4 Distributions

Laurent Schwartz introduced the notion of distributions two talk about generalized solutions to differential equations. For his work he got the fields medal.

4.1 Space of Distributions

Definition 4.1

\(\mathcal{D}(\Omega ) = C_c^\infty (\Omega )\) is the set of test functions together with a topology determined by its converging sequences : \(\phi _n \to \phi \) in \(\mathcal{D}(\Omega )\) if

there exists a compact subset \(K \subset \Omega \) such that \(\operatorname{Supp}(\phi _n) \subset K\).
For all multiindices \(\alpha \) we have \(\partial _\alpha \phi _n \to \partial _\alpha \phi \) in uniformly.

Convention: if \(x \in \mathbb {R}^d \setminus \Omega \), then \(\phi (x) := 0\).
\(\mathcal{D}'(\Omega )\) is the topological dual space, i.e. the space of continuous linear functionals \(D(\Omega ) \to \mathbb {C}\) with the weak-*-convergence, i.e. pointwise convergence.

Remark 4.2

A notion of converging sequence on a set \(X\) is

The constant function on \(x\) converges to \(x\)
A sequence converges to \(x\) iff any subsequence has a subsequence converging to \(x\).

Note, that any subsequence of a converging sequence converges to the same limit. It induces a closure operator on \(X\) (for \(\overline{A \cup B} \subset \bar A \cup \bar B\) you use, that if a sequence in \(A \cup B\) converges to \(x\), then it has a subsequence (converging to \(x\)) lying in \(A\) or in \(B\)).

Example 4.3

Every locally integrable function \(f \in L^1_{loc}(\Omega )\) gives us a distribution \(\Lambda f \in D'(\Omega )\) defined by

\[ \Lambda f (\phi ) = \int _{\Omega } f(x) \phi (x) \ \mathrm{d} x \]

which is well-defined because \(\phi \) has compact support.

Example 4.4

Let \(\mu \) be a Radon measure on \(\Omega \) (or more generally a signed Borel measure which is finite on compact subsets of \(\Omega \)). Then it defines a distribution:

\[ T_\mu (\phi ) = \int _{\Omega } \phi \ \mathrm{d} \mu \]

Proof ▶

As Borel sets are \(\mu \)-measurable, every continuous function is \(\mu \)-measurable. Let \(K := \operatorname{Supp}\phi \subset \Omega \). Because \(\mu (K) {\lt} \infty \) and \(\max \phi (K) {\lt} \infty \) it follows that \(\phi \) is \(\mu \)-integrable. Linearity follows from linearity of the integral. If \(\phi _j \to \phi _*\) unifomrly, then \(\operatorname{Supp}\phi _j \subset K\) for all \(j\) and \(T_\mu \phi _j \to \Lambda _\mu \phi _*\).

We have the following important special case:

Example 4.5 Dirac-\(\delta \)

We have \(\delta \in D'(\Omega )\) given by

\[ \delta (\phi ) := \phi (0) \]

4.1.1 Convolution

Notation 1

For \(\phi \in D\) define \(\phi ^R \in D\) as \(\phi ^R(x) = \phi (-x)\) and for \(x \in \mathbb {R}^d\) we have the shift \(\tau _x(\phi ) \in D\) given by \(\tau _x(\phi )(y) = \phi (y - x)\) .

Example 4.6

For \(f \in L^1_{loc}(\Omega ), g \in D(\Omega )\) we have

\[ (f * g)(x) = \Lambda f (\tau _x (\psi ^R)) \]

Proposition 4.7

Let \(F \in \mathcal{D}'(\Omega ) , \psi \in D\). The following two distributions coincide:

The distribution determined by the smooth function \(x \mapsto F(\tau _x (\psi ^R))\).
The distribution \(\phi \mapsto F(\psi ^R * \phi )\).

We write this distribution as \(F * \psi \).

