. 329–358 MR0262646 Zbl 0193.06603 Zbl 0191.10901 Zbl 0188.40901
[16]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1–4 , Springer (1983–1985) MR2512677 MR2304165 MR2108588 MR1996773 MR1481433 MR1313500 MR1065993 MR1065136 MR0961959 MR0925821 MR0881605 MR0862624 MR1540773 MR0781537 MR0781536 MR0717035 MR0705278 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

References

[a1]	V. Guillemin, S. Sternberg, "Geometric asymptotics" , Amer. Math. Soc. (1977) MR0516965 Zbl 0364.53011
[a2]	V.I. Arnol'd, "Singularities of caustics and wave fronts" , Kluwer (1990) Zbl 0734.53001