Why is $\lvert x:\lvert T(g)(x)\rvert>\alpha \rvert\le\alpha^{-p}\lvert\lvert T(g)\rvert\rvert^p_{L^p}$
Is this a consequence of Chebyshev inequality, Does it not matter how $T$ is defined (here $T$ is an operator given by concolution with some function), Is it always true that $\lvert\lvert T(g)\rvert\rvert=\sup\limits_{\lvert x\rvert=1}\frac{\lvert T(g)(x)\rvert}{\lvert x \rvert}=\sup\limits_{\lvert x\rvert=1}\lvert T(g)(x)\rvert$
or should it be $\lvert g(x)\rvert$ instead of $\lvert x \rvert$ in the denominator ?
What if $\lvert g(x)\rvert=1$ is never reached ? Is it $0$ then ?
I'm not sure what $T$ is from the context of your post, but for a measurable function $f$ we have that $$ \vert \{ \vert f \vert > \alpha\} \vert = \int_{\{\vert f \vert > \alpha\}} 1 \le \int \frac{\vert f\vert^p}{\alpha^p} $$ for all $p < \infty$.