Dimension free estimates for the vector-valued Hardy--Littlewood maximal function on