A functional Loomis-Whitney type inequality in the Heisenberg group and projection theorems over finite fields
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2025
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| author | Cheong, Daewoong Pham, Thang Tran, Dung The |
| author_facet | Cheong, Daewoong Pham, Thang Tran, Dung The |
| contents | We establish functional Loomis--Whitney type inequalities in the finite Heisenberg group $\mathbb{H}^n(\mathbb{F}_q)$. For $n=1$, we determine the sharp region of exponents $(u_1,u_2)$ for which the Heisenberg Loomis--Whitney inequality \[ \frac{1}{q^3}\sum_{(x,t)\in \mathbb{H}^1(\mathbb{F}_q)} f_1(π_1(x,t))\,f_2(π_2(x,t)) \;\lesssim\; \|f_1\|_{L^{u_1}(\mathbb{F}_q^2,dx)}\|f_2\|_{L^{u_2}(\mathbb{F}_q^2,dx)} \] holds uniformly in $q$, namely \[ \frac{1}{u_1}+\frac{2}{u_2}\le 2 \quad\text{and}\quad \frac{2}{u_1}+\frac{1}{u_2}\le 2, \] which includes the endpoint estimate $L^{\frac{3}{2}}\times L^{\frac{3}{2}}\to L^1$. For general $n$, we prove the symmetric multilinear estimate at the endpoint exponent $
u=\frac{n(2n+1)}{n+1}, $
using an induction on $n$ that exploits the Heisenberg fiber structure together with a multilinear interpolation scheme. Specializing to indicator functions yields a sharp Loomis--Whitney type set inequality bounding $|K|$ for every finite $K\subset \mathbb{H}^n(\mathbb{F}_q)$ in terms of the sizes of its $2n$ Heisenberg projections $\{π_j(K)\}_{j=1}^{2n}$, and in particular, \[ \max_{1\le j\le 2n} |π_j(K)| \;\gtrsim_n\; |K|^{\frac{2n+1}{2(n+1)}}\,q^{-\frac{1}{2(n+1)}}. \] This result is optimal up to absolute constants. Moreover, when $n=1$ and $|K|>q$, we obtain a stronger statement via Vinh's point--line incidence theorem. We also discuss connections to a boundedness problem for multilinear forms/operators over finite fields studied by Bhowmik, Iosevich, Koh, and Pham (2025), and to orthogonal projection/covering questions in $\mathbb{F}_q^{2n+1}$ studied by Chen (2018). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A functional Loomis-Whitney type inequality in the Heisenberg group and projection theorems over finite fields Cheong, Daewoong Pham, Thang Tran, Dung The Combinatorics Classical Analysis and ODEs Number Theory We establish functional Loomis--Whitney type inequalities in the finite Heisenberg group $\mathbb{H}^n(\mathbb{F}_q)$. For $n=1$, we determine the sharp region of exponents $(u_1,u_2)$ for which the Heisenberg Loomis--Whitney inequality \[ \frac{1}{q^3}\sum_{(x,t)\in \mathbb{H}^1(\mathbb{F}_q)} f_1(π_1(x,t))\,f_2(π_2(x,t)) \;\lesssim\; \|f_1\|_{L^{u_1}(\mathbb{F}_q^2,dx)}\|f_2\|_{L^{u_2}(\mathbb{F}_q^2,dx)} \] holds uniformly in $q$, namely \[ \frac{1}{u_1}+\frac{2}{u_2}\le 2 \quad\text{and}\quad \frac{2}{u_1}+\frac{1}{u_2}\le 2, \] which includes the endpoint estimate $L^{\frac{3}{2}}\times L^{\frac{3}{2}}\to L^1$. For general $n$, we prove the symmetric multilinear estimate at the endpoint exponent $ u=\frac{n(2n+1)}{n+1}, $ using an induction on $n$ that exploits the Heisenberg fiber structure together with a multilinear interpolation scheme. Specializing to indicator functions yields a sharp Loomis--Whitney type set inequality bounding $|K|$ for every finite $K\subset \mathbb{H}^n(\mathbb{F}_q)$ in terms of the sizes of its $2n$ Heisenberg projections $\{π_j(K)\}_{j=1}^{2n}$, and in particular, \[ \max_{1\le j\le 2n} |π_j(K)| \;\gtrsim_n\; |K|^{\frac{2n+1}{2(n+1)}}\,q^{-\frac{1}{2(n+1)}}. \] This result is optimal up to absolute constants. Moreover, when $n=1$ and $|K|>q$, we obtain a stronger statement via Vinh's point--line incidence theorem. We also discuss connections to a boundedness problem for multilinear forms/operators over finite fields studied by Bhowmik, Iosevich, Koh, and Pham (2025), and to orthogonal projection/covering questions in $\mathbb{F}_q^{2n+1}$ studied by Chen (2018). |
| title | A functional Loomis-Whitney type inequality in the Heisenberg group and projection theorems over finite fields |
| topic | Combinatorics Classical Analysis and ODEs Number Theory |
| url | https://arxiv.org/abs/2510.05022 |