Basic Query On Sobolev Space

Question

Okay, I am very new to the premise of Sobolev Spaces and there is one exercise here that's really grinding my gears. The premise is this- to what Sobolev spaces for each real number $\alpha$ does $|x|^{\alpha}e^{-x^{2}}$ belong? I've tried heading down the weak derivative route but it seems to me that the strong derivative is far and away easier to compute. The exponential factor eats away everything at either extreme (+- infinity). Best answer gets a hug or equivalent.

Answer 1

I'm assuming you're starting with the one-dimensional case, which is what I'll address here. The one-dimensional case is a little bit special, because some of your intuition about what a weak derivative 'should be' breaks down. For instance, a step function has a jump discontinuity and does not have a weak derivative at that point. On the other hand, something like $|x|$ has a weak derivative.

The second thing to note is that if a classical derivative exists, then the weak derivative exists and the two must be equal. So go ahead and compute stuff all you like -- that's good.

Now what do you have to worry about? If $\alpha \in \mathbb{N}$, you are good -- forever. This guy is in $W^{p,\infty}$ for all $p \in [1,\infty]$.

If $\alpha \not\in \mathbb{N}$, what happens? I'll give you some thinking points:

• For which $\alpha > 0$ is this guy in $L^p$?
• You don't have to worry about cusps, but what about jump discontinuities?
• What happens when $\alpha < 0$? Specifically, what happens at 0?

It's a really good idea to compute the derivatives and plot functions, so I'll let you try before giving you an answer. Remember, all you really need is to satisfy the integration by parts and be in $L^p$. There are two good examples at the beginning of Chapter 5 in Evans which explain how to get a contradiction to integration by parts if there is a jump discontinuity, and you basically want to apply this reasoning for situations where $\alpha < 0$. [Hint: take a sequence of functions in $C^\infty$ which approach 0 everywhere except...]