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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

This is a guest post kindly contributed by Noam Lifshitz.

My short introduction: There is nothing like a new hypercontractivity inequality to cheer you up in difficult times and this post describes an amazing new hypercontractivity inequality.  The post describes a recent hypercontractive inequality by Peter Keevash, Noam Lifshitz, Eoin Long and Dor Minzer (KLLM) from their paper: Hypercontractivity for global functions and sharp thresholds. (We reported on this development in this post. By now, there are quite a few important applications.) And for Talagrand’s generic chaining inequality, see this beautiful blog post by Luca Trevisan.

Image may be NSFW.
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Barry Simon coined the term “hypercontractivity” in the 70s.  (We asked about it here and Nick Read was the first to answer.) A few months ago Barry told us about the early history of hypercontractivity inequalities, and, in particular, the very entertaining story on William Beckner’s Ph. D. qualifying exam.

And now to Noam Lifshitz’s guest post.

Hypercontractivity on product spaces

Analysis of Boolean functions (ABS) is a very rich subject. There are many works whose concern is generalising some of the results on analysis of Boolean functions to other (product) settings, such as functions on the multicube Image may be NSFW.
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{\left[m\right]^{n},}
where Image may be NSFW.
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{m}
is very large. However, in some of these cases the fundemental tools of AOBF seem to be false for functions on the multicube Image may be NSFW.
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{f\colon\left[m\right]^{n}\rightarrow\mathbb{R}.}
However, in the recent work of Keevash, Long, Minzer, and I. We introduce the notion of global functions. These are functions that do not strongly depend on a small set of coordinates. We then show that most of the rich classical theory of AOBF can in fact be generalised to these global functions. Using our machinery we were able to strengthen an isoperimetric stability result of Bourgain, and to make progress on some Erdos-Ko-Rado type open problem.

We now discuss some background on the Fourier analysis on functions on the multicube Image may be NSFW.
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{f\colon\left\{ 1,\ldots,m\right\} ^{n}\rightarrow\mathbb{R}.}

Derivatives and Laplacians

There are two fundemental types of operators on Boolean functions Image may be NSFW.
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{f\colon\left\{ 0,1\right\} ^{n}\rightarrow\mathbb{R}.}
The first ones are the discrete derivatives, defined by Image may be NSFW.
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{D_{i}[f]=\frac{f_{i\rightarrow1}-f_{i\rightarrow0}}{2},}
where Image may be NSFW.
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{f_{i\rightarrow x}}
denotes the we plug in the value Image may be NSFW.
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{x}
for the Image may be NSFW.
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{i}
th coordinate. The other closely related ones are the laplacians defined by Image may be NSFW.
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{L_{i}f\left(x\right):=f\left(x\right)-\mathbb{E}f\left(\mathbf{y}\right),}
where Image may be NSFW.
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{\mathbf{y}}
is obtained from Image may be NSFW.
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{x}
by resampling its Image may be NSFW.
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{i}
th coordinate.

The laplacians and the derivatives are closely related. In fact, when we plug in Image may be NSFW.
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{1}
in the Image may be NSFW.
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{i}
th coordinate, we obtain Image may be NSFW.
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{L_{i}[f]_{i\rightarrow1}=D_{i}[f]}
, and when we plug in Image may be NSFW.
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{0}
in it, we obtain Image may be NSFW.
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{L_{i}[f]_{i\rightarrow0}=-D_{i}[f].}

The 2-norm of the Image may be NSFW.
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{i}
th derivative is called the Image may be NSFW.
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{i}
th influence of Image may be NSFW.
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{f}
as it measures the impact of the Image may be NSFW.
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{i}
th coordinate on the value of Image may be NSFW.
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{f}
. It’s usually denoted by Image may be NSFW.
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{\mathrm{Inf}_{i}[f]}
.

Generalisation to functions on the multicube

For functions on the multicube we don’t have a very good notion of a discrete derivative, but it turns out that it will be enough to talk about the laplacians and their restrictions. The Laplacians are again defined via Image may be NSFW.
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{L_{i}f\left(x\right):=f\left(x\right)-\mathbb{E}f\left(\mathbf{y}\right),}
where Image may be NSFW.
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{\mathbf{y}}
is obtained from Image may be NSFW.
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{x}
by resampling its Image may be NSFW.
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{i}
th coordinate. It turns out that in the continuous cube it’s not enough to talk about Laplacians of coordinate, and we will also have to concern ourselves with Laplacians of sets. We define the generalised Laplacians of a set Image may be NSFW.
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{S}
by composing the laplacians corresponding to each coordinate in Image may be NSFW.
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{S}
Image may be NSFW.
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{L_{\left\{ i_{1},i_{2},\ldots,i_{r}\right\} }\left[f\right]:=L_{i_{1}}\circ\cdots\circ L_{i_{r}}\left[f\right].}

We now need to convince ourselves that these laplacians have something to do with the impact of Image may be NSFW.
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{S}
on the outcome of Image may be NSFW.
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{f.}
In fact, the following notions are equivalent

  1. For each Image may be NSFW.
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    {x,y\in\left[m\right]^{S}}
    we have Image may be NSFW.
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    {\|f_{S\rightarrow x}-f_{S\rightarrow y}\|_{2}<\delta_{1}}
  2. For each Image may be NSFW.
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    {x\in\left[m\right]^{S}}
    we have Image may be NSFW.
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    {\|L_{S}[f]_{S\rightarrow x}\|_{2}<\delta_{2},}

in the sense that if (1) holds then (2) holds with Image may be NSFW.
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{\delta_{2}=C^{\left|S\right|}\delta_{1}}
and conversely if (2) holds, then (1) holds with Image may be NSFW.
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{\delta_{1}=C^{\left|S\right|}\delta_{2}.}

The main theme of our work is that one can understand global function on the continuous cube, and these are functions that satisfy the above equivalent notions for all small sets Image may be NSFW.
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{S}
.

