Flag Hardy spaces and partial differential equations
Michael G Cowling, Ji Li
TL;DR
This work situates flag geometry on the Heisenberg group $H^n$ and analyzes flag atoms through a PDE lens, proving Stein's conjecture for the flag Hardy space $H^1_F(H^n)$ using Baldi–Franchi–Pansu precursor results. It develops an atomic framework via particles associated to tubes, and investigates whether flag atoms admit a factorization as a convolution of a one-level atom and a one-dimensional atom; naive cancellation fails, prompting a second-order derivative representation across the two flag directions. By constructing a flag kernel from smooth mean-zero pieces, decomposing into four scale regions, and leveraging Journé's lemma, the authors obtain robust bounds for flag singular integrals on the Hardy space, yielding equivalence with the CCLLO24 definition. The results unify atomic, maximal, square, and area-function characterizations of $H^1_F(H^n)$ and connect Hardy space theory with PDE techniques on stratified groups, notably via the Baldi–Franchi–Pansu framework. The work thus reinforces the view that geometric PDE methods are essential for a deep understanding of flag Hardy spaces and their operators.
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.