The Fourier transform of a function supported on $B_1$ is essentially constant on $B_1$?

Question

I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a little to make it more transparent.

Let $f_1, \dots, f_n$ be $n$ functions whose Fourier transform is supported on $B(1)$. Bourgain claimed that by the heuristics he and Guth used in their paper we can think that $f_i$ is "essentially constant" on $B_1$. This seems to allow the following "reverse Holder" inequality

$$\prod_{i=1}^n \|f_i\|_{L^p(B_1)}^{1/n}\ll_{n,p} \|\prod_{i=1}^n f_i^{1/n}\|_{L^p(B_1)}. $$

But I skimmed that paper and only found an estimate morally like the following:

$$ f(x)=f*\eta(x)\le \|f\|_{L^1}\|\eta\|_{L^\infty}, $$

where $\hat\eta$ is compactly supported and equals 1 on $B_1$. How do I deduce the reverse Holder inequality from that?

Added: Just for reference, the inequality (43) that I referred to is

$$ \prod_{i=1}^n \|\widehat{g_id\sigma}\|_{p,\delta,B_{\delta^{-1/2}}}^{1/n}\ll_{n,p} \|\prod_{i=1}^n (\sum_{\theta\subset\tau_i:\delta^{1/2}\text{ cap}} |\widehat{g_{i,\theta}d\sigma}|^2)^{1/2n}\|_{L^p(w_{B_{\delta^{-1/2}}})}. $$

In the above, the $f_i$ are supposed to be $\widehat{g_{i,\theta}d\sigma}$, and I'm assuming $\tau_i=\text{supp }g_i$ contains only one $\delta^{1/2}$ cap, which is a slab of size $\delta^{1/2}\times\cdots\times\delta^{1/2}\times\delta$. $\tau_i$ are assumed to be $\nu-$transverse, but I feel this is unimportant in this inequality. $w_{B_{\delta^{-1/2}}}$ is just a smooth cutoff weight that equals 1 on $B_{\delta^{-1/2}}$. I have normalized $\delta$ to 1 in the above.

Answer 1

I checked with Jean and Ciprian about this, and there is indeed a small issue here; the bound indicated is "morally" correct, and may possibly even be true (using the weights $w_B$ rather than a sharp truncation $1_B$), but it does not quite follow from the standard device of representing a Fourier-localised function by a convolution with a Schwartz function. In later papers, most notably in the Vinogradov conjecture paper with Guth, the issue was addressed by inserting an additional $L^p$ norm at small scales, replacing expressions such as $\| \widehat{g d\sigma} \|_{p,\delta,\Delta}$ with somewhat more complicated looking expressions $A_p(u,B^n,u)$. From what I understand, it is possible to adapt the arguments in these later papers (which are more streamlined than the original arguments in some respects) to reprove the hypersurface decoupling estimate rigorously. The authors plan to update the arXiv version of the original decoupling paper with some additional remarks addressing these issues.