History of the Hypercontractivity Theorem

Hope you're having a good spring break. I put up the lecture notes for Lecture 16, on the Hypercontractivity Theorem. I find the history of the theorem quite fascinating, so I've put a slightly briefer version below. If you read French, you may also want to look at Djalil Chafai's version. Gross also has a nice survey about the history, but unfortunately I can't find it right now...

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The history of the Hypercontractivity Theorem is quite involved, and
interesting. To understand the history, it's important to first note
that it has a "Gaussian'' version. In this setting it applies to functions $f : \R^n \to \R$, where $\R^n$ is thought of as having the Gaussian distribution and the $T_\rho$ operator is an appropriately generalized linear operator (specifically, the "Ornstein-Uhlenbeck operator" from mathematical physics, which we'll encounter later). Gross observed in 1975 that this Gaussian version follows from our boolean version, by a straightforward argument using the Central Limit Theorem.

An early version of the theorem in the Gaussian setting was proved by Edward Nelson of Princeton in the mid-60's [Nel:66]; I think it was the $q = 4$, $p = 2$ case, possibly with a constant bigger than $1$ on the right side. This was in an important paper on quantum field theory. Several works in subsequent years (e.g., Glimm '68, Federbush '69) improved on the result, and the culmination was the complete proof of the theorem in the Gaussian setting, due to Nelson again [Nel:73]. Nelson's two papers won him the Steele Prize. He is an interesting character, having gone on to work on foundations of mathematics, bounded arithmetic, and automatic proof verification; he is now well-known for having invented Internal Set Theory, a partial axiomatization of Nonstandard Analysis.

In 1973, Leonard Gross proved a limiting version of the theorem called a Logarithmic Sobolev Inequality, and deduced Nelson's Hypercontractive theorem from it [Gro:75]. His proof was in the boolean setting, getting the Gaussian setting via the Central Limit Theorem. However, it turned out that the proof of the Hypercontractivity Theorem (in the boolean setting) was not new; it was first proved by Aline Bonami [Bon:70] in 1979. In fact, Bonami had proved the $q = 4$, $p = 2$ case in the boolean setting even earlier [Bon:68].

The history of the theorem can be traced even further back; Bonami's work was informed by that of Walter Rudin, who proved [Rud:60] similar inequalities in the setting of $\Z_n$ rather than $\{-1,1\}^n$ (``one of my favorite papers'' -- Rudin). Further, a version of the log-sobolev inequality in the Euclidean (rather than Gaussian) setting was proved by A. J. Stam [Sta:59] in 1959, in work on Fisher/Shannon information theory -- much closer to the world of computer science!

Finally, the Hypercontractivity Theorem was introduced to the world of theoretical computer science in the groundbreaking work of Kahn, Kalai, and Linial [KKL:88]. Unfortunately, they attributed the theorem to William Beckner [Bec:75], which I'm not sure is exactly an accurate accreditation. Beckner's work was in fact important followup work on the work of Nelson and Gross, making extensions to Euclidean and complex settings.

In the computer science theory literature, the Hypercontractivity Theorem is often called "Beckner's Theorem". Lately there has been a move towards "Bonami-Beckner Theorem", although "Bonami Theorem" would respect the original discoverer and "Bonami-Gross Theorem" might more properly respect the independent discovery. To sidestep the issue, we will simply call it the Hypercontractivity Theorem.