A Fourier-analytic inequality used by Jean Bourgain

Question

I am currently reading Jean Bourgain's 1986 paper A Szemerédi type theorem for sets of positive density in $R^k$ and would appreciate some help in understanding a Fourier-analytic estimate used in that article. I suspect that my question is relatively elementary, but my knowledge of Fourier analysis is not very strong and there are no experts in the topic at my current place of work, so I would very much appreciate a pointer.

In Bourgain's argument, $f \colon \mathbb{R}^d \to [0,1]$ is a nonzero measurable function supported in a fixed bounded measurable set $A$, and the $L^2$ norm of $f$ is fixed. For each $\lambda>0$ we define $P_\lambda \colon \mathbb{R}^d \to \mathbb{R}$ to be the function whose Fourier transform $\hat{P}_\lambda(\xi):=\int_{\mathbb{R}^d} e^{-2\pi i \langle x,\xi\rangle} P_\lambda(x)dx$ is given by $\hat{P}_\lambda(\xi)=e^{-\lambda\|\xi\|}$ for all $\xi \in \mathbb{R}^d$. Parameters $\delta, t>0$ are introduced, and the parameter $\delta$ is subsequently fixed at some small value which depends on $\|f\|_2$ (and possibly on $A$) but not on the precise choice of $f$. It is then claimed that by taking $t$ small enough, the quantity $$\|(f * P_{\delta t}) - (f * P_{\delta^{-1}t})\|_2$$ can be made arbitrarily small in a manner which is uniform with respect to $f$. It is clear to me that this quantity must converge to zero as $t \to 0$ when $\delta$ and $f$ are fixed, but it is not clear to me why a single value $t$ can be chosen which works simultaneously for all $f$ (where $\|f\|_2$ is fixed and the support of $f$ lies in $A$). Bourgain's paper seems to use a quantitative bound which I infer to resemble $$\|(f * P_{\delta t}) - (f * P_{\delta^{-1}t})\|_2 \leq C\|f\|_2\frac{\log (1/\delta)}{\log (1/t)}.$$ Certainly it is stated that in order to make the above difference small (relative to $\delta^{1/4}$ and $\|f\|_2$) it is sufficient that $\log (1/t)$ should be a large multiple of $\log (1/\delta)$. Can anyone see more precisely what estimate is being used here, or at least how the above quantity can be bounded uniformly with respect to $f$?

Thanks!

Answer 1

I assume you are referring to the argument in page 313 of Jean's paper Link . The point here is that the bound does not hold for all $t$, but for a single $t$ (out of $J$ possible choices $t_1,\dots,t_J$); note that Jean crucially refers in the paper to a "suitable" $t$ rather than an arbitrary $t$. This is a pigeonholing argument, based on the estimation of

$$ \sum_{j=1}^J \| f * P_{\delta t_j} - f * P_{\delta^{-1} t_j} \|_{L^2}^2$$

which can be done by Plancherel's theorem and routine computations (if the $t_j$ are lacunary, as noted in Jean's paper).

The use of pigeonholing to turn qualitative results (such as dominated convergence) to quantitative ones (at the cost of losing some control on the parameter for which the bound is attained) is an important trick in the subject; I discuss it at https://terrytao.wordpress.com/2007/05/23/soft-analysis-hard-analysis-and-the-finite-convergence-principle/ . Another key trick displayed here is to always be aware whether one needs to control the worst-case choice of parameter (i.e. uniform bounds), average-case choice of parameter (e.g. integrated or probabilistic bounds), or best-case choice of parameter (e.g. what comes from the pigeonhole principle). In this case, because one only needs the bound for a single t, best-case analysis suffices, and one can use many more tricks in this setting than in worst-case or average-case analysis.

Incidentally, I found the reading of Jean's papers as a graduate student to be simultaneously extremely frustrating and extremely rewarding. Decoding an offhand remark or a mysterious step in his paper was often as instructive (and as time-consuming) as reading several pages of arguments by some other authors. (But his papers do become much easier to read once one has internalised enough of his "box of tools"...)