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

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 $p>2,$ $\epsilon >0,$ and $\hat{I}¸=\hat{I}¸(\epsilon ,p)\in (0,1)$ satisfying \[ \frac{1}{p}=\fracθ{p+\varepsilon}+\frac{1-θ}{2} \] 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 \[\|Δ^k f\|_{p} \le C(p,\varepsilon)^k \,d^k\, \|f\|_{2}^{1-θ}\|f\|_{p+\varepsilon}^θ,\qquad k\in \mathbb{N}.\] Analogous inequalities are also proved for $p\in (1,2)$ with $p-\epsilon$ replacing $p+\epsilon .$ As a corollary, if $f$ is Boolean-valued or $f:\{-1,1{\}}^n\to \{-1,0,1\},$ we obtain the bounds \[\|Δ^k f\|_{p} \le C(p)^k \,d^k\, \|f\|_p,\qquad k\in \mathbb{N}.\] 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 $$\Vert e^{-t\hat{I}”}f{\Vert }_p\le \exp (-c(p,\epsilon )td)\Vert f{\Vert }_2^{1-\hat{I}¸}\Vert f{\Vert }_{p+\epsilon }^{\hat{I}}¸,t>0,$$ and an analogous estimate for $p\in (1,2).$ The constants $c(p,\epsilon )$ and $C(p,\epsilon )$ depend only on $p$ and $\epsilon$; crucially, they are independent of the dimension $n$.

10 pages, incporporating suggestions from referees reports, accepted for publication in Studia Mathematica