Document Type

Article

Publication Date

1-31-2025

Abstract

The main result of this paper are dimension-free Lp inequalities, 12, ε>0, and θ=θ(ε,p)∈(0,1) satisfying 1/p=θ/(p+ε)+(1−θ)2

we obtain, for any function f:{−1,1}n→C whose spectrum is bounded from above by d, the Bernstein–Markov type inequalities ∥Δkf∥p≤C(p,ε)kdk∥f∥1−θ2∥f∥θp+ε,k∈N.

Analogous inequalities are also proved for p∈(1,2) with p−ε replacing p+ε. As a corollary, if f is Boolean-valued or f:{−1,1}n→{−1,0,1}, we obtain the bounds ∥Δkf∥p≤C(p)kdk∥f∥p,k∈N.

At the endpoint p=∞ 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: ∥e−tΔf∥p≤exp(−c(p,ε)td)∥f∥1−θ2∥f∥θp+ε,t>0,

and an analogous estimate for p∈(1,2).

The constants c(p,ε) and C(p,ε) depend only on p and ε; crucially, they are independent of the dimension n.

Comments

This article was originally published in Studia Mathematica, volume 280, in 2025. https://doi.org/10.4064/sm240417-27-11

Peer Reviewed

1

Copyright

Instytut Matematyczny PAN

Creative Commons License

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.

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