Editing Distribution (mathematics) (section)

====Differential operators acting on smooth functions====

A linear differential operator in <math>U</math> with smooth coefficients acts on the space of smooth functions on <math>U.</math> Given such an operator
<math display=inline>P := \sum_\alpha c_\alpha \partial^\alpha,</math>
we would like to define a continuous linear map, <math>D_P</math> that extends the action of <math>P</math> on <math>C^\infty(U)</math> to distributions on <math>U.</math> In other words, we would like to define <math>D_P</math> such that the following diagram [[Commutative diagram|commutes]]:
<math display=block>\begin{matrix}
\mathcal{D}'(U) & \stackrel{D_P}{\longrightarrow} & \mathcal{D}'(U) \\[2pt]
\uparrow & & \uparrow \\[2pt]
C^\infty(U) & \stackrel{P}{\longrightarrow} & C^\infty(U)
\end{matrix}</math>
where the vertical maps are given by assigning <math>f \in C^\infty(U)</math> its canonical distribution <math>D_f \in \mathcal{D}'(U),</math> which is defined by: 
<math display=block>D_f(\phi) = \langle f, \phi \rangle := \int_U f(x) \phi(x) \,dx \quad \text{ for all } \phi \in \mathcal{D}(U).</math> 
With this notation, the diagram commuting is equivalent to:
<math display=block>D_{P(f)} = D_PD_f \qquad \text{ for all } f \in C^\infty(U).</math>

To find <math>D_P,</math> the transpose <math>{}^{t} P : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> of the continuous induced map <math>P : \mathcal{D}(U)\to \mathcal{D}(U)</math> defined by <math>\phi \mapsto P(\phi)</math> is considered in the lemma below. 
This leads to the following definition of the differential operator on <math>U</math> called {{em|the '''formal transpose''' of <math>P,</math>}} which will be denoted by <math>P_*</math> to avoid confusion with the transpose map, that is defined by
<math display=block>P_* := \sum_\alpha b_\alpha \partial^\alpha \quad \text{ where } \quad b_\alpha := \sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \partial^{\beta-\alpha} c_\beta.</math>

{{math theorem|name=Lemma|math_statement= Let <math>P</math> be a linear differential operator with smooth coefficients in <math>U.</math> Then for all <math>\phi \in \mathcal{D}(U)</math> we have
<math display=block>\left\langle {}^{t}P(D_f), \phi \right\rangle = \left\langle D_{P_*(f)}, \phi \right\rangle,</math>
which is equivalent to:
<math display=block>{}^{t}P(D_f) = D_{P_*(f)}.</math>}}

{{collapse top|title=Proof|left=true}}
As discussed above, for any <math>\phi \in \mathcal{D}(U),</math> the transpose may be calculated by:
<math display=block>\begin{align}
\left\langle {}^{t}P(D_f), \phi \right\rangle &= \int_U f(x) P(\phi)(x) \,dx \\
&= \int_U f(x) \left[\sum\nolimits_\alpha c_\alpha(x) (\partial^\alpha \phi)(x) \right] \,dx \\
&= \sum\nolimits_\alpha \int_U f(x) c_\alpha(x) (\partial^\alpha \phi)(x) \,dx \\
&= \sum\nolimits_\alpha (-1)^{|\alpha|} \int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,d x
\end{align}</math>