Proof ▶

\(\zeta := \psi ^R\) The function \(x \mapsto F(\tau _x(\psi ^R))\) is smooth :

It is continuous: If \(x_n \to x\), then \(\tau _x (\zeta ) - \tau _{x_n}(\zeta ) \to 0 \) uniformly and the same holds for partial derivatives. Now \(F\) is continuous hence \(F(\tau _{\bullet }(\psi ^R))\) is continuous and \(F\) preserves the difference quotients, hence it will be also smooth.
The two distributions coincide:

\begin{align*} F(\psi ^R * \phi ) & = F \left(\int (\tau _x \psi ^R(\bullet ) \phi (x) \ \mathrm{d} x) \right) \\ & \overset {!}{=} \int F(\tau _x \psi ^R(\bullet ) \phi (x)) \ \mathrm{d} x \\ & = \int F(\tau _x \psi ^R) \phi (x) \ \mathrm{d} x \\ & = \Lambda (F(\tau _{\bullet } \psi ^R))(\phi ) \end{align*}

Where we are allowed to pull out the integral by 4.8

Lemma 4.8

Let \(\phi \in C_c^{\infty }(\Omega \times \Omega )\). Then for any \(F \in D'(\Omega )\) we have

\[ F \left(\int _\Omega \phi (x,\_ ) \ \mathrm{d} x \right) = \int _\Omega F(\phi (x ,\_ )) \ \mathrm{d} x \]

Proof ▶

Consider \(S_\varepsilon \in D\) defined by

\[ S_\varepsilon ^\phi (y) = \varepsilon ^d \sum _{n \in \mathbb {Z}^d} \phi (n \varepsilon ,y) \]

which is a finite sum as \(\phi \) has compact support. Then for all \(\phi \in C_c^\infty (\Omega \times \Omega )\) one has a limit in \(\mathcal{D}\)

\[ \int _\Omega \phi (x,\_ ) \ \mathrm{d} x = \lim _{\varepsilon \to 0} S^\phi _\varepsilon \]

Hence by continuity of \(F\)

\begin{align*} F(\int _\Omega \phi (x,\_ ) \ \mathrm{d} x) & = F(\lim _{\varepsilon \to 0} S^\phi _\varepsilon ) \\ & = \lim _{\varepsilon \to 0} F(S^{\phi }_\varepsilon ) \\ & = \lim _{\varepsilon \to 0} \varepsilon ^d \sum _{n \in \mathbb {Z}^d} F(\phi (n \varepsilon ,\_ )) \\ & = \lim _{\varepsilon \to 0} S^{F \circ \phi }_\varepsilon \\ & = \int _\Omega F(\phi (x,\_ )) \ \mathrm{d} x \end{align*}

Using the first definition we learn the following two things

Example 4.9

From the description 4.6: Writing \(\Lambda f * g\) is unambiguous.

Example 4.10

We have \(\delta _0 * f = \Lambda f\) for all \(f \in D\).

Lemma 4.11

Convolution \(F * \psi \) is continuous in both variables.

Proof ▶

Continuity in the distribution variable is clear by pointwise convergence.
For the continuity in the test function variable one uses that convolution with a fixed test function is a continuous function \(\mathcal{D}\to \mathcal{D}\) and distributions are continuous.

Proposition 4.12

There exists a sequence \(\psi _n \in C_c^{\infty }(\Omega )\) such that \(\Lambda \psi _n \to \delta _0\) in \(D'(\Omega )\).

Proof ▶

Fix some \(\psi \in D\) with \(\int \psi (x) \ \mathrm{d} x = 1\). Define \(\psi _n (x) := n^d \psi (nx)\). Then

\[ (\Lambda \psi _n - \delta _0) (\phi ) = \int n^d \psi (n x) \phi (x) \ \mathrm{d} x - \phi (0) = \int \psi (x) \cdot (\phi (x / n) - \phi (0)) \ \mathrm{d} x \to 0 \]

where we used that \(\phi (x/n) - \phi (0) \to 0\) uniformly.

The following is how we view \(L^1_{loc}(\Omega ) \subset D'(\Omega )\).

