Dimension-free estimates for low degree functions on the Hamming cube
Volume 280 / 2025
Abstract
The main result of this paper are dimension-free inequalities, 1 \lt p \lt \infty , for low degree scalar-valued functions on the Hamming cube. More precisely, for any p \gt 2, \varepsilon \gt 0, and \theta =\theta (\varepsilon ,p)\in (0,1) satisfying \frac{1}{p}=\frac{\theta}{p+\varepsilon}+\frac{1-\theta}{2} we obtain, for any function f\colon \{-1,1\}^n\to \mathbb C whose spectrum is bounded from above by d, the Bernstein–Markov type inequalities \|\Delta^k f\|_{p} \le C(p,\varepsilon )^k \,d^k\, \|f\|_{2}^{1-\theta }\|f\|_{p+\varepsilon }^{\theta },\quad k\in \mathbb N. Analogous inequalities are also proved for p\in (1,2) with p-\varepsilon replacing p+\varepsilon . As a corollary, if f is Boolean-valued or f\colon \{-1,1\}^n\to \{-1,0,1\}, we obtain the bounds \|\Delta^k f\|_{p} \le C(p)^k \,d^k\, \|f\|_p,\quad k\in \mathbb N. At the endpoint p=\infty 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\colon \{-1,1\}^n\to \mathbb C whose spectrum is bounded from below by d satisfies the following upper bound on the decay of the heat semigroup: \|e^{-t\Delta }f\|_{p} \le \exp (-c(p,\varepsilon ) td) \|f\|_{2}^{1-\theta }\|f\|_{p+\varepsilon }^{\theta },\quad t \gt 0, and an analogous estimate for p\in (1,2).
The constants c(p,\varepsilon ) and C(p,\varepsilon ) depend only on p and \varepsilon ; crucially, they are independent of the dimension n.
Published in Open Access (under CC-BY license).
