Distributions & Currents

1. Distributions

The theory of distributions frees differential calculus from certain difficulties that arise because nondifferentiable functions exist. This is done by extending it to the class of distributions or generalized functions.

Set $\mathcal{D}({\mathbb{R}}^n)=C_0^{\infty }({\mathbb{R}}^n)$, and then $\int f\varphi$ exists for every locally integrable function $f$ and for every $\varphi \in \mathcal{D}({\mathbb{R}}^n)$.
If $f$ is smooth, by integral by parts we have:
$$\int f^{(k)}\varphi =(-1{)}^k\int f{\varphi }^{(k)},k=1,2,3,...$$
Observe that the integrals on the right sides make sense whether $f$ is differentiable or not. Also we can assign a “$k$-th derivative” to every $f$ that is locally integrable: $f^{(k)}$ is the linear functional on $\mathcal{D}$ that sends $\varphi$ to $(-1{)}^k\int f{\varphi }^{(k)}$.
The distributions will be those linear functionals on $\mathcal{D}({\mathbb{R}}^n)$ that are continuous with respect to a certain topology which we just simply conclude here:
we define the $C^k$-norm on $\mathcal{D}({\mathbb{R}}^n)$, and then we have the induced topology as well as the convergence theory.

$\text{Definition }1.1$ Let $\Omega \subset {\mathbb{R}}^n$. A linear functional on $\mathcal{D}(\Omega )$ which is continuous with respect to the topology defined in [Rudin Definition 6.3] is called a distribution in $\Omega$. The space of all distributions in $\Omega$ is denoted by ${\mathcal{D}}^{\prime }(\Omega )$.

For example, each $x\in \Omega$ determines a linear functional ${\delta }_x$ on $\mathcal{D}(\Omega )$, by the formula
$${\delta }_x(\varphi )=\varphi (x).$$ Specially, if $x$ is the origin of ${\mathbb{R}}^n$, the functional $\delta ={\delta }_0$ is frequently called the Dirac measure on ${\mathbb{R}}^n$.

We can also consider functions and measures as distributions. For example, suppose $f$ is a locally integrable complex function in $\Omega$, which means $f$ is Lebesgue measurable and ${\int }_K|f(x)|dx<\infty$ for every compact $K\subset \Omega$; $dx$ denotes Lebesgue measure. Define
$${\Lambda }_f(\varphi )={\int }_{\Omega }\varphi (x)f(x)dx,\varphi \in \mathcal{D}(\Omega ).$$
In this way, ${\Lambda }_f\in {\mathcal{D}}^{\prime }(\Omega )$.
Also, we can define the differentiation of distributions:
if $\alpha$ is a muti-index and $\Lambda \in {\mathcal{D}}^{\prime }(\Omega )$, the formula:
$$(D^{\alpha }\Lambda )(\varphi )=(-1{)}^{|\alpha |}\Lambda (D^{\alpha }\varphi ),\varphi \in \mathcal{D}(\Omega ).$$
As a result, we have $D^{\alpha }\Lambda \in {\mathcal{D}}^{\prime }(\Omega )$.

One interesting question is if $D^{\alpha }f$ exists in the classical sense and is locally integrable whether do we have the equation
$$D^{\alpha }{\Lambda }_f={\Lambda }_{D^{\alpha }f}$$
which means
$$(-1{)}^{|\alpha |}{\int }_{\Omega }f(x)(D^{\alpha }\varphi )(x)dx={\int }_{\Omega }(D^{\alpha }f)(x)\varphi (x)dx.$$
As a result, if $f$ has continuous partial derivatives of all orders up to $N\ge |\alpha |$, integrations by part assure it is true. However, in general, the result may be false.

We can also multiplicate the distribution by functions: suppose $\Lambda \in {\mathcal{D}}^{\prime }(\Omega )$ and $f\in C^{\infty }(\Omega )$, we can define the distribution $f\Lambda$ acts on a function $\varphi \in \mathcal{D}(\Omega )$ as
$$(f\Lambda )(\varphi )=\Lambda (f\varphi ).$$

Since ${\mathcal{D}}^{\prime }(\Omega )$ is the space of all continuous linear functions on $\mathcal{D}(\Omega )$, we can endow it the weak*-topology. Thus, we can discuss the sequence convergence in ${\mathcal{D}}^{\prime }(\Omega )$:
if $f_i$ is a sequence of locally integrable functions in $\Omega$, “$f_i\to \Lambda$ in ${\mathcal{D}}^{\prime }(\Omega )$” means
$$\underset{i\to \infty }{\lim }{\int }_{\Omega }\varphi (x)f_i(x)dx=\Lambda \varphi ,$$
for every $\varphi \in \mathcal{D}(\Omega )$.
We have the following two theorems:

$\text{Theorem }1.2$ Suppose ${\Lambda }_i\in {\mathcal{D}}^{\prime }(\Omega )$ for $i=1,2,3,...$, and
$$\underset{i\to \infty }{\lim }{\Lambda }_i\varphi =\Lambda \varphi$$
exists for every $\varphi \in \mathcal{D}(\Omega )$. Then $\Lambda \in {\mathcal{D}}^{\prime }(\Omega )$, and
$$D^{\alpha }{\Lambda }_i\to D^{\alpha }\Lambda \text{ in }{\mathcal{D}}^{\prime }(\Omega ),$$
for every multi-index $\alpha$.

