Dimension-free estimates for low degree functions on the Hamming cube
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929557659975680 |
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| author | Domelevo, Komla Durcik, Polona Fragkiadaki, Valentia Klein, Ohad Silva, Diogo Oliveira e Slavíková, Lenka Wróbel, Błażej |
| author_facet | Domelevo, Komla Durcik, Polona Fragkiadaki, Valentia Klein, Ohad Silva, Diogo Oliveira e Slavíková, Lenka Wróbel, Błażej |
| contents | 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 $θ=θ(\varepsilon,p)\in (0,1)$ 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 $p-\varepsilon$ 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 \[\|Δ^k f\|_{p} \le C(p)^k \,d^k\, \|f\|_p,\qquad k\in \mathbb{N}.\] 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
$$ \|e^{-tΔ}f\|_{p} \le \exp(-c(p,\varepsilon) td) \|f\|_{2}^{1-θ}\|f\|_{p+\varepsilon}^θ,\qquad t>0,$$ 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 $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07699 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimension-free estimates for low degree functions on the Hamming cube Domelevo, Komla Durcik, Polona Fragkiadaki, Valentia Klein, Ohad Silva, Diogo Oliveira e Slavíková, Lenka Wróbel, Błażej Functional Analysis Classical Analysis and ODEs Primary 42C10, Secondary 41A17, 41A63, 47A60 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 $θ=θ(\varepsilon,p)\in (0,1)$ 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 $p-\varepsilon$ 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 \[\|Δ^k f\|_{p} \le C(p)^k \,d^k\, \|f\|_p,\qquad k\in \mathbb{N}.\] 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 $$ \|e^{-tΔ}f\|_{p} \le \exp(-c(p,\varepsilon) td) \|f\|_{2}^{1-θ}\|f\|_{p+\varepsilon}^θ,\qquad t>0,$$ 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 $n$. |
| title | Dimension-free estimates for low degree functions on the Hamming cube |
| topic | Functional Analysis Classical Analysis and ODEs Primary 42C10, Secondary 41A17, 41A63, 47A60 |
| url | https://arxiv.org/abs/2401.07699 |