Heisenberg 群上最大算子定理的新证明
经典分析与常微分方程
2026-05-19 v2
摘要
给定 0 ≤ α < 1 0\leq\alpha<1 0 ≤ α < 1 M α f ( u , v , t ) = sup R ∋ ( 0 , 0 , 0 ) v o l { R } α − 1 ∭ R ∣ f [ ( u , v , t ) ⊙ ( ξ , η , τ ) − 1 ] ∣ d ξ d η d τ \begin{array}{lr} \mathbf{M}_\alpha f(u,v,t) = \sup_{ \mathbf{R} \ni (0,0,0)} {\rm vol} \{\mathbf{R}\}^{\alpha-1} \iiint_\mathbf{R}\left|f [(u,v,t)\odot(\xi,\eta,\tau)^{-1}]\right|d\xi d\eta d\tau \end{array} M α f ( u , v , t ) = sup R ∋ ( 0 , 0 , 0 ) vol { R } α − 1 ∭ R f [( u , v , t ) ⊙ ( ξ , η , τ ) − 1 ] d ξ d η d τ R ⊂ R 2 n + 1 \mathbf{R}\subset\mathbb{R}^{2n+1} R ⊂ R 2 n + 1 ⊙ \odot ⊙ M 0 \mathbf{M}_0 M 0 L p \mathbf{L}^p L p M α : L p ( R 2 n + 1 ) → L q ( R 2 n + 1 ) \mathbf{M}_\alpha :\mathbf{L}^p(\mathbb{R}^{2n+1}) \to \mathbf{L}^q(\mathbb{R}^{2n+1}) M α : L p ( R 2 n + 1 ) → L q ( R 2 n + 1 ) α = 1 p − 1 q , 1 < p ≤ q < ∞ \alpha={1\over p}-{1\over q},~ 1<p\leq q<\infty α = p 1 − q 1 , 1 < p ≤ q < ∞
引用
@article{arxiv.2605.14961,
title = {A new proof of maximal theorem on Heisenberg groups},
author = {Chuhan Sun and Zipeng Wang},
journal= {arXiv preprint arXiv:2605.14961},
year = {2026}
}