Zero extension of W01 ($W_0^{1,p}(U)\hookrightarrow W^{1,p}({\mathbb{R}}^n)$)

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.