Upper bound for the operator norm
I know that the operator and its adjoint have the same norm but what if you have to find an upper bound for the adjoint, i.e. say $T$ sends $L^p$ to $L^p$ and the following inequality holds; $$\lvert\lvert T\rvert\rvert\le M\cdot f(p)$$ for some function $f$ depending on $p$, Why is then the following true ?
$$\lvert\lvert T^*\rvert\rvert\le M\cdot f(p')$$
where $p'$ is the conjugate of $p$
Just replacing $p$ by $p'$ doesn't convince me, it needs a justification.
In the below text the author states that $T^*$ maps $L^{p'}$ to $L^{p'}$ with bound at most $C_n'(A_2+B)(p-1)^{-\frac 1p}$ then by duality he deduces that $T$ sends with bound at most $C_n'(A_2+B)(p-1)^{1-\frac 1p}$ as I said above he replaces $p$ by $p'$ without any reasoning, can you maybe explain this step ?
