Extension By zero maps functions in $L^P(\Omega)$ into functions in $L^P(\mathbb{R^N})$

Question

Extension By zero maps functions in $L^P(\Omega)$ into functions in $L^P(\mathbb{R^N})$

let $u:\Omega \to \mathbb{R}$ Define $\tilde{u}:\mathbb{R^n}\to \mathbb{R}$ by zero zero outside $\Omega$

$$\tilde{u}=\left\{\begin{matrix} u(x) & if \; x\in \Omega \\ 0& if \;x \notin \Omega \end{matrix}\right.$$

how to prove this Extension By zero maps functions in $L^P(\Omega)$ into functions in $L^P(\mathbb{R^N})$

Answer 1

All you have to show is that $\bar{u}$ satisfies

$$\int_{\Bbb R^n} |\bar{u}(x)|^p\,dV <\infty.$$

Split the integral into two (fairly obvious) pieces to prove the result.