Sharp hypercontractivity for global functions

Abstract

For a function $f:\{0,1{\}}^n\to \mathbb{R}$ with Fourier expansion $f={\sum }_{S\subset \{1,\dots ,n\}}\hat{f}(S){\chi }_S$, the hypercontractive inequality for the noise operator allows bounding norms of $T_{\rho }f={\sum }_S{\rho }^{|S|}\hat{f}(S){\chi }_S$ in terms of norms of $f$. If $f$ is Boolean-valued, the level-$d$ inequality allows bounding the norm of $f^{=d}={\sum }_{|S|=d}\hat{f}(S){\chi }_S$ in terms of $\mathbb{E}[f]$. These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube $\{0,1{\}}^n$ is endowed with the uniform measure, it is easy to show that they do not hold for more general discrete product spaces, and finding a ‘natural’ generalization was a long-standing open problem. Keevash, Lifshitz, Long, and Minzer [J. Amer. Math. Soc. 37, 245–279 (2024)] obtained a hypercontractive inequality for general discrete product spaces, that holds for functions which are ‘global’ – namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group $S_n$ and to other settings where sharpness of the bound is crucial. Also, no sharp level-$d$ inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level-$d$ inequality for global functions over discrete product spaces. Our inequalities open the way for diverse applications to extremal set theory, group theory, theoretical computer science, and number theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications that were obtained using our results. Those contain applications to the study of functions over the symmetric group $S_n$ – including hypercontractivity and level-$d$ inequalities, character bounds, variants of Roth’s theorem and of Bogolyubov’s lemma, and diameter bounds, as well as an application to the Furstenberg–Sárközy problem on the maximal size of a subset of $\{1,\dots ,n\}$ which does not contain two elements that differ by a perfect square.