If we consider $T$ as the convolution of $f \in L^2$ with the indicator function $\mathbb{1}_{[0,1]}$, e.g. $Tf:=f * \mathbb{1}_{[0,1]}$. I want to calculate the Operator Norm of $T$, $||T||$.
Using Young, i get $||f * \mathbb{1}_{[0,1]}||_p \leq ||f|| ||\mathbb{1}_{[0,1]}||_1 = ||f||_p$, thus $||T|| = \sup_{||f||=1} ||Tf|| \leq \sup_{||f||=1} ||f||_p = 1$.
I also think that $||T|| = 1$, but don't know how to show that $||T|| \geq 1$.