A functional Loomis-Whitney type inequality in the Heisenberg group and projection theorems over finite fields

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 $$\frac{1}{q^3}\sum _{(x,t)\in {\mathbb{H}}^1({\mathbb{F}}_q)}{f}_1({\pi }_1(x,t))f_2({\pi }_2(x,t))\lesssim \mathrm{\parallel }{f}_1{\mathrm{\parallel }}_{L^{u_1}({\mathbb{F}}_q^2,dx)}\mathrm{\parallel }{f}_2{\mathrm{\parallel }}_{L^{u_2}({\mathbb{F}}_q^2,dx)}$$ holds uniformly in $q$, namely $$\frac{1}{u_1}+\frac{2}{u_2}\le 2\text{and}\frac{2}{u_1}+\frac{1}{u_2}\le 2,$$ 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 $\mathrm{\mid }K\mathrm{\mid }$ for every finite $K\subset {\mathbb{H}}^n({\mathbb{F}}_q)$ in terms of the sizes of its $2n$ Heisenberg projections $\{{\pi }_j(K){\}}_{j=1}^{2n}$, and in particular, $$\underset{1\le j\le 2n}{\max }\mathrm{\mid }{\pi }_j(K)\mathrm{\mid }{\gtrsim }_n\mathrm{\mid }K{\mathrm{\mid }}^{\frac{2n+1}{2(n+1)}}{q}^{-\frac{1}{2(n+1)}}\mathrm{.}$$ This result is optimal up to absolute constants. Moreover, when $n=1$ and $\mathrm{\mid }K\mathrm{\mid }>q$, 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).