Sobolev space of a function

Question

Let $f(x) = √x$ denote the square root function. For what m ∈ {1, 2, 3 . . . } and $p ∈ [1,∞)$ values is it true that $$f ∈ W^{m,p}(1,∞)?$$

I just tried like this by using integration by parts: $\int_{1}^\infty\varphi'(x)\sqrt{x}\,dx = [\varphi(x)\sqrt x]_1^\infty - \int_1^\infty \varphi(x)\cdot\frac 1 {2\sqrt x}\,dx = -\int_1^\infty \varphi(x)\cdot\frac 1 {2\sqrt x}\,dx.$

So, the weak derivative of $√x$ is $1/(2\sqrt x)$, which is an element of $L^p(1,\infty)$ and $f$ is differentiable for all orders of m. What's wrong here?

Thanks!

Answer 1

$f$ is not in $ L^p(1,\infty)$, so it cannot be in $W^{m,p}(1,\infty)$