simple exercise sobolev spaces

Question

let B a open Ball in R^n . Let $\psi \in C_{o}^{\infty} (B)$ . Define $\varphi = max ( 0 , \psi ) = {\psi}^{+}$. let $1 < p <q < \infty$ . Consider $s = q/p$ . Then ${\varphi}^{s} \in W^{1,p} (B) $

The affirmation is true or false?

Answer 1

Depends on the interpretation of "$\psi\in C_0^\infty(B)$". If it simply means that $\psi$ is an infinitely differentiable function that is zero on the boundary, then the statement is false. Indeed, $\psi$ can oscillate rapidly at small scale near the boundary, which will make its derivative non-integrable, and consequently the derivatives of $\varphi$ and $\varphi^s$ as well.

But if $\psi\in C_0^\infty(B)$ includes the continuity of derivatives up to the boundary, then the statement is true. Indeed, since the derivatives of $\psi$ are bounded, it is a Lipschitz function. The maximum of Lipschitz functions is also Lipschitz. Finally, raising a bounded Lipschitz function to a power greater than $1$, we get a Lipschitz function again. Thus $\varphi^s\in W^{1,\infty}(B)\subset W^{1,p}(B)$.