nLab
compactly supported distribution

Contents

Contents

Idea

A compactly supported distribution is a distribution whose support of a distribution is a compact subset.

This implies that compactly supported distributions may be evaluated not just on bump functions, but in fact on the larger space of all smooth functions. Indeed it turns out that the compactly supported distributions exhaust the continuous linear functionals on the space of smooth functions (Hörmander 90, theorem 2.3.1).

Therefore and since Schwartz’s notation for the space C (X)C^\infty(X) of all smooth functions on the given smooth manifold is (X)\mathcal{E}(X), the space of compactly supported distributions is often denoted by (X)\mathcal{E}'(X) (Hörmander 90, p. 45).

Definition

As continuous linear duals to smooth functions

Definition

(topological vector space of smooth functions on a Cartesian space)

For n nn \in \mathbb{R}^n, write C ( n)C^\infty(\mathbb{R}^n) for the \mathbb{R}-vector space of smooth functions n\mathbb{R}^n \to \mathbb{R}.

Then the vector space C ( n)C^\infty(\mathbb{R}^n) becomes a Fréchet vector space induced by the family of seminorms

C c ( n) p K α [0,) Φ supxK| αΦ(x)|, \array{ C^\infty_c(\mathbb{R}^n) &\overset{p_{K}^\alpha}{\longrightarrow}& [0,\infty) \\ \Phi &\mapsto& \underset{x \in K}{sup} {\vert \partial^\alpha \Phi(x) \vert} } \,,

indexed by K nK \subset \mathbb{R}^n a compact subset and α n\alpha \in \mathbb{N}^n a tuple encoding the degrees of the partial derivative α\partial^\alpha.

Hence the seminorm p K αp_K^\alpha sends a bump function Φ\Phi to the supremum over the points xKx \in K of the absolute value of the partial derivative αΦ\partial^\alpha \Phi; and the open subsets defined thereby are the unions of translations of the base open balls

B K,α,ϵ (0)={v n|p K α(v)<ϵ} B^\circ_{K,\alpha,\epsilon}(0) = \left\{ v \in \mathbb{R}^n \;\vert\; p_K^\alpha(v) \lt \epsilon \right\}

for ϵ(0,)\epsilon \in (0,\infty).

We write

( n)TopVect \mathcal{E}(\mathbb{R}^n) \in TopVect_{\mathbb{R}}

for the resulting Fréchet topological vector space.

Definition

(compactly supported distibutions as continuous linear duals to bump functions)

The space 𝔼( n)\mathbb{E}'(\mathbb{R}^n) of compactly supported distributions on n\mathbb{R}^n is the dual space

( n)(( n)) * \mathcal{E}'(\mathbb{R}^n) \;\coloneqq\; \left(\mathcal{E}(\mathbb{R}^n)\right)^\ast

to the topological vector space of bump functions from def. .

e.g. (Klainerman 08, p. 9)

This means the following

Proposition

(characterization of continuity for compactly supported distributions)

A linear function

u:C ( n) u \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}

is continuous with respect to the topology of def. , hence is a compactly supported distribution (def. )

( n) \mathcal{E}(\mathbb{R}^n) \longrightarrow \mathbb{R}

precisely if the following condition holds

  • there exists a compact subset K nK \subset \mathbb{R}^n and kk \in \mathbb{N} and C(0,)C \in (0,\infty) such that

    (1)ΦC ( n)(|u(Φ)|C|α|ksupxK| αK|) \underset{\Phi \in C^\infty(\mathbb{R}^n)}{\forall} \left( \vert u(\Phi) \vert \;\leq\; C \underset{ {\vert \alpha \vert \leq k} }{\sum} \underset{x \in K}{sup} \vert \partial^\alpha K \vert \right)

(see also Hörmander 90, (2.3.2) and theorem 2.3.1)

Proof

By this prop. the continuity of LL is equivalent to there being an inhabited finite set {(K 1,α 1),,(K r,α r)}\{ (K_1, \alpha_1), \cdots, (K_r, \alpha_r) \} and C(0,)C \in (0,\infty) such that

(2)|u(Φ)|Cmaxi=1,,n(supxK i| α iΦ|) {\vert u(\Phi)\vert} \;\leq\; C \underset{i = 1, \cdots, n}{max} \left( \underset{x \in K_i}{sup} {\vert \partial^{\alpha_i} \Phi \vert} \right)

We need to see that this is equivalent to a bound of the form (1).

