$(a)$ Suppose $T:L^p \to L^p$ is a linear operator(might not be bounded) , $A$ a dense subset of $L^p$ , $T|_A $ is a bounded operator , then $T|_A$ can be extend to all of $L^p$ which we call it $T'$ . Can we show that $T=T'$ on $L^p$ ?
$(b)$ If we only assume $T$ is weak $(p,p)$ and sublinear , can we show that $T'=T$
The above question frequently appear when I apply Marcinkiewicz Interpolation theorem , when I show that an operator $T$ is weak $(p,p)$ and weak $(q,q)$ on schwartz space , then $T|_S$ can be extend to $T'$ which is strong $(r,r)$ when $p\lt r \lt q$ . Then I need to show $T'=T$ by using the original definition of the operater $T$ . However , can we only use the property that $T$ is sublinear to show $T=T'$ ?