We know that if we have an operator that is (restricted) weak type $(p,p)$ and (strong) type $(\infty,\infty)$ with norm 1, then it's also of strong type $(q,q)$ for all $p<q<\infty$ by the real interpolation method, and the operator norm there is bounded by $C/(q-p)$.
My question is, is there any counter example in the other direction?
That is, does there exist an operator $T$ bounded from $L^q$ to $L^q$ with norm like $1/(q-p)$ as $q$ goes to $p$, for all $q>p$, but $T$ is not restricted weak type $(p,p)$ at the endpoint?
I want to remark that such an example does exist if we replace the "restricted weak type" by just "weak type", due to the fact that there exist operators that are restricted weak type but not weak type.
Any comments and references are welcome.
