Let $S$ denote Schwartz space , $T:L^p \to L^p$ be a linear operator . If the restriction of $T$ on $S$ is bounded , can we show that $T$ is bounded?

Question

Let $S$ denote Schwartz space $1 \le p \lt \infty$, can we construst a linear (sublinear) operator $T:L^p \to L^p$ which is unbounded. However , the restriction $T|_S$ is bounded on $S$ ?

Definition: If $|T(f+g)(x)| \le |T(f)(x)|+|T(g)(x)|$ for $a.e.\,\,\,x$ . We say that $T$ is sublinear .

By extension theorem , every bounded operator on $S$ can be extended to a bounded operator on $L^p$. However, if the operator $T$ on $L^p$ is given , and we know that it is bounded on a dense subset , then can we show $T$ is bounded ?