$\text{Theorem }1.3$ If ${\Lambda }_i\to \Lambda$ in ${\mathcal{D}}^{\prime }(\Omega )$ and $g_i\to g$ in $C^{\infty }(\Omega )$, then $g_i{\Lambda }_i\to g\Lambda$ in ${\mathcal{D}}^{\prime }(\Omega )$.

The above two theorems are not hard to prove, with some knowledge of functional analysis, I omit here.

Besides, it should be mentioned that it’s possible to describe a distribution globally if its local behavior is known.
At last, the theory of distributions enlarges the concept of function in such a way that partial differentiations can be carried out unrestrictedly. Conversely, every distribution is $D^{\alpha }f$ for some continuous function $f$ and some multi-index $\alpha$.

$\text{Theorem }1.4$ Suppose $\Lambda \in {\mathcal{D}}^{\prime }(\Omega )$, and $K$ is a compact subset of $\Omega$. Then there is a continuous function $f$ in $\Omega$ and there is a multi-index $\alpha$ such that
$$\Lambda \varphi =(-1{)}^{|\alpha |}{\int }_{\Omega }f(x)(D^{\alpha }\varphi )(x)dx$$
for every $\varphi \in \mathcal{D}(K)$.

2. Currents

We also denote $\Omega$ as an open subset of ${\mathbb{R}}^n$.

A differential form $\alpha$ of degree $p$ on $\Omega$ with locally integrable coefficients,
$$\alpha =\sum _{|I|=p}{\alpha }_{I}d{x}_I,$$
acts as a linear form on the space of continuous test forms of complementary degree $q=n-p$: if $\psi =\chi d{x}_K$ is a smooth test form of degree $q$ with compact support, then
$$<\alpha ,\psi >=\sum _{|I|=p}{ϵ}_{I,K}{\int }_{\Omega }\chi {\alpha }_{I}dV,$$
where ${ϵ}_{I,K}$ is such that $d{x}_I\wedge d{x}_K={ϵ}_{I,K}dV$, with ${ϵ}_{I,K}=0$ unless $K=I^c$ complements $I$ in $[1,N]$, in which case ${ϵ}_{I,K}=\pm 1$.

$\text{Definition 1.5}(\text{Current})$ A current $S$ of degree $p$ is a continuous linear form on the space ${\mathcal{D}}_{n-p}(\Omega )$ of test forms (i.e., smooth differential forms with compact support). We let ${\mathcal{D}}_{n-p}^{\prime }(\Omega )$ denote the space of currents of degree $p$. the action of $S$ on a test form $\Psi \in {\mathcal{D}}_{n-p}(\Omega )$ is denoted by $<S,\Psi >$.

If $\alpha$ is a smooth form of degree $q$, the wedge product of $\alpha$ and $S$ is defined as follows:

$\text{Definition 1.6}$ For a test form $\Phi$ of degree $n-p-q$, we set
$$<S\wedge \alpha ,\Psi >:=<S,\alpha \wedge \Psi >.$$
Here $S\in {\mathcal{D}}_{n-p}(\Omega )$, which is a current of degree $p$.
We also define $\alpha \wedge S:=(-1{)}^{pq}S\wedge \alpha$.

Specially, the current $S\wedge \alpha \wedge \Psi$ is a current of maximal degree with compact support, and it can be identified with a distribution with compact support. One can similarly interpret a current of degree $p$ as a differential form of degree $p$ with distribution coefficients.

Next, we can extend differential calculus on forms to currents.
$\text{Definition 1.7}$ If $S$ is a current of degree $p$, then $dS$ is the current of degree $(p+1)$ defined by
$$<dS,\varphi >=(-1{)}^{(p+1)}<S,d\varphi >,$$
for any test form $\varphi$ of degree $n-p-1$.

For $\Omega \subset {\mathbb{C}}^n$, a domain in the complex space, we can define the current $T$ of bidegree $(p,q)$ is a differential form of bidegree $(p,q)$ with distribution coefficients.
And then, we have the positivity of forms and currents of bidegree $(p,p),\forall$ integer $p,1\le p\le n$.

Then we states some relations between current and plurisubharmonic functions:

$\text{Proposition 1.8}$ A function $\varphi$ is plurisubharmonic if and only if the current $\sqrt{-1}\partial \bar{\partial }\varphi$ is positive.