Existence and Uniqueness of Distributional Primitive

Theorem
Let $T \in \map {\DD'} \R$ be a distribution.

Then there exists a distribution $S \in \map {\DD'} \R$ such that in the distributional sense:


 * $S' = T$

Furthermore, $S$ is unique up to an arbitrary constant.

Proof
Let $\mathbf 0 : \R \to 0$ be the zero mapping.

Let $\phi_0 \in \map \DD \R \setminus \set {\mathbf 0}$ be a test function.

Let $\psi \in \map \DD \R$ be a test function.

Let $\phi : \R \to \R$ be a real function such that:


 * $\ds \phi := \psi - \frac {\int_{-\infty}^\infty \map \psi x \rd x} {\int_{-\infty}^\infty \map {\phi_0} x \rd x} \phi_0$

Existence of Unique Primitive of $\phi$
Since $\phi_0, \psi \in \map \DD \R$, by the closure of the test function vector space $\phi \in \map \DD \R$ too.

Moreover:

Thus:


 * $\ds \phi \in Y := \set {\phi \in \map \DD \R : \int_{-\infty}^\infty \map \phi x \rd x = 0}$

By Characterization of Derivative of Test Function, there exists a unique $\Phi \in \map \DD \R$ such that $\Phi' = \phi$.

Let $T \in \map {\DD'} \R$ be a distribution.

Let $S: \map \DD \R \to \C$ be a mapping such that $\map S \psi = - \map T \Phi$.

Linearity of $S$
Let $\psi_1, \psi_2 \in \map \DD \R$.

Let $\Phi_1, \Phi_2 \in \map \DD \R$ be such that:


 * $\ds \Phi_1' = \phi_1 := \psi_1 - \frac {\int_{-\infty}^\infty \map {\psi_1} x \rd x} {\int_{-\infty}^\infty \map {\phi_0} x \rd x} \phi_0$


 * $\ds \Phi_2' = \phi_2 := \psi_2 - \frac {\int_{-\infty}^\infty \map {\psi_2} x \rd x} {\int_{-\infty}^\infty \map {\phi_0} x \rd x} \phi_0$

Then:


 * $\ds \paren {\Phi_1 + \Phi_2}' = \psi_1 + \psi_2 - \frac {\int_{-\infty}^\infty \paren {\map {\psi_1} x + \map {\psi_2} x} \rd x} {\int_{-\infty}^\infty \map {\phi_0} x \rd x} \phi_0$

Thus:

Similarly:

Continuity of $S$
Let $\sequence {\psi_n}_{n \mathop \in \N}$ be a sequence in $\map \DD \R$ such that $\psi_n \stackrel {\DD} {\longrightarrow} \mathbf 0$.

By definition of the convergent sequence in the test function space, there exists $a \in \R_{> 0}$ such that:


 * $\forall n \in \N : \forall x \notin \closedint {-a} a : \map {\psi_n} x = 0$

and $\sequence {\psi_n}_{n \mathop \in \N}$ converges uniformly to $\mathbf 0$.

Hence:

Let:


 * $\ds \phi_n := \psi_n - \frac {\int_{-\infty}^\infty \map {\psi_n} x \rd x} {\int_{-\infty}^\infty \map {\phi_0} x \rd x} \phi_0$

$\phi_n$ is composed of $\psi_n$ and $\phi_0$, which have compact supports.

Then $\phi_n$ also has a compact support.

Thus:


 * $\forall n \in \N : \exists b \in \R_{> 0} : \forall x \notin \closedint {-b} b : \map {\phi_n} x = 0$

For $k \in \N_{> 0}$ we have that:


 * $\ds \phi_n^{\paren k} = \psi_n^{\paren k} - \frac {\int_{-\infty}^\infty \map {\psi_n} x \rd x} {\int_{-\infty}^\infty \map {\phi_0} x \rd x} \phi_0^{\paren k}$

Since $\sequence {\psi_n}_{n \mathop \in \N}$ converges in the test function space, it holds that for all $k \in \N_{> 0}$ $\sequence {\psi_n^{\paren k}}_{n \mathop \in \N}$ converges uniformly to $\mathbf 0$.

Also, for all $k \in \N_{> 0}$ $\phi_0^{\paren k}$ is fixed.

Then:

Thus, for all $k \in \N_{> 0}$ the sequence $\sequence {\phi_n^{\paren k}}_{n \mathop \in \N}$ converges uniformly to the same mapping as $\sequence {\psi_n^{\paren k}}_{n \mathop \in \N}$, that is, $\mathbf 0$.

Hence, $\phi_n$ converges in the test function space to $\mathbf 0$:


 * $\phi_n \stackrel {\DD} {\longrightarrow} \mathbf 0$

For every $n \in \N$ let $\Phi_n$ be the unique element in $\map \DD \R$ such that $\Phi'_n = \phi_n$.

From Conditions for Preservation of Covergence in Test Function Space under Differentiation we conclude that:


 * $\Phi_n \stackrel {\DD} {\longrightarrow} \mathbf 0$

Consequently:

Thus, $S \in \map {\DD'} \R$.

$S' = T$
Let $\Phi \in \map \DD \R$.

Then:

Thus:

Hence, $S' = T$.

$S$ is unique up to a constant
Suppose $S, R \in \map {\DD'} \R$ are distributions such that for some test function $\rho \in \map \DD \R$ we have $\map {S'} \rho = \map T \rho$, $\map {R'} \rho = \map T \rho$.

Then:

We have that:


 * Distributional Derivative on Distributions is Linear Operator
 * Distribution Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space

By Vanishing Distributional Derivative of Distribution implies Distribution is Constant, $S - R$ must be a constant distribution $F_c$ with $c \in \R$.