It is possible to extend a continuous operator defined in Schwartz space $S$ to $L^p,\ 1\leq p<\infty$?
Example: Let $T$ a bounded operator defined on $L^p,\ 1\leq p<\infty$ invariant under traslation. By a theorem, exists a tempered distribution $u$ such that $T(f)=f*u$ for all $f\in S$ (schwartz space).
Now, as $f$ is bounded (continuous) and S dense in $L^p$ for $1\leq p<\infty$, then $T(f)=f*u$ for all $f\in L^p$?