higher weak derivatives of the hat function

Question

The hat function is defined as: $$ \phi(x) = (1-|x|)_{+} $$ where $(y)_{+} = \max\{y, 0\}$. I know that the hat function is a member of the Sobolev space $W^{1,p}(-1, 1)$ for $p \in [1, \infty]$. The question is: $\phi(x) \in W^{k, p}(-1, 1)$? for $k > 1$ and $ p \in [1, \infty]$.

Answer 1

The key word is Sobolev Embedding. In fact in dimension $ 1 $, it implies that if $$ k-\frac{1}{p} > 1 $$ then $ W^{k,p}(-1,1) \subset C^1 $.

The answer to your question is than $ \phi $ is in $ W^{k,p} $ if
$$ k-\frac{1}{p} < 1. $$ If $ k \in \mathbb{N} $, we proved for $ k \in \mathbb{R} $ you need an extra argument.