Question

Let T be a tempered distribution. Then for all multi-indices a, we define ∂^(a)T to be the linear functional on S by (∂^(a)T)(φ) = (-1)^(|a|)T(∂^(a)φ), φES. Prove that ∂^(a)T is a tempered distribution.

Answer 1

According to the definition of a tempered distribution, we need to prove that $\partial^a T$ is a linear and continuous functional in $S$. Linearity means that for any $\varphi_1 \in S , \varphi_2 \in S$ and any numbers $c_1, c_2$, one has $\partial^a T ( c_1 \varphi_1 + c_2 \varphi_2) = c_1 \partial^a T ( \varphi_1) + c_2 \partial^a T (\varphi_2)$. Continuity means that, for any sequence $\varphi_n \to \varphi$ in $S$, the numerical sequence $\partial^a T (\varphi_n) \to \partial^a T (\varphi)$. We prove both properties using the properties of linearity and continuity of $T$ itself.

Proof of linearity:

$\partial^a T \left( c_1 \varphi_1 + c_2 \varphi_2 \right) = (-1)^{|a|} T \left( \partial^a ( c_1 \varphi_1 + c_2 \varphi_2 ) \right) \\ = (-1)^{|a|} T \left( c_1 \partial^a \varphi_1 + c_2 \partial^a \varphi_2 \right) \\ = c_1 (-1)^{|a|} T \left( \partial^a \varphi_1 \right) + c_2 (-1)^{|a|} T \left( \partial^a \varphi_2 \right) \\ = c_1 \partial^a T (\varphi_1) + c_2 \partial^a T (\varphi_2) \, .$

Proof of continuity:

For any sequence $\varphi_n \to \varphi$ in $S$, and for any multi-index $a$, we have $\partial^a \varphi_n \to \partial^a \varphi$ in $S$. Then

$\partial^a T (\varphi_n) = (-1)^{|a|} T \left( \partial^a \varphi_n \right) \to (-1)^{|a|} T \left( \partial^a \varphi \right) \\ = \partial^a T (\varphi) \, .$

Both defining properties are proved. Therefore, $\partial^a T$ is a tempered distribution.