What is the difference between $H^1_{loc}$ and $H^1$?

I have started studying Sobolev spaces and I came across a space referred to as $H^1_{loc}$. I am not sure what the $loc$ subscript infers? What is it that makes this space different from $H^1$? Why would you every specify this space instead of $H^1$?

• Usually loc means local. So I guess $H^1_{loc}$ would mean the set of functions that are locally $L^2$ and have weak derivatives that are locally $L^2$ too. Being locally $L^2$ means that $\int_K |f|^2 < \infty$ for compact sets $K$. – desos Mar 18 '16 at 9:42
• Where did you find such notation? Can you give us the source material? – wroobell Mar 18 '16 at 10:34
• @jooi What is the significance of the functions being integrable on compact sets? – Riggs Mar 18 '16 at 10:42
• @jooi: If you're working in a domain in $\Bbb R^n$, then it just means that around every point $x$ in the domain, there's a neighborhood on which the function is $H^1$. It's just about demanding sufficient integrability/differentiability hypotheses on your function locally without making such demands globally (so, eg, the integral of your function over the whole domain needn't be finite). – user98602 Mar 18 '16 at 14:48
• Sorry, that ping was supposed to be @Riggs. – user98602 Mar 18 '16 at 17:03

For J.L. Lions and others, $H^m_{loc}(\Omega)$ spaces are made of functions of $H^m(\Theta)$ where $\Theta$ represents any open subset such that $\bar{\Theta}\subset \Omega$. This allows to define a family of seminorms, the norms in $H^m(\Theta)$, and in the end these spaces are Frechet spaces.