Corollary 4.13

Let \(f,g \in L^1_{loc}(\Omega )\) such that \(\Lambda _f = \Lambda _g\). Then \(f = g\) almost everywhere.

Proof ▶

We have \(0 = \Lambda (f-g)(\tau _{\bullet } \psi _n^R) = \psi _n * (f - g) \to \delta _0 * (f-g) = f-g\) in \(L^1_{loc}\), hence \(f = g\) almost everywhere.

Example 4.14

\(\delta \) is not a function! I.e. its not of the form \(\Lambda f\) for some \(f \in L^1_{loc}\)

Proof ▶

We have \(\Delta |_{\Omega \setminus \{ 0\} }= \Lambda 0\) so if it would be a function, then it would be zero almost everywhere, hence 0. But this is a contradiction.

Corollary 4.15

Then \(C^\infty (\mathbb {R}^d)\) is dense in \(\mathcal{D}'(\mathbb {R}^d)\).

Proof ▶

We know by 4.12 that there exists \(\Lambda \psi _n \to \delta _0\) in \(\mathcal{D}'\). Let \(F\) be a distribution on \(\mathbb {R}^d\). Now setting \(F_n := F * \psi _n^R\) , yields pointwise

\[ F_n(\phi ) = F (\psi _n * \phi ) \to F(\delta _0 * \phi ) = F (\phi ) \]

by 4.11 hence \(F_n \to F\) in \(\mathcal{D}'\).

4.1.2 Derivatives

For \(\phi , \psi \in D\) we have

\[ \int _\Omega \partial ^\alpha \phi \psi \ \mathrm{d} x = (-1)^{|\alpha |} \int _\Omega \phi \partial ^\alpha \psi \ \mathrm{d} x \]

This motivates the definition

Definition 4.16

For a multiindex \(\alpha \) and a distribution \(F\) define the distribution

\[ \partial ^\alpha F (\phi ) = (-1)^{|\alpha |}(F \partial ^\alpha \phi ) \]

Remark 4.17

let \(f \in L^1_{loc}(\Omega )\). If there exists some \(f' \in L^1_{loc}(\Omega )\), such that \(\Lambda f' = \partial ^\alpha \Lambda f\) as distributions, then we call \(f'\) the weak derivative of \(f\) with respect to \(\alpha \).

Proposition 4.18

For \(F \in D', \phi \in D \) , We have

\[ \partial ^\alpha (F * \phi ) = (\partial ^\alpha F) * \phi = F * \partial ^\alpha \phi \]

Proof ▶

First note, that holds in the case where \(F\) is a test function, so that we have ordinary convolution. Then check pointwise. After erasing the sign \((-1)^{|\alpha |}\)

\[ F(\phi ^R * \partial ^\alpha \psi ) = F(\partial ^\alpha (\phi ^R * \psi )) = F((\partial ^\alpha \phi ) * \psi ) \]

4.1.3 Support

The Support of a distribution \(F\) is the complement of the largest open subset \(U\), such that \(F(\phi ) = 0 \forall \phi \in D, \operatorname{Supp}(\phi ) \subset U\). This definition is unambiguous: If \(F\) vanishes on each \(U_i\) for some index set \(I\) and \(\phi \in D\) such that the compact set \(\operatorname{Supp}\phi \subset U = \bigcup U_i\), we may assume that \(I\) is finite and choose a partition of unity \(\operatorname{Supp}\eta _i \subset U_i\) and \(\sum \eta _i = 1\). Then \(F(\phi ) = \sum F(\phi \eta _i) = 0\).

4.1.4 Tempered Distributions

We no enlarge our test space to the schwartz space \(\mathcal{S}= \mathcal{S}(\mathbb {R}^d)\) consisting of smooth functions that are rapidly decreasing at \(\infty \) with all derivatives.

Definition 4.19

Consider the increasing sequence of norms on \(C^\infty (\Omega )\) defined by

\[ \| \phi \| _N = \sup \{ | x^\beta (\partial _x^\alpha \phi (x)) | \mid x \in \mathbb {R}^d , |\alpha |,\| \beta \| \le N\} \]

\[ \mathcal{S}= \{ \phi \in C^\infty (\mathbb {R}^d) \mid \| \phi \| _N {\lt} \infty \forall N\} \]

with the obvious notion of convergence.

