Talagrand Inequality at Second Order and Application to Boolean Analysis

Abstract

This note is concerned with an extension, at second order, of an inequality on the discrete cube $C_n=\{-\,1,1\}^n$ (equipped with the uniform measure) due to Talagrand (Ann Probab 22:1576–1587, 1994). As an application, the main result of this note is a theorem in the spirit of a famous result from Kahn et al. (cf. Proceedings of 29th Annual Symposium on Foundations of Computer Science, vol 62. Computer Society Press, Washington, pp 68–80, 1988) concerning the influence of Boolean functions. The notion of the influence of a couple of coordinates $(i,j)\in \{1,\ldots ,n\}^2$ is introduced in Sect. 2, and the following alternative is obtained: For any Boolean function $f\,:\, C_n\rightarrow \{0,1\}$, either there exists a coordinate with influence at least of order $(1/n)^{1/(1+\eta )}$, with $\, 0<\eta <1$ (independent of f and n), or there exists a couple of coordinates $(i,j)\in \{1,\ldots ,n\}^2$ with $i\ne j$, with influence at least of order $(\log n/n)^2$. In Sect. 4, it is shown that this extension of Talagrand inequality can also be obtained, with minor modifications, for the standard Gaussian measure $\gamma _n$ on ${\mathbb {R}}^n$; the obtained inequality can be of independent interest. The arguments rely on interpolation methods by semigroup together with hypercontractive estimates. At the end of the article, some related open questions are presented.