This is a guest post kindly contributed by Noam Lifshitz. Here is a pdf version. This post is a continuation of the post To cheer you up in difficult times 3: A guest post by Noam Lifshitz on the new hypercontractivity inequality of Peter Keevash, Noam Lifshitz, Eoin Long and Dor Minzer, and it gives the proof of the new hypercontractive inequality. We plan a third post where various applications will be mentioned.
Before we get to the post I want to mention that there are a lot of activities on the web. I may devote a special post to links and discussion (and contributing links in the comment section is very welcome.) but meanwhile a few links: 1) Advances in Boolean Function Analysis Lecture Series (thanks to Avishay Tal and Prasad Raghavendra for letting me know); 2) Online course in Foundations of Algebraic Geometry Given by Ravi Vakil from Stanford. You can take the course at varying levels of involvement. (Thanks to Tami Ziegler for telling me) A very very interesting way of online teaching. 3) A site with online mathematical lectures.
Aline Bonami with Szilard Revesz and me (2006). Aline Bonami first proved the 2-point hypercontractive inequality which is very useful in the analysis of Boolean functions. (Leonard Gross proved it independently a few years later and William Beckner found important applications to harmonic analysis.)
Proof of the new hypercontractivity inequality
Our aim is to prove the hypercontractivity theorem for global functions. The proof here is taken from a joint paper with David Ellis and Guy Kindler that’ll soon be out on the Arxiv.
Here we use the notations given in the last blog post. Let us first get a feel for our hypercontractivity theorem by proving the case. Here the RHS is
1. Proof of the case
We will prove the following slightly stronger version of Theorem 1 for the case.
Let Let Then
Proof: Let us write for Rearranging, we have
The noise operator in the case is by definition equal to where is the expectation over operator, and is the identity operator. Hence,
Now when expanding the 4-norm of the function , we obtain
where we used the fact that the expectation of is 0. When looking at the right hand side of the global hypercontractivity theorem, we see most of the above terms except for the one involving the third norm of the Laplacian. Indeed we have
Hence we see that the only term in the left hand side that doesn’t appear with a greater coefficient in the left hand side is the term and by AM-GM we have
which allows us to upper bound the only term appearing in the left hand side but not in the right hand side by corresponding terms that do appear in the right hand side.
2. Tensorisation lemma
Next we are going to prove a theorem that doesn’t seem to fit to our setting, but we’re going to fit it in by force. Let be finite sets. Let us write for the linear space of complex valued functions on . The space can be identified with the space where a pair of function is identified with the function
Given two operators , the operator is the unique operator sending to . We write for The operator can also be defined more explictly in terms of its values on functions. The operator can be understood more explicitly by noting that it is the composition of the operators and Now the operator is given by where
Lemma 3: Let be measure spaces with finite underlying sets. Let be operators satisfying
for all functions
Here the spaces and are equipped with the product measure, where the measure of an atom is the product of the measures of its coordiates.
Proof: For each let be given by As mentioned Hence by hypothesis, we have
We may now repeat the same process on each of the other coordinates to replace the s by s one by one.
3. The main idea: Fourifying the 2-norms.
The strategy of our proof is to take the theorem
which we established in the case for , and to turn it into an essentially equivalent statement about 4-norms. We will then get a tensorised statement for general , which we will be able to convert back into our hypercontractivity theorem for global functions. Our idea is to encode our function as a function satisfying
The benefit of working with rather than is that in one may move between 4-norms and 2-norms by appealing to the hypercontractivity theorem there, which gives
at the cost of some noise.
To define we use Fourier analysis of Abelian groups. Let us briefly recall it. For simplicity let us assume that where is a prime. Let be a th root of unity. For any we have a character given by The are an orthonormal basis of and we write , where Note that is the constant function, and so we have
Our mission will first be to convert the -norm of a function to the norm of a different function.
We define an encoding operator by setting
as the are orthonormal and so are the Moreover, by the Fourier formula for Since -norms are always smaller than 4-norms on probability spaces, we’ve got the following corollary of Proposition 2.
Lemma 4. For all and all we have
We now reach the final little trick. We define a measure space whose underlying set is and where the measure is given by for and for We let be given by on and letting it be on This way Lemma 4 takes the form
4. Tensorised operators
The operator on satisfies where the latter refers to the noise operator on The characters satisfy and so we have the Fourier formula
We also have
This will allow us to conclude that
We will also encounter the operator which by abusing notation we also call encodes
as the function on
Now finally we can get to the understanding of the operator The space is the disjoint union of spaces of the form
By definition of the tensor product, for is the function
5. Finishing the proof
Proof: Lemmas 3 and 4 yield:
for any We now have
The first equality follows from the formula and the fact that commutes with the encoding. The inequality used hypercontractivity on the discrete cube. The last equality follows from the fact that the operator preserves 2-norms.