For the last line we used [[integration by parts]] combined with the fact that <math>\phi</math> and therefore all the functions <math>f (x)c_\alpha (x) \partial^\alpha \phi(x)</math> have compact support.<ref group="note">For example, let <math>U = \R</math> and take <math>P</math> to be the ordinary derivative for functions of one real variable and assume the support of <math>\phi</math> to be contained in the finite interval <math>(a,b),</math> then since <math>\operatorname{supp}(\phi) \subseteq (a, b)</math>
<math display=block>\begin{align}
\int_\R \phi'(x)f(x)\,dx &= \int_a^b \phi'(x)f(x) \,dx \\
&= \phi(x)f(x)\big\vert_a^b - \int_a^b f'(x) \phi(x) \,d x \\
&= \phi(b)f(b) - \phi(a)f(a) - \int_a^b f'(x) \phi(x) \,d x \\
&=-\int_a^b f'(x) \phi(x) \,d x
\end{align}</math>
where the last equality is because <math>\phi(a) = \phi(b) = 0.</math></ref> Continuing the calculation above, for all <math>\phi \in \mathcal{D}(U):</math>
<math display=block>\begin{align}
\left\langle {}^{t}P(D_f), \phi \right\rangle &=\sum\nolimits_\alpha (-1)^{|\alpha|} \int_U \phi(x) (\partial^\alpha(c_\alpha f))(x) \,dx && \text{As shown above} \\[4pt]
&= \int_U \phi(x) \sum\nolimits_\alpha (-1)^{|\alpha|} (\partial^\alpha(c_\alpha f))(x)\,dx \\[4pt]
&= \int_U \phi(x) \sum_\alpha \left[\sum_{\gamma \le \alpha} \binom{\alpha}{\gamma} (\partial^{\gamma}c_\alpha)(x) (\partial^{\alpha-\gamma}f)(x) \right] \,dx && \text{Leibniz rule}\\
&= \int_U \phi(x) \left[\sum_\alpha \sum_{\gamma \le \alpha} (-1)^{|\alpha|} \binom{\alpha}{\gamma} (\partial^{\gamma}c_\alpha)(x) (\partial^{\alpha-\gamma}f)(x)\right] \,dx \\
&= \int_U \phi(x) \left[ \sum_\alpha \left[ \sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \left(\partial^{\beta-\alpha}c_{\beta}\right)(x) \right] (\partial^\alpha f)(x)\right] \,dx && \text{Grouping terms by derivatives of } f \\
&= \int_U \phi(x) \left[\sum\nolimits_\alpha b_\alpha(x) (\partial^\alpha f)(x) \right] \, dx && b_\alpha:=\sum_{\beta \geq \alpha} (-1)^{|\beta|} \binom{\beta}{\alpha} \partial^{\beta-\alpha}c_{\beta} \\
&= \left\langle \left(\sum\nolimits_\alpha b_\alpha \partial^\alpha \right) (f), \phi \right\rangle
\end{align}</math>
{{collapse bottom}}

The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, <math>P_{**}= P,</math>{{sfn|Trèves|2006|pp=247-252}} enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator <math>P_* : C_c^\infty(U) \to C_c^\infty(U)</math> defined by <math>\phi \mapsto P_*(\phi).</math> We claim that the transpose of this map, <math>{}^{t}P_* : \mathcal{D}'(U) \to \mathcal{D}'(U),</math> can be taken as <math>D_P.</math> To see this, for every <math>\phi \in \mathcal{D}(U),</math> compute its action on a distribution of the form <math>D_f</math> with <math>f \in C^\infty(U)</math>:

<math display=block>\begin{align}
\left\langle {}^{t}P_*\left(D_f\right),\phi \right\rangle &= \left\langle D_{P_{**}(f)}, \phi \right\rangle && \text{Using Lemma above with } P_* \text{ in place of } P\\
&= \left\langle D_{P(f)}, \phi \right\rangle && P_{**} = P
\end{align}</math>

We call the continuous linear operator <math>D_P := {}^{t}P_* : \mathcal{D}'(U) \to \mathcal{D}'(U)</math> the '''{{em|differential operator on distributions extending <math>P</math>}}'''.{{sfn|Trèves|2006|pp=247-252}} Its action on an arbitrary distribution <math>S</math> is defined via:
<math display=block>D_P(S)(\phi) = S\left(P_*(\phi)\right) \quad \text{ for all } \phi \in \mathcal{D}(U).</math>

If <math>(T_i)_{i=1}^\infty</math> converges to <math>T \in \mathcal{D}'(U)</math> then for every multi-index <math>\alpha, (\partial^\alpha T_i)_{i=1}^\infty</math> converges to <math>\partial^\alpha T \in \mathcal{D}'(U).</math>