Recently I've been working on a project for which I've needed to reference Iwaniec's paper Fourier coefficients of cusp forms and the Riemann zeta function, where the short fourth moment estimate $$ \tag{1} \int\limits_{T}^{T+T^{2/3}} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^4 dt\ll T^{2/3+\epsilon} $$ is proved. It is claimed that this implies Heath-Brown's classic twelfth moment estimate $$ \tag{2} \int\limits_{T}^{2T} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^{12} dt\ll T^{2+\epsilon}, $$ but no details are given. I have worked for quite some time, but cannot seem to work out how this computation goes. Thus my question:
How does one deduce the estimate (2) from (1)?
Here are some thoughts/ideas I've had:
It seems that inevitably, one has to apply the Weyl subconvexity estimate $$ \zeta\left(\textstyle{\frac{1}{2}}+it\right) \ll t^{1/6+\epsilon}, $$ however no matter how I try to do this, I can't seem to push this through (I'm aware that this is not the strongest subconvexity, but if I need to apply a subconvexity estimate, it should certainly be sufficient).
It suffices to bound the integral $$ \int\limits_{\substack{T\\ \left|\zeta(\textstyle{\frac{1}{2}}+it) \right| > T^{1/8}}}^{2T} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^{12} dt, $$ since the remaining integral can be bounded using the fourth moment estimate: $$ \int\limits_{\substack{T\\ \left|\zeta(\textstyle{\frac{1}{2}}+it) \right| \leq T^{1/8}}}^{2T} \left| \zeta\left(\textstyle{\frac{1}{2}}+it\right)\right|^{12} dt\leq T \int\limits_{T}^{2T} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^4 dt\ll T^{2+\epsilon}. $$
One can estimate the number of large values of $\zeta$ on short intervals via $$ \left|\left\{ t\in[T,T+T^{2/3}] : \left|\zeta(\textstyle{\frac{1}{2}}+it) \right| > T^{1/8}\right\} \right| \leq \int\limits_{T}^{T+T^{2/3}} \left(\frac{\left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|}{T^{1/8}} \right)^4 dt\ll T^{1/6+\epsilon}. $$
It seems that one needs some sort of clever decomposition of the interval $[T,2T]$ into intervals of size $T^{2/3}$, perhaps by distinguishing those intervals on which the twelfth moment is large, i.e. intervals $[T_0,T_0+T^{2/3}]$ such that $$ \int\limits_{T_0}^{T_0+T^{2/3}} \left| \zeta\left({\textstyle\frac{1}{2}}+it\right)\right|^{12} > W $$ for some parameter $W > 0$ to be optimized.
In the language of $L_p$ norms, in general, if $p < q$, then one cannot hope to control an $L_q$-norm by an $L_p$-norm (and the moments are essentially the $L_4$ and $L_{12}$ norms). However, if one understands the $L_p$ on a "local" level (i.e. on a short interval), then the $L_q$-norm should be able to be controlled by the $L_p$-norm in some way.
Any ideas or help is most appreciated.
My answer to your question is that what you are conjecturing is false. (1) does NOT imply (2). You said that Iwaniec claimed it does... but if you look back at his paper carefully you will see that he did not make this claim. Reason why your conjecture is false: Suppose $|\zeta(1/2+it)|=T^{1/6}$ for $|t-(T+kT^{2/3})|<T^{\epsilon}$ for $0\le k\le T^{1/3}$ and $|\zeta(1/2+it)|=0$ elsewhere. This is a possible scenario under (1), but it contradicts (2).