In one direction, assume that (2) holds.

The union Ki=1,,nK iK \coloneqq \underset{i = 1, \cdots, n}{\cup} K_i is still a compact subset (this prop.). Hence (2) implies that

|u(Φ)| Cmaxi=1,,nsupxK| α iΦ| \begin{aligned} {\vert u(\Phi)\vert} & \leq C \underset{i = 1, \cdots, n}{max} \underset{x \in K}{sup} {\vert \partial^{\alpha_i} \Phi \vert} \end{aligned}

and hence with kmaxi=1,,n|α i|k \coloneqq \underset{i = 1, \cdots, n}{max} {\vert \alpha_i\vert} that

|u(Φ)|C|α|ksupxK| αΦ|, {\vert u(\Phi)\vert} \;\leq\; C \underset{{\vert \alpha\vert} \leq k}{\sum} \underset{x \in K}{sup} {\vert \partial^{\alpha} \Phi \vert} \,,

which is of form (1).

Conversely, assume that a bound of the form (1) holds. Then take the finite set of pairs (K i,α i)(K_i, \alpha_i) where all K iKK_i \coloneqq K and with all |α i|k{\vert\alpha_i\vert} \leq k. With NN denoting the number of nn-tuples α\alpha with |α|k{\vert \alpha \vert} \leq k we then get a bound as in (2) with coefficient NCN C.

As smooth linear duals to smooth functions

Given that distributions are concerned with smooth functions it is sometimes more natural to regard them not as continuous linear functionals as in def. , but as smooth linear functionals. Indeed this turns out to be equivalent (prop. below), if one considers an ambient context of suitably generalized smooth spaces, namely diffeological spaces or more generally smooth sets or formal smooth sets. We will write H\mathbf{H} for any of these categories of generalized smooth spaces.

We may canonially regard any smooth manifold such as the Cartesian space n\mathbb{R}^n as an object of H\mathbf{H}. For XHX \in \mathbf{H} any object, we write [X,][X,\mathbb{R}] for the mapping space (the internal hom). The underlying set is C (X)C^\infty(X). If XX itself has \mathbb{R}-linear structure, we write

[X,] [[X,],] [X,\mathbb{R}]_{\mathbb{R}} \hookrightarrow [[X,\mathbb{R}], \mathbb{R}]

for the subobject of \mathbb{R}-linear maps.

Concretely, for UU a smooth manifold (or just a Cartesian space), then the sheaf [X,][X,\mathbb{R}] assigns (see at closed monoidal structure on presheaves for details)

[X,](U)=C (U×X) [X,\mathbb{R}](U) = C^\infty(U \times X)

and [X,](U)C (U×X)[X,\mathbb{R}](U) \subset C^\infty(U \times X) is the subset of those functions Φ ()()\Phi_{(-)}(-) such that for all uUu \in U the function Φ u:X\Phi_u \colon X \to \mathbb{R} is \mathbb{R}-linear. The global elements Γ()\Gamma(-) of the mapping space constitute the ordinary hom set

Γ[X,]H(X,). \Gamma [X,\mathbb{R}] \simeq \mathbf{H}(X,\mathbb{R}) \,.
Proposition

(compactly supported distributions are the smooth linear functionals)

