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

The main result of this paper are dimension-free $L^p$ inequalities, $1<p<\mathrm{\infty }$, for low degree scalar-valued functions on the Hamming cube. More precisely, for any $p>2,$ $\epsilon >0,$ and $\theta =\theta (\epsilon ,p)\in (0,1)$ satisfying $$\frac{1}{p}=\frac{\theta }{p+\epsilon }+\frac{1-\theta }{2}$$ 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 $$\mathrm{\parallel }{\mathrm{\Delta }}^{k}f{\mathrm{\parallel }}_p\le C(p,\epsilon {)}^k{d}^k\mathrm{\parallel }f{\mathrm{\parallel }}_2^{1-\theta }\mathrm{\parallel }f{\mathrm{\parallel }}_{p+\epsilon }^{\theta },k\in \mathbb{N}\mathrm{.}$$ Analogous inequalities are also proved for $p\in (1,2)$ with $p-\epsilon$ replacing $p+\epsilon \mathrm{.}$ As a corollary, if $f$ is Boolean-valued or $f\text{\hspace{0.17em}⁣}:\{-1,1{\}}^n\to \{-1,0,1\},$ we obtain the bounds $$\mathrm{\parallel }{\mathrm{\Delta }}^{k}f{\mathrm{\parallel }}_p\le C(p{)}^k{d}^k\mathrm{\parallel }f{\mathrm{\parallel }}_p,k\in \mathbb{N}\mathrm{.}$$ At the endpoint $p=\mathrm{\infty }$ 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 $$\mathrm{\parallel }{e}^{-t\mathrm{\Delta }}f{\mathrm{\parallel }}_p\le \exp (-c(p,\epsilon )td)\mathrm{\parallel }f{\mathrm{\parallel }}_2^{1-\theta }\mathrm{\parallel }f{\mathrm{\parallel }}_{p+\epsilon }^{\theta },t>0,$$ and an analogous estimate for $p\in (1,2)\mathrm{.}$ The constants $c(p,\epsilon )$ and $C(p,\epsilon )$ depend only on $p$ and $\epsilon$; crucially, they are independent of the dimension $n$.