For any $s \geq 0$ (integer or non-integer), the Sobolev space $H^s(\mathbb{R}^3)$ is defined as $$H^s(\mathbb{R}^3) := \{u \in L^2(\mathbb{R}^3) : \|u\|_{H^s}^2 < \infty\}$$ where the norm is calculated in Fourier space: $$\|u\|_{H^2}^2 := \int_{\mathbb{R}^3} (1 + |\xi|^{2s})|\hat{u}(\xi)|^2 d\xi.$$
The homogenous Sobolev space $\dot{H}^s(\mathbb{R}^3)$ is defined as: $$\dot{H}^s(\mathbb{R}^3) = \Big\{u \in \mathcal{S}' : \hat{u} \in L^1_{\text{loc}} \quad \text{and} \quad \int_{\mathbb{R}^3}|\xi|^{2s}|\hat{u}(\xi)|^2 d\xi < \infty \Big\}$$ where $\mathcal{S}'$ is the set of tempered distributions (dual of Schwartz space). So in words it seems this is the set of all tempered distributions whose Fourier transforms are $L^1_{\text{loc}}$ functions with finite "norm".
I have two questions about this definition.
- Why are we going from $L^2$ to $S'$ when defining the space itself? Is this to overcome some technical difficulty?
- What is the motivation for such a space and how is it used in practice? As I understand it, the whole point of Sobolev spaces is to have complete spaces containing certain functions and some number of its (weak) derivatives. How does generalizing to distributions help us, especially in the context of PDE?