For nn \in \mathbb{N}, there is a natural bijection between the underlying sets of compactly supported distributions on n\mathbb{R}^n (def. ) and the \mathbb{R}-linear mapping space formed in the category H\mathbf{H} of either diffeological space or smooth sets or formal smooth sets:

()˜:( n)H([ n,],) \widetilde{(-)} \;\colon\; \mathcal{E}'(\mathbb{R}^n) \overset{\simeq}{\longrightarrow} \mathbf{H}([\mathbb{R}^n, \mathbb{R}], \mathbb{R})_{\mathbb{R}}

given by sending μ( n)\mu \in \mathcal{E}'(\mathbb{R}^n) to the natural transformation which on a test space UU \in CartSp takes a smoothly UU-parameterized function Φ ()():U× n\Phi_{(-)}(-) \colon U \times \mathbb{R}^n \to \mathbb{R} to its evaluation in μ\mu pointwise in UU:

μ˜(Φ ())(u)μ,Φ u. \tilde \mu(\Phi_{(-)})(u) \;\coloneqq\; \langle \mu, \Phi_{u}\rangle \,.

(Moerdijk-Reyes 91, chapter II, prop. 3.6)

Proof

First consider this for the case that H=\mathbf{H} = SmoothSet (which immediately subsumes the case that H=\mathbf{H} = DiffelogicalSpace).

To see that ()˜\widetilde{(-)} is well defined, we need to check that the function

U μ˜(Φ ()) u μ,Φ u \array{ U &\overset{ \tilde \mu(\Phi_{(-)})}{\longrightarrow}& \mathbb{R} \\ u &\mapsto& \langle \mu, \Phi_u\rangle }

is smooth. But this follows immediately since μ,\langle \mu,-\rangle by definition is linear and continuous (Hörmander 90, theorem 2.1.3).

To see that ()˜\widetilde{(-)} is indeed a bijection for each UU it remains that every \mathbb{R}-linear smooth functional (morphisms of smooth sets) of the form

A:[ n,] A \;\colon\; [\mathbb{R}^n,\mathbb{R}] \longrightarrow \mathbb{R}

when restricted on global elements to a function of sets

A(*):C ( n) A(\ast) \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}

is continuous with respect to the topological vector space structure from def. on the left.

Now by definition of the internal hom AA is actually “path-smooth” (this def.) and hence the statement is implied by this prop.

Finally to see that this argument generalizes to H=\mathbf{H} = FormalSmoothSet observe that the Weil algebra of every infinitesimally thickened point is a quotient ring of an algebra of smooth functions on some Cartesian space (by the Hadamard lemma). The previous argument now applies to representatives under this quotient coprojection and one checks that it is independent of the representative chosen.

Properties

Fourier-Laplace transform

Proposition/Definition

For nn \in \mathbb{N}, let u( n)u \in \mathcal{E}'(\mathbb{R}^n) be a compactly supported distribution on Cartesian space n\mathbb{R}^n. Then its Fourier transform of distributions is the function

n u^ ζ u(e i,ζ) \array{ \mathbb{R}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R} \\ \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right) }

where on the right we have the application of uu, regarded as a linear function u:C ( n)u \colon C^\infty(\mathbb{R}^n) \to \mathbb{R}, to the exponential function applied to the canonical inner product ,\langle -,-\rangle on n\mathbb{R}^n.

This same formula makes sense more generally for complex numbers ζ n\zeta \in \mathbb{C}^n. This is then called the Fourier-Laplace transform of uu, still denoted by the same symbol:

n u^ ζ u(e i,ζ) \array{ \mathbb{C}^n &\overset{\hat u}{\longrightarrow}& \mathbb{R} \\ \zeta &\mapsto& u\left(e^{-i\langle -,\zeta \rangle} \right) }

This is an entire analytic function on n\mathbb{C}^n.