Noise operator, hypercontractivity, and small set expansion

For Image may be NSFW.
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{\rho\in\left(0,1\right),}
the noise operator is given by Image may be NSFW.
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{\mathrm{T}_{\rho}\left[f\right]\left(x\right)=\mathbb{E}f\left(\mathbf{y}\right)}
when Image may be NSFW.
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{\mathbf{y}}
is obtained from Image may be NSFW.
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{x}
by independently setting each coordinate Image may be NSFW.
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{\mathbf{y}_{i}}
to be Image may be NSFW.
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{\mathbf{x}_{i}}
with probability Image may be NSFW.
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{\rho}
and resampling it with uniformly out of Image may be NSFW.
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{\left\{ -1,1\right\} }
otherwise. The process which given Image may be NSFW.
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{x}
outputs Image may be NSFW.
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{\mathbf{y}}
is called the Image may be NSFW.
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{\rho}
-noisy process, and we write Image may be NSFW.
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{\mathbf{y}\sim N_{\rho}\left(x\right).}

The Bonami hypercontractivity theorem, which was then generalised by Gross and Beckner states that the noise operator Image may be NSFW.
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{T_{\frac{1}{\sqrt{3}}}}
is a contraction from Image may be NSFW.
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{L^{2}\left(\left\{ 0,1\right\} ^{n}\right)}
to Image may be NSFW.
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{L^{4}\left(\left\{ 0,1\right\} ^{n}\right),}
i.e.

Image may be NSFW.
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\displaystyle \|\mathrm{T}_{\frac{1}{\sqrt{3}}}f\|_{4}\le\|f\|_{2}

for any function Image may be NSFW.
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{f.}

One consequence of the hypercontractivity theorem is the small set expansion theorem of KKL. It concerns fixed Image may be NSFW.
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{\rho\in\left(0,1\right)}
and a sequence of sets Image may be NSFW.
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{A_{n}\subseteq\{0,1\}^{n}}
with Image may be NSFW.
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{\left|A_{n}\right|=o\left(2^{n}\right).}
The small set expansion theorem states that if we choose Image may be NSFW.
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{\mathbf{x}\sim A_{n}}
uniformly and a noisy Image may be NSFW.
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{\mathbf{y}\sim N_{\rho}\left(\mathbf{x}\right),}
then Image may be NSFW.
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{\mathbf{y}}
will reside outside of Image may be NSFW.
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{A_{n}}
almost surely.

The Generalisation to the multicube:

The small set expansion theorem and the hypercontractivity theorem both fail for function on the multicube that are of a very local nature. For instance, let Image may be NSFW.
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{A}
be the set of all Image may be NSFW.
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{x\in\left\{ 1,\ldots,m\right\} ^{n},}
such that Image may be NSFW.
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{x_{1}}
is Image may be NSFW.
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{m.}
Then Image may be NSFW.
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{A}
is of size Image may be NSFW.
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{m^{n-1},}
which is Image may be NSFW.
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{o\left(m^{n}\right)}
if we allow Image may be NSFW.
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{m}
to be a growing function of Image may be NSFW.
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{n}
. However, the Image may be NSFW.
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{\rho}
-noisy process from the set stays within the set with probability Image may be NSFW.
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{\rho.}
For a similar reason the hypercontractivity theorem fails as is for functions on Image may be NSFW.
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{\left\{ 1,\ldots,m\right\} ^{n}.}
However we were able to generalise the hypercontractivity theorem by taking the globalness of Image may be NSFW.
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{f}
into consideration.

Our main hypercontractive inequality is the following

Theorem 1.

Image may be NSFW.
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\displaystyle \|\mathrm{T}_{\frac{1}{100}}f\|_{4}^{4}\le\sum_{S\subseteq\left[n\right]}\mathbb{E}_{\mathbf{x}\sim\left\{ 1,\ldots,m\right\} ^{m}}\left(\|L_{S}\left[f\right]_{S\rightarrow\mathbf{x}}\|_{2}^{4}\right).

The terms Image may be NSFW.
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{\|L_{S}\left[f\right]_{S\rightarrow x}\|_{2}}
appearing on the right hand side are small whenever Image may be NSFW.
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{f}
has a small dependency on Image may be NSFW.
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{S}
and it turns out that you have the following corrolary of it, which looks a bit more similar to the hypercontractive intequality.

 

Corollary 2.

Let Image may be NSFW.
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{f\colon\left\{ 1,\ldots,m\right\} ^{n}\rightarrow\mathbb{R}}
, and uppose that Image may be NSFW.
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{\|L_{S}[f]_{S\rightarrow x}\|_{2}\le4^{\left|S\right|}\|f\|_{2}}
for all sets Image may be NSFW.
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{S.}

Then Image may be NSFW.
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{\mathrm{\|\mathrm{T}_{\frac{1}{1000}}f\|_{4}\le\|f\|_{2}.}}

Finally, one might ask wonder why this globalness notion appears only when we look at large values of Image may be NSFW.
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{m}
and not when Image may be NSFW.
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{m=2.}
I think the corollary is a good explanation for that as Image may be NSFW.
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{\|f\|_{2}^{2}\ge\left(\frac{1}{2}\right)^{\left|S\right|}\|f_{S\rightarrow x}\|_{2}^{2}}
holds trivially for any Boolean function Image may be NSFW.
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{f\colon\left\{ 0,1\right\} ^{n}\rightarrow\mathbb{R}.}


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