Dimension-free estimates for low degree functions on the Hamming cube (2401.07699v2)

Abstract: The main result of this paper are dimension-free $L^p$ inequalities, $1<p<\infty$, for low degree scalar-valued functions on the Hamming cube. More precisely, for any $p\>2,$ $\varepsilon>0,$ 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 $f:{-1,1}^n\to \mathbb{C}$ 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},\qquad 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,\qquad 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 $p>2$ we prove that any function $f:{-1,1}^n\to \mathbb{C}$ whose spectrum is bounded from below by $d$ satisfies the upper bound on the decay of the heat semigroup $$ |e^{-t\Delta}f|{p} \le \exp(-c(p,\varepsilon) td) |f|{2}^{1-\theta}|f|{p+\varepsilon}^{\theta},\qquad t>0,$$ 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 $n$.