(Hörmander 90, theorem 7.1.14)

Theorem

(Paley-Wiener-Schwartz theorem)

For nn \in \mathbb{N} the vector space C c ( n)C^\infty_c(\mathbb{R}^n) of compactly supported smooth functions (bump functions) on Euclidean space n\mathbb{R}^n is (algebraically and topologically) isomorphic, via the Fourier-Laplace transform (prop. ), to the space of entire functions FF on n\mathbb{C}^n which satisfy the following estimate: there is a positive real number BB such that for every integer N>0N \gt 0 there is a real number C NC_N such that:

ξ n(F(ξ)C N(1+|ξ|) Ne B|Im(ξ)|). \underset{\xi \in \mathbb{C}^n}{\forall} \left( {\Vert F(\xi) \Vert} \le C_N \left( 1 + {\vert \xi\vert } \right)^{-N} e^{B \, |Im(\xi)|} \right) \,.

More generally, the space of compactly supported distributions on n\mathbb{R}^n of order NN is isomorphic via Fourier transform of distributions to those entire functions on n\mathbb{C}^n for which there exists positive real numbers CC and BB such that

ξ n(F(ξ)C N(1+|ξ|) Ne B|Im(ξ)|). \underset{\xi \in \mathbb{C}^n}{\forall} \left( {\Vert F(\xi) \Vert} \le C_N (1 + {\vert \xi\vert })^{N} e^{ B \; |Im(\xi)|} \right) \,.

(Notice that the Fourier transform of a compactly supported distribution is guaranteed to be a smooth function, by this prop..)

(e.g. Hoermander 90, theorem 7.3.1)

Proposition

(decay of Fourier transform of compactly supported functions)

A compactly supported distribution u( n)u \in \mathcal{E}'(\mathbb{R}^n) is a non-singular distribution (def. ), hence given by a compactly supported function bC cp ( n)b \in C^\infty_{cp}(\mathbb{R}^n) via u(f)=b(x)f(x)dvol(x)u(f) = \int b(x) f(x) dvol(x), precisely if its Fourier transform u^\hat u (this def.) satisfies the following decay property:

For all NN \in \mathbb{N} there exists C N +C_N \in \mathbb{R}_+ such that for all ξ n\xi \in \mathbb{R}^n we have that the absolute value |v^(ξ)|{\vert \hat v(\xi)\vert} of the Fourier transform at that point is bounded by

(3)|v^(ξ)|C N(1+|ξ|) N. {\vert \hat v(\xi)\vert} \;\leq\; C_N \left( 1 + {\vert \xi\vert} \right)^{-N} \,.

(e.g. Hoermander 90, around (8.1.1))

Singular support and Wave front set

Definition

(singular support of a compactly supported distribution)

For nn \in \mathbb{N} and u( n)u \in \mathcal{E}'(\mathbb{R}^n) a compactly supported distribution, its singular support is the subset of the Cartesian space n\mathbb{R}^n of those points which have no neighbourhood on which uu restricts to a non-singular distribution:

supp sing(u){u n|¬(Unbhd{x}(u| UC cp (U)))}. supp_{sing}(u) \;\coloneqq\; \left\{ u \in \mathbb{R}^n \,\vert\, \not \left( \underset{U \underset{\text{nbhd}}{\supset} \{x\}}{\exists} \left( u\vert_U \in C^\infty_{cp}(U) \right) \right) \right\} \,.

By prop. the singular support of a distribution (def. ) consists of those points around which the Fourier transform of the distribution receives large high-frequency (“UV”) contributions. But in fact prop. allows to say more precisely which high frequency Fourier modes make the distribution singular at a given point. These are said to be part of the wave front of the distribution, and the collection of all of them is called the wave front set of the distribution:

Definition

(wavefront set)

For nn \in \mathbb{N} let u( n)u \in \mathcal{E}'(\mathbb{R}^n) be a compactly supported distribution.

For bC cp ( n)b \in C^\infty_{cp}(\mathbb{R}^n) a compactly supported smooth function, write bu( n)b u \in \mathcal{E}'(\mathbb{R}^n) for the corresponding product (this example).