Lemma 4.20

We have a continuous inclusions \(\mathcal{D}\subset \mathcal{S}\), hence \(\mathcal{S}' \subset \mathcal{D}'(\mathbb {R}^d)\).
Moreover, this inclusion is dense. Hence being tempered is a property of distributions.

Lemma 4.21

Let \(f \in L^1_{loc}(\mathbb {R}^d)\) such that there exists \(N \ge 0\) with

\[ \int _{|x| {\lt} R} |f(x)| \ \mathrm{d} x = O(R^N) , \text{ as } R \to \infty \]

The distribution \(\Lambda _f\) is tempered.

Example 4.22

This condition holds for functions in \(L^p(\mathbb {R}^d)\) for \(p \in [1,\infty ]\)

Lemma 4.23

If \(F \in \mathcal{D}'\) has compact support then it is tempered: Choose \(\eta \in \mathcal{D}\) such that \(\eta |_U = 1\) on some neighborhood \(U \supset \operatorname{Supp}(F)\). Then \(F(\eta \phi ) = F(\phi )\) for all \(\phi \in \mathcal{D}\) and \(\phi \mapsto F (\eta \phi )\) defines a continuous functional on \(\mathcal{S}\). It does not depend on the choice of \(\eta \), because given another such \(\eta '\) and any \(\phi \in \mathcal{S}\) we have \(\operatorname{Supp}((\eta - \eta ') \cdot \phi ) \cap \operatorname{Supp}(F) = \varnothing \).

Let \(F\) denote a tempered distribution.

Lemma 4.24

All \(\partial ^\alpha F\) are tempered.
Let \(\psi \in C^\infty \) be slowly increasing, i.e. for each \(\alpha \) exists \(N_\alpha \) such that \(\partial _x^\alpha \psi (x) = O(|x|^{N_\alpha })\). Then \(\psi F\), defined by \((\psi F)(\phi )(F(\psi \phi )\) is tempered.

Example 4.25

If \(\psi \in \mathcal{S}\), then \(F(\tau _\bullet (\psi ^R))\) is slowly increasing. The other formulation is still valid because \(\mathcal{S}\) is stable under \(*\).

What is the point of the Schwartz space?

Definition 4.26

The fourier transformation is a continuous bijection

\begin{align*} \mathcal{S}& \to \mathcal{S}\\ \phi & \mapsto \hat\phi = (\xi \mapsto \int _{\mathbb {R}^d} \phi (x) e^{-2\pi i x \xi } \ \mathrm{d} x) \end{align*}

Lemma 4.27

We have

\[ \Lambda _{\hat\psi }(\phi ) = \int _{\mathbb {R}^d} \hat\psi (x) \phi (x) \ \mathrm{d} x = \int _{\mathbb {R}^d} \psi (x) \hat\phi (x) \ \mathrm{d} x = \Lambda _{\psi }(\hat\phi ) \]

So its easy to define a compatible generalization to the tempered distributions:

Definition 4.28

Define

\[ \hat F (\phi ) = F (\hat\phi ) \]

and similarly for the inverse transform \(f \mapsto \check{f}\).

We automatically have the inversion theorem for distributions, because it holds for test functions.

Example 4.29

If \(1 \in S\) denotes the constant function at \(1\), then

\[ \widehat{\delta } = \Lambda _1 \]

because

\[ \hat\delta (\phi ) =\hat\phi (0) = \int 1 \phi (x) \ \mathrm{d} x = \Lambda _1(\phi ) \]

4.2 Fundamental solutions

In this chapter we may omit the \(\Lambda \). Fix a partial differential operator \(L\)

\[ L = \sum _{|\alpha |\le m } a_\alpha \partial ^\alpha \text{ on } \mathbb {R}^d \]

with \(a_\alpha \in \mathbb {C}\).

