Dimension-free estimates for low degree functions on the Hamming cube

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F25%3A10510998" target="_blank" >RIV/00216208:11320/25:10510998 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=1rr42urJiJ" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=1rr42urJiJ</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4064/sm240417-27-11" target="_blank" >10.4064/sm240417-27-11</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Dimension-free estimates for low degree functions on the Hamming cube

Popis výsledku v původním jazyce

The main result of this paper are dimension-free Lp inequalities, 1&lt;p&lt;infinity, for low degree scalar-valued functions on the Hamming cube. More precisely, for any p&gt;2, epsilon&gt;0, and theta=theta(epsilon,p)is an element of(0,1) satisfying 1/p=theta/p+epsilon+1-theta/2 we obtain, for any function f:{-1,1}n -&gt; C whose spectrum is bounded from above by d, the Bernstein-Markov type inequalities parallel to Delta(k)f parallel to p &lt;= C(p,epsilon)(k)d(k)parallel to f parallel to(1-theta)(2)parallel to f parallel to(theta)(p+epsilon),k is an element of N. Analogous inequalities are also proved for p is an element of(1,2) with p-epsilon replacing p+epsilon. As a corollary, if f is Boolean-valued or f:{-1,1}n -&gt;{-1,0,1}, we obtain the bounds parallel to Delta(k)f parallel to p &lt;= C(p)(k)d(k)parallel to f parallel to p,k is an element of N. At the endpoint p=infinity we provide counterexamples for which a linear growth in d does not suffice when k=1. We also obtain a counterpart of this result on tail spaces. Namely, for p&gt;2 we prove that any function f:{-1,1}n -&gt; C whose spectrum is bounded from below by d satisfies the following upper bound on the decay of the heat semigroup: parallel to e(-t Delta)f parallel to(p)&lt;= exp(-c(p,epsilon)td)parallel to f parallel to(1-theta)(2)parallel to f parallel to(theta)(p+epsilon),t&gt;0, and an analogous estimate for p is an element of(1,2). The constants c(p,epsilon) and C(p,epsilon) depend only on p and epsilon; crucially, they are independent of the dimension n.

Název v anglickém jazyce

Dimension-free estimates for low degree functions on the Hamming cube

Popis výsledku anglicky

The main result of this paper are dimension-free Lp inequalities, 1&lt;p&lt;infinity, for low degree scalar-valued functions on the Hamming cube. More precisely, for any p&gt;2, epsilon&gt;0, and theta=theta(epsilon,p)is an element of(0,1) satisfying 1/p=theta/p+epsilon+1-theta/2 we obtain, for any function f:{-1,1}n -&gt; C whose spectrum is bounded from above by d, the Bernstein-Markov type inequalities parallel to Delta(k)f parallel to p &lt;= C(p,epsilon)(k)d(k)parallel to f parallel to(1-theta)(2)parallel to f parallel to(theta)(p+epsilon),k is an element of N. Analogous inequalities are also proved for p is an element of(1,2) with p-epsilon replacing p+epsilon. As a corollary, if f is Boolean-valued or f:{-1,1}n -&gt;{-1,0,1}, we obtain the bounds parallel to Delta(k)f parallel to p &lt;= C(p)(k)d(k)parallel to f parallel to p,k is an element of N. At the endpoint p=infinity we provide counterexamples for which a linear growth in d does not suffice when k=1. We also obtain a counterpart of this result on tail spaces. Namely, for p&gt;2 we prove that any function f:{-1,1}n -&gt; C whose spectrum is bounded from below by d satisfies the following upper bound on the decay of the heat semigroup: parallel to e(-t Delta)f parallel to(p)&lt;= exp(-c(p,epsilon)td)parallel to f parallel to(1-theta)(2)parallel to f parallel to(theta)(p+epsilon),t&gt;0, and an analogous estimate for p is an element of(1,2). The constants c(p,epsilon) and C(p,epsilon) depend only on p and epsilon; crucially, they are independent of the dimension n.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2025

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Studia Mathematica

ISSN

0039-3223

e-ISSN

1730-6337

Svazek periodika

280

Číslo periodika v rámci svazku

2

Stát vydavatele periodika

PL - Polská republika

Počet stran výsledku

14

Strana od-do

1-14

Kód UT WoS článku

001418795300001

EID výsledku v databázi Scopus

2-s2.0-105022900608