Which functions lies in $H^{loc}_{s}\setminus H_{s}$?

Question

We put $H^{s}=$The Sobolev spaces, and $H^{loc}_{s}=$The localized Sobolev spaces.

We note that, $H_{s}\subset H^{loc}_{s};$ also this. Bit roughly speaking, I am interested in knowing that how big $H^{loc}_{s}$ can be than $H_{s}.$

My Questions are:

(1) How we find few examples of functions $f$ so that $f\in H^{loc}$ but $f\notin H_{s}$ ?

(2) Can we think of some well-known function space say $E,$ so that $E\subset H^{loc}\setminus H_{s}$? What is $H^{loc}_{s}\setminus H_{s}$(= The space of functions which belongs to localized Sobolev space $H^{loc}_{s}$, but not in Sobolev space $H_{s}$ ) intuitively ? Furthermore, Can we characterize the set $H^{loc}_{s}\setminus H_{s}$ ?

Thanks,

Answer 1

1. The only difference between $H^s_{\rm loc}(\mathbb R^n)$ and $H^s(\mathbb R^n)$ is the decay at infinity (of the function itself, and appropriate derivatives). So, if you want functions to not belong to $H^s(\mathbb R^n)$, make them not decay at infinity. Adding a nonzero constant will do the trick. Or add a polynomial, etc.

2. Yes: the space of smooth periodic functions, $C^k_{\rm per}(\mathbb R)$ [with some period $L$], and its analogues in higher dimensions. If $k\ge s$, these are in $H^s_{\rm loc}$, but do not contain any elements of $H^s$ other than the zero function.

Can we characterize the set...

Sure: a function belongs to this set if it belongs to $H^s_{\rm loc}(\mathbb R^n)$ but does not belong to $H^s_{ }(\mathbb R^n)$. What simpler characterization could there be? It says exactly what is in the space.

intuitively

People use the word intuition for the most strange things