Definition 4.30

A fundamental solution of \(L\) is a distribution \(F\) such that \(L(F) = \delta \)

The reason why this is interesting:

Lemma 4.31

The operator

\[ f \mapsto T(f) := F * f \]

defines an inverse to \(L\)

Proof ▶

because

\[ \partial ^\alpha (F * f) = (\partial ^\alpha F) * f = F * (\partial ^\alpha f) \]

Summing over \(\alpha \) gives

\[ L(F * f) = \delta * f = F * L f \]

Now recall \(\delta * f = f\).

Definition 4.32

The characteristic polynomial of \(L\) is

\[ P(\xi ) = \sum _{|\alpha |\le m} a_\alpha (2 \pi i \xi )^\alpha \]

It is defined it such a way, that \(\widehat{L f} = P \cdot \hat f\). So our hope is

\[ F := \widecheck {1 / (P (\xi ))} = \int \frac{1}{P(\xi )} e^{2 \pi i x \xi } \ \mathrm{d} \xi \]

The problem is, that the zeros of \(P\) result in difficulties to define \(F\), even as a distribution.

4.2.1 Laplacian

If \(L = \Delta = \sum _{i=1}^d \frac{\partial ^2}{\partial ^2 x_i}\) So \(1 / P (\xi ) = 1 / (-4 \pi ^2 |\xi |^2) \) which lies in \(L^1_{loc}\) for \(d \ge 3\). To calculate our distribution the following is helpful

Theorem 4.33

For \(\lambda {\gt} -d\), Let \(H_\lambda \) be the tempered distribution associated to \(|x|^\lambda \in L^1_{loc}\).
If \(-d {\lt} \lambda {\lt} 0\) then

\[ \widehat{H_\lambda } = c_\lambda H_{-d-\lambda } \]

with

\[ c_\lambda = \frac{\Gamma ((d + \lambda )/2)}{\Gamma (\lambda / 2)} \pi ^{-d/2-\lambda } \]

Set \(\lambda :=-d+2\). Hence we can find an appropriate constant \(C_d\) such that \(F(x) = C_d |x|^{-d +2}\) is a fundamental solution, because \(\hat{F}(\xi ) = 1 / (-4\pi ^2|\xi |^2)\). So writing out \(\widehat{\Delta F} = 1\).

Proposition 4.34

If \(d = 2\), the function \(F := 1/(2\pi ) \log |x| \in L^1_{loc}\) is a fundamental solution of \(\Delta \)

Proof ▶

Sketch. One can actually compute \(\hat F = -1 / (4\pi ^2) \left[\frac{1}{|x|^2} \right] - c' \delta \) for some constant \(c'\), where \(\left[\frac{1}{|x|^2} \right]\) is a distribution, that replaces the (non locally integrable) function \(1 / |x|^2\) in an appropriate way:

\[ \left[\frac{1}{|x|^2} \right] = \int _{|x| \le 1} \frac{\phi (x) - \phi (0)}{|x|^2} \ \mathrm{d} x + \int _{|x| {\gt} 1} \frac{\phi (x)}{|x|^2} \ \mathrm{d} x \]

Notice, that on the complement of zero, this distribution coincides with \(1 / |x|^2\).

Notation 2

If \(\phi \in C^\infty (\Omega )\) slowly increasing (i.e. all derivatives are bounded by polynomials), define \(\phi F \in \mathcal{S}'\) as \(\phi F(\psi ) = F(\phi \psi )\).

\begin{align*} \widehat{ \Delta F} & = -4 \pi ^2 |x|^2 \hat F\\ & = |x|^2 \left[\frac{1}{|x|^2} \right] - 4\pi ^2 c’ \underbrace{|x|^2 \delta }_{=0} \\ & = 1 \end{align*}

