Zero extension of W01
= Zero extension of W01
{title2=$W_0^{1,p}(U)\hookrightarrow W^{1,p}(\mathbb R^n)$}
By the definition of $W_0^{1,p}(U)$ as the closure of $C_c^\infty(U)$, extending a function by zero outside $U$ gives a bounded map into $W^{1,p}(\mathbb R^n)$ and does not create a boundary distribution.