Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.02390

Flag Hardy spaces and partial differential equations

Michael G Cowling·Ji Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

After a quick introduction to flag geometry on the Heisenberg group, we discuss the problem of identifying flag atoms, which boils down to a question in PDE. We establish a conjecture of E.M.~Stein for the flag Hardy space on the Heisenberg group. The key result on which our arguments hinge is due to Baldi, Franchi and Pansu.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael G Cowling, Ji Li. 2025-04-03. Flag Hardy spaces and partial differential equations. https://arxiv.org/abs/2504.02390

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Exponential decay of threshold correlations for symmetric periodic functions

We study how often $n$ distinct integer dilates of a periodic function can all lie above a fixed threshold or all lie below its negative. We consider continuous functions $f:\mathbb R\to[-1,1]$ that are $1$-periodic, even, strictly decreasing on $[0,1/2]$, and satisfy $f(0)=1$ and $f(t+1/2)=-f(t)$. This class includes the cosine and the triangle wave. For $0\leqα<1$, let $ρ_α\in(0,1/4]$ be determined by $f(ρ_α)=α$, and let $F_α(n)$ denote the infimum of these proportions over all sets of $n$ distinct positive integer frequencies. We prove that $F_α(n)\geq 4ρ_α^n$ and that the limit $\lim_{n\to\infty}F_α(n)^{1/n}$ exists for every threshold. When $ρ_α^{-1}$ is an integer, we obtain \begin{align} F_α(n) &\leq C_αn^{\log_2((1-ρ_α)/(1-2ρ_α))}ρ_α^n, \end{align} and hence the limit equals $ρ_α$. Otherwise, the limit belongs to the interval $[ρ_α,\lfloorρ_α^{-1}\rfloor^{-1}]$. The upper bound in the reciprocal case uses a cyclic covering construction previously studied in the context of circular graphs. We give a self-contained proof of the corresponding optimal modulus.

math.CA↗
arXiv2026

Endpoint estimates for spherical maximal operators acting on radial functions in hyperbolic spaces

In this article we investigate spherical maximal operators $\mathfrak m^α$ of real order $α$ acting on radial functions in the hyperbolic space $\mathbb H^n$. We show that for every $n\geq 2$, $1 (1-n)/2$, $\mathfrak m^α$ is bounded on $L^p_{\mathrm{rad}}(\mathbb H^n)$ if and only if \begin{align*} \begin{cases} α>1-n+\frac np, & 1 \frac1p-\frac{n-1}2, & 2 \frac{2n}{n-1}. \end{cases} \end{align*} Moreover, we show that for $1<p\leq 2n/(n-1)$ and \begin{align*} α= \begin{cases} 1-n+\frac np, & 1<p\leq 2, \frac1p-\frac{n-1}2, & 2<p\leq\frac{2n}{n-1}, \end{cases} \end{align*} $\mathfrak m^α$ is bounded from $L^{p,1}_{\mathrm{rad}}(\mathbb H^n)$ to $L^{p,\infty}_{\mathrm{rad}}(\mathbb H^n)$.

math.CA↗
arXiv2026

On $q$-middle convolution and generalized $q$-hypergeometric equation

We investigate higher-order $q$-difference equations obtained by successive applications of a $q$-analogue of addition and the $q$-middle convolution. We construct an $n$-dimensional system of linear $q$-difference equations and derive a single $q$-difference equation whose coefficients are polynomials of degree at most $1$ in the independent variable. Under certain conditions on the parameters, the resulting equation is a generalized $q$-hypergeometric equation of order $n$. We also obtain a multiple Jackson integral representation of solutions and give sufficient conditions for its convergence. By suitably choosing the functions appearing in the integrand and specializing the parameters, we obtain $q$-hypergeometric series solutions of type ${}_n ϕ_{n-1}$.

math.CA↗