arXiv subjects

Sheng-Chen Mao

Publications and source records attributed to Sheng-Chen Mao.

8 recordsLinked to original sources

arXiv2026

Dimension-free estimates for the full discrete Euclidean ball maximal function

Let $M_t$ denote the normalized average over the lattice points in the Euclidean ball of radius $t$ in $\mathbb{Z}^d$. We prove that the full maximal operator $f\mapsto\sup_{t\geq0}\lvert M_t f\rvert$ is bounded on $\ell^p(\mathbb{Z}^d)$, for every $1<p\leq\infty$, with a constant independent of the dimension. In particular, this resolves a question of E.M. Stein from the mid 1990s. The principal ingredient in our proof is that, when $t\lesssim d$ with $t$ sufficiently large, the associated multiplier $\mathfrak{m}_{\sqrt{\lfloor t^2\rfloor}}(ξ)$ admits an asymptotic expansion of arbitrary prescribed order, uniform in $ξ$, whose resulting maximal operators can be controlled by the discrete normalized Gaussian maximal function studied by Mirek--Szarek--Wróbel \cite{MSW25}.

math.CA↗

arXiv2026

Loomis-Whitney inequalities on Reiter-Heisenberg groups

We establish a Loomis-Whitney inequality for the Reiter-Heisenberg groups $\mathbb{G}_{qp}$, a family of step-two Carnot groups that includes the Heisenberg groups when $q=1$. The proof is based on the duality between Brascamp-Lieb inequalities and entropy subadditivity: we first derive the result for $\mathbb{G}_{q1}$ from the known inequality on the first Heisenberg group, using conditional entropy and the invariance of differential entropy under volume-preserving diffeomorphisms; then we pass from $\mathbb{G}_{q1}$ to $\mathbb{G}_{qp}$ via a stability principle for Loomis-Whitney inequalities under finite central sums, which generalizes the argument in (Zhang, 2024 arXiv:2402.02749v2). As consequences, we obtain the associated geometric projection inequality, a Gagliardo-Nirenberg-Sobolev inequality, and an isoperimetric inequality.

math.CA↗

arXiv2026

Spectral asymptotics of sub-Riemannian Laplacians on compact Heisenberg manifolds

Let $N_M(λ)$ be the spectral counting function of the sub-Laplacian on the compact Heisenberg manifold $M=Γ\backslash\mathbb H_d$, where $Γ$ is a lattice subgroup of the Heisenberg group $\mathbb H_d$. In 2016, Strichartz \cite[\textit{J. Geom. Anal.}]{Str16} proved the Weyl law with remainder $R_M(λ)=N_M(λ)-A_d\operatorname{vol}(M)λ^{d+1} = O_M(λ^d\logλ)$, and conjectured the optimal remainder to be $O_M(λ^d)$. In this work, we establish a new upper bound and the first two-sided lower bounds $$ R_M(λ)=O_M\!\left(λ^d(\logλ)^{2/3}\right), \qquad R_M(λ)=Ω_{M,\pm}\!\left(λ^d\log\logλ\right). $$ As a result, this implies that the sharp polynomial order is $d$, and disproves Strichartz's conjecture.

math.SP↗

arXiv2026

A $p = 2$ dichotomy for uniform Riesz transform bounds on stratified Lie groups

Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step, and the underlying group structure of $\mathbb{G}$. Our result provides a noncommutative generalization of the dimension-free Euclidean theorem of Ouyang, Spector, and Stockdale arXiv:2608.18068, with the same universal constant. Our proof relies upon a fractional obstacle problem adapted to stratified Lie groups by using the functional calculus of $\mathcal L$ instead of the Fourier transform. As a consequence, by interpolation we obtain uniform $L^p$ bounds for the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ for $p \in (1,2]$. By contrast, for every $p > 2$, we construct a sequence of stratified Lie groups with fixed horizontal dimension $5$ and steps tending to infinity for which the $L^p$ norms of the horizontal Riesz transforms diverge.

