Bohnenblust--Hille inequality for cyclic groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913633626226688 |
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| author | Slote, Joseph Volberg, Alexander Zhang, Haonan |
| author_facet | Slote, Joseph Volberg, Alexander Zhang, Haonan |
| contents | For any $K>2$ and the multiplicative cyclic group $Ω_K$ of order $K$, consider any function $f:Ω_K^n\to\mathbf{C}$ and its Fourier expansion $f(z)=\sum_{α\in\{0,1,\ldots,K-1\}^n}a_αz^α$, with $d:=\text{deg}(f)$ denoting its degree as a multivariate polynomial.
We prove a Bohnenblust--Hille (BH) inequality in this setting: the $\ell_{2d/(d+1)}$ norm of the Fourier coefficients of $f$ is bounded by $C(d,K)\|f\|_\infty$ with $C(d,K)$ independent of $n$.
This is the interpolating case between the now well-understood BH inequalities for functions on the poly-torus ($K =\infty$) and the hypercube ($K=2$) but those extreme cases of $K$ have special properties whose absence for intermediate $K$ prevent a proof by the standard BH framework.
New techniques are developed exploiting the group structure of $Ω_K^n$.
By known reductions, the cyclic group BH inequality also entails a noncommutative BH inequality for tensor products of the $K \times K$ complex matrix algebra (or in the language of quantum mechanics, systems of $K$-level qudits). These new BH inequalities generalize several applications in harmonic analysis and statistical learning theory to broader classes of functions and operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_10560 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Bohnenblust--Hille inequality for cyclic groups Slote, Joseph Volberg, Alexander Zhang, Haonan Functional Analysis Analysis of PDEs Classical Analysis and ODEs 43A75, 47A30, 81P45, 06E30 F.2.2 For any $K>2$ and the multiplicative cyclic group $Ω_K$ of order $K$, consider any function $f:Ω_K^n\to\mathbf{C}$ and its Fourier expansion $f(z)=\sum_{α\in\{0,1,\ldots,K-1\}^n}a_αz^α$, with $d:=\text{deg}(f)$ denoting its degree as a multivariate polynomial. We prove a Bohnenblust--Hille (BH) inequality in this setting: the $\ell_{2d/(d+1)}$ norm of the Fourier coefficients of $f$ is bounded by $C(d,K)\|f\|_\infty$ with $C(d,K)$ independent of $n$. This is the interpolating case between the now well-understood BH inequalities for functions on the poly-torus ($K =\infty$) and the hypercube ($K=2$) but those extreme cases of $K$ have special properties whose absence for intermediate $K$ prevent a proof by the standard BH framework. New techniques are developed exploiting the group structure of $Ω_K^n$. By known reductions, the cyclic group BH inequality also entails a noncommutative BH inequality for tensor products of the $K \times K$ complex matrix algebra (or in the language of quantum mechanics, systems of $K$-level qudits). These new BH inequalities generalize several applications in harmonic analysis and statistical learning theory to broader classes of functions and operators. |
| title | Bohnenblust--Hille inequality for cyclic groups |
| topic | Functional Analysis Analysis of PDEs Classical Analysis and ODEs 43A75, 47A30, 81P45, 06E30 F.2.2 |
| url | https://arxiv.org/abs/2305.10560 |