Strong type inequality $(p,q)$ implies weak type inequality $(p,q)$?

Question

Recall that a weak type $(p,q)$ is when you have a $C>0$ such that $\mathrm{measure}(|T_f(x)|> \alpha)\leq \frac{c ||f||^q_{L^p}}{\alpha^q}$.

Also, we have a strong type $(p,q)$ if $\Vert T_f\Vert_{L^q} \leq c \Vert f\Vert_{L^p}$

I have seen here, in the comment section,that a strong type inequality $(p,p)$ implies a weak type inequality $(p,p)$. Why is this the case? Does it also hold for $(p,q)$?

Answer 1

\begin{align} \mathrm{measure}(|T_f(x)|> \alpha)&=\int_{|T_f(x)|> \alpha}1\\ &\le\int_{|T_f(x)|> \alpha}\left(\frac{|T_f(x)|}{\alpha}\right)^q\\ &\le \frac{1}{\alpha^q}\int|T_f(x)|^q\\ &\le \frac{1}{\alpha^q}\,c\,\|f\|_{L^p}^q. \end{align}