math.CA↗

arXiv2026

Lattice point counting problems on step-two nilpotent Lie groups

We develop the theory of lattice point counting on connected and simply connected nilpotent Lie groups of step-two, endowed with the parabolic type dilation and a family of homogeneous norms $ \mathcal{N}_{α,M}(x, t)=\left(|M_1x|^α+ |M_2t|^{α/ 2}\right)^{1 / α}$ adapted to the dilation structure, where $α>0$ and $M_1,M_2$ are invertible matrices. With appropriate notions of lattices, the domains to be counted are balls associated to these norms, and explicit counting discrepancy estimates are deduced for all possible dimensions and all $α>0$. The bounds are sharp when the group center is unidimensional and $α=2$, in certain rational sense. Our study also generalizes and even quantitatively improves previous results on Heisenberg groups obtained by Garg--Nevo--Taylor \cite[\textit{Ann. Inst. Fourier}, 2015]{GNT15}: (i) In dimension $5$, the exponent of logarithmic factor is lowered from $2/3$ to ${1}/{3}$ if $α\in(3,4] $ or $α=1$; and the factor $\log ^{2/3} R $ is dropped if $α\in(2,3]$. (ii) In dimension $3$, the estimation is upgraded from $O_ε(R^{ 5/2+ε})$ to $O(R^{2}\log^{ 1/2} R)$ for $α=1$, and to $O(R^{{19}/{8}})$ for $α\in (1,2)$; and the factor $\log R$ is removed for $α>4$. Moreover, as a byproduct, we extend the lattice counting near Heisenberg spheres, recently considered by Campolongo--Taylor \cite[\textit{Matematica}, 2023]{CT23} and Srivastava--Taylor \cite[\textit{J. Fourier Anal. Appl.}, 2026]{ST26}, to the above step-two group setting with arbitrary dimensional group center, where some quantitative improvements are also attained. Our method relies upon Poisson's summation formulas, oscillatory integral estimates and asymptotic properties as well as recursion formulas of Bessel functions.

math.CA↗

arXiv2026

Lattice point counting in Cygan--Korányi balls on Heisenberg groups

Lattice point counting in gauge balls on the Heisenberg group $\mathbb{H}^q$ is a non-commutative analogue of the Euclidean multidimensional sphere problem, initiated by Garg, Nevo and Taylor \cite[\textit{Ann. Inst. Fourier}, 2015]{GNT15}. The case of particular interest is when the gauge is taken as the Cygan--Korányi norm and the error term reads: $$\mathcal{E}_q(t)=\#\left(\mathbb{Z}^{2 q+1} \cap \mathcal{B}_t\right)-\operatorname{vol}(\mathcal{B}_1) \, t^{2 q+2},$$ with $\mathcal{B}_t=\{(v,w)\in\mathbb{H}^q: (|v|^4 + w^2)^{1/4} \le t \}$, which is closely related to the Gauss circle problem. When $q\ge3$, Gath \cite[\textit{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 2022]{Gat22} improved upon \cite[]{GNT15} by showing that $ |\mathcal{E}_q(t)|\lesssim t^{2q-1+ 1/3}$ and proposed the conjecture that the optimal order should be $2q-1$. In this paper, through Landau's formula and the $5,6$-th Derivative Tests of van der Corput, we arrive at that $|\mathcal{E}_q(t)| \lesssim t^{2 q-1 + 241/753} $ for any $ q \geq 4$, and recover the bound of Gath for $q=3$ up to a logarithmic factor. This, via a simpler method, provides the first progress towards Gath's conjecture.

math.NT↗

arXiv2026

On gradient estimates of the heat semigroups on step-two Carnot groups

In this work, we give a sufficient condition for a step-two Carnot group to satisfy the quasi Bakry-Émery curvature condition. As an application, we establish the gradient estimate for the heat semigroup on the free step-two Carnot group with three generators $N_{3,2}$. Moreover, high order gradient estimates and the Riemannian counterparts are also deduced under an extra condition.

math.AP↗

arXiv2023

Heat kernel asymptotics on the free step-two Carnot group with $3$ generators

In this work, we establish the uniform heat kernel asymptotics as well as sharp bounds for its derivatives on the free step-two Carnot group with $3$ generators. As a by-product, on this highly non-trivial toy model, we completely solve the Gaveau-Brockett problem, in other words, we obtain the expression of the squared Carnot-Carathéodory distance, as explicitly as one can possibly hope for. Furthermore, the precise estimates of the heat kernel, and its small-time asymptotic behaviors are deduced.

math.AP↗