Let $T$ be a linear mapping from $L^{p}\cap L^{q}$ into itself and $1\leq p<q<\infty$. Suppose $T$ is of weak-type $(p,p)$ and weak-type $(q,q)$ on functions $f$ in $L^{p}\cap L^{q}$, that is, $\omega_{Tf}(t)\leq t^{-p}C_{p}\|f\|_{p}^{p}$ and $\omega_{Tf}(t)\leq t^{-q}C_{q}\|f\|_{q}^{q}$ for all $f\in L^{p}\cap L^{q}$, where $C_{p}, C_{q}$ are constants and $\omega_{Tf}(t)$ is the distribution function of $Tf$.
My question is: Is there a way to extend $T$ to $L^{r}$ into itself, where $p<r<q$, such that $T$ is of weak-type $(p,p)$ and weak-type $(q,q)$ on $L^{r}$?
(That is, there are constants $T_{p}, T_{q}$ such that $\omega_{Tf}(t)\leq t^{-p}T_{p}\|f\|_{p}^{p}$ for all $f\in L^{r}\cap L^{p}$ and $\omega_{Tf}(t)\leq t^{-q}T_{q}\|f\|_{q}^{q}$ for all $f\in L^{r}\cap L^{q}$)
Any help is appreciated.