Dimension-free estimates for low degree functions on the Hamming cube
Streszczenie
The main result of this paper are dimension-free $L^p$ 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 $n$.