4.2.2 Heat operator

\(L = \frac{\partial }{\partial t} - \Delta _x\) taken over \((x,t) \in \mathbb {R}^{d+1} = \mathbb {R}^d \times \mathbb {R}\) i.e. we want to solve the homogeneous initial value problem

\[ \begin{cases} L (u) = 0 & , t {\gt} 0 \\ u(x,0) = f(x) & , t = 0 \end{cases} \]

for some initial value \(f \in \mathcal{S}\).
We have

\[ (\frac{\partial }{\partial t} \hat{\mathcal{H}_t})(\xi ) = \widehat{\frac{\partial }{\partial t} \mathcal{H}_t}(\xi ) = \widehat{\Delta _x \mathcal{H}_t}(\xi ) = -4\pi ^2 |\xi |^2 \hat{\mathcal{H}_t} (\xi ) \]

and this is obviously solved by \(\mathcal{H}_t = e^{-4\pi ^2 |\xi |^2 t}\). We may call this the heat kernel

\[ \hat{\mathcal{H}_t}(\xi ) = e^{-4\pi ^2 |\xi |^2 t} \]

Note that for \(t = 0\), we have \(\hat{\mathcal{H}_0} = 1\), hence \(\mathcal{H}_0 = \delta \), so \(u(x,t) = (\mathcal{H}* f)(x)\) solves the equation \(L(u) = 0\) and \(u(x,t) \to f(x)\) in \(\mathcal{S}\) as \(t \to 0\).

Remark 4.35

\(\mathcal{H}_t \to \delta \) in \(\mathcal{S}'\) as \(t \to 0\) and \(\int _{\mathbb {R}^d} \mathcal{H}_t(x) \ \mathrm{d} x = 1\) for all \(t\)

Now define

\[ F(x,t) := \begin{cases} \mathcal{H}_t(x) & , \ if \ t {\gt} 0 , \\ 0 & , t \le 0 \end{cases} \]

\(F\) is locally integrable on \(\mathbb {R}^{d+1}\) and it actually holds

\[ \int _{|t|\le R} \int _{\mathbb {R}^d} F(x,t) \ \mathrm{d} x \ \mathrm{d} t \le R \]

so \(F\) defines a tempered distribution by 4.21.

Theorem 4.36

\(F\) is a fundamental solution of \(L = \frac{\partial }{\partial t} - \Delta _x\).

Proof ▶

Denote \(L' = - \frac{\partial }{\partial t} - \Delta _x\), then we have to see the last equation

\[ LF (\phi ) = F(L'(\phi )) = \lim _{\varepsilon \to 0} \int _{t \ge \varepsilon } \int _{\mathbb {R}^d} F(x,t) ( -\frac{\partial }{\partial t} - \Delta _x) \phi (x,t) \ \mathrm{d} x \ \mathrm{d} t \overset {!}{=} \delta (\phi ) \]

Integration by parts

\begin{align*} & \int _{t \ge \varepsilon } \int _{\mathbb {R}^d} F(x,t) ( -\frac{\partial }{\partial t} - \Delta _x) \phi (x,t) \ \mathrm{d} x \ \mathrm{d} t \\ & = - \int _{\mathbb {R}^d} \left(\int _{t \ge \varepsilon } \mathcal{H}_t \frac{\partial }{\partial t} + (\Delta _x \mathcal{H}_t) \phi \ \mathrm{d} t \right) \ \mathrm{d} x \\ & = - \int _{\mathbb {R}^d} \left(\int _{t \ge \varepsilon } \mathcal{H}_t \frac{\partial }{\partial t} + ( \frac{\partial }{\partial t} \mathcal{H}_t) \phi \ \mathrm{d} t \right) \ \mathrm{d} x \\ & = \int _{\mathbb {R}^d} \mathcal{H}_\varepsilon (x) \phi (x,\varepsilon ) \ \mathrm{d} x & & \mid |\phi (x,\varepsilon ) - \phi (x,0)| \le O(\varepsilon ) \text{ uniformly in } x \\ & = \int _{\mathbb {R}^d} \mathcal{H}_\varepsilon (x) (\phi (x,0) + O(\varepsilon )) \ \mathrm{d} x & & \ref{remark:approxId}\\ & \to \phi (0,0) \end{align*}

where in the last line we let \(\varepsilon \to 0\)