How to prove the existence of u?

Question

Given $T \in D^{'}(R^{1})$,prove the existence of $u$, satisfying $\frac{du}{dx}=T$.

I totally have no idea of how to prove this. Could anybody give me a hint?

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Remark:$D^{'}(R^{1})$ is the dual space of $C_{c}^{\infty}(R^{1})$.

Answer 1

Let $\eta \in C_c^{\infty}(\mathbb{R}^1)$ be any function such that $\int_{\mathbb{R}} \eta = 1$, then define

$$u(\varphi) = - T(G(\varphi))$$

where

$$G: C_c^{\infty}(\mathbb{R}) \to C_c^{\infty}(\mathbb{R}), \qquad G(\varphi) = \int_{-\infty}^x [\varphi(y) - \eta(y) \cdot I(\varphi)] dy$$

where $I(\varphi) = \int_{-\infty}^{\infty} \varphi$. Then because $\varphi$ and $\eta$ both have compact support, then so will $G(\varphi)$, and smoothness will hold because our two functions are smooth. Now note that

$$\forall \varphi \in C_c^{\infty}(\mathbb{R}) \qquad \frac{du}{dx}(\varphi) := - u(\varphi') = T(G(\varphi')) = T(\varphi)$$

having noted that $G(\varphi')(x) = \varphi(x)$ because $\int_{\mathbb{R}} \varphi'(x) = \varphi(\infty) - \varphi(-\infty) = 0$

I actually have a follow up question: can we prove uniqueness of $u$ given some condition of $u(\psi_0) = a_0$? If so, how