Dimension-free estimates for low degree functions on the Hamming cube
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Dimension-free estimates for low degree functions on the Hamming cube
Original language description
The main result of this paper are dimension-free Lp inequalities, 1<p<infinity, for low degree scalar-valued functions on the Hamming cube. More precisely, for any p>2, epsilon>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 -> C whose spectrum is bounded from above by d, the Bernstein-Markov type inequalities parallel to Delta(k)f parallel to p <= 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 ->{-1,0,1}, we obtain the bounds parallel to Delta(k)f parallel to p <= 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>2 we prove that any function f:{-1,1}n -> 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)<= exp(-c(p,epsilon)td)parallel to f parallel to(1-theta)(2)parallel to f parallel to(theta)(p+epsilon),t>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.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
—
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace
Others
Publication year
2025
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Studia Mathematica
ISSN
0039-3223
e-ISSN
1730-6337
Volume of the periodical
280
Issue of the periodical within the volume
2
Country of publishing house
PL - POLAND
Number of pages
14
Pages from-to
1-14
UT code for WoS article
001418795300001
EID of the result in the Scopus database
2-s2.0-105022900608