For xsupp(b) nx\in supp(b) \subset \mathbb{R}^n, we say that a unit covector ξS(( n) *)\xi \in S((\mathbb{R}^n)^\ast) is regular if there exists a neighbourhood US(( n) *)U \subset S((\mathbb{R}^n)^\ast) of ξ\xi in the unit sphere such that for all cξ( n) *c \xi' \in (\mathbb{R}^n)^\ast with c +c \in \mathbb{R}_+ and ξUS(( n) *)\xi' \in U \subset S((\mathbb{R}^n)^\ast) the decay estimate (3) is valid for the Fourier transform bu^\widehat{b u} of bub u; at cξc \xi'. Otherwise ξ\xi is non-regular. Write

Σ(bu){ξS(( n) *)|ξnon-regular} \Sigma(b u) \;\coloneqq\; \left\{ \xi \in S((\mathbb{R}^n)^\ast) \;\vert\; \xi \, \text{non-regular} \right\}

for the set of non-regular covectors of bub u.

The wave front set at xx is the intersection of these sets as bb ranges over bump functions whose support includes xx:

Σ x(u)bC cp ( n)xsupp(b)Σ(bu). \Sigma_x(u) \;\coloneqq\; \underset{ { b \in C^\infty_{cp}(\mathbb{R}^n) } \atop { x \in supp(b) } }{\cap} \Sigma(b u) \,.

Finally the wave front set of uu is the subset of the sphere bundle S(T * n)S(T^\ast \mathbb{R}^n) which over x nx \in \mathbb{R}^n consists of Σ x(U)T x * n\Sigma_x(U) \subset T^\ast_x \mathbb{R}^n:

WF(u)x nΣ x(u)S(T * n) WF(u) \;\coloneqq\; \underset{x \in \mathbb{R}^n}{\cup} \Sigma_x(u) \;\subset\; S(T^\ast \mathbb{R}^n)

Usually this is considered as the full conical set inside the cotangent bundle generated by the unit covectors under multiplication with positive real numbers.

(Hörmander 90, def. 8.1.2)

Hence the wave front set of a compactly supported distribution consists of all those directions of wave vectors along which the Fourier transform of the distribution is not a rapidly decreasing function.

Proposition

(empty wave front set corresponds to ordinary smooth functions)

The wave front set (def. ) of a compactly supported distribution is empty precisely if the distribution is non-singular (example ).

This is effectively the Paley-Wiener-Schwartz theorem (Hörmander 90, below (8.1.1)).

Examples

Definition

(non-singular compactly supported distributions)

For nn \in \mathbb{N}, a compactly supported smooth function bC cp ( n)b \in C^\infty_{cp}(\mathbb{R}^n) (a bump function) induces a compactly supported distribution

n()bdvol n:C ( n) \int_{\mathbb{R}^n} (-) b dvol_{\mathbb{R}^n} \;\colon\; C^\infty(\mathbb{R}^n) \longrightarrow \mathbb{R}

by integration of smooth functions against bdvolb dvol.

This construction defines a linear inclusion

C cp ( n)( n). C^\infty_{cp}(\mathbb{R}^n) \hookrightarrow \mathcal{E}'(\mathbb{R}^n) \,.

The compactly supported distributions arising this way are called the non-singular distributions.

References

Traditional

Textbook accounts include

  • Lars Hörmander, section 2.3 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

Lecture notes include

In terms of smooth toposes

Discussion of compactly supported distributions in terms morphisms out of internal homs in a smooth topos is in

and specifically for the Cahiers topos of formal smooth sets in

using results of

and following the general conception of “intensive and extensive” in

Generalization to non-compactly supported distributions is in

and sheaf theoretic discussion of distributions as morphisms of smooth spaces is in

Last revised on August 1, 2018 at 08:05:44. See the history of this page for a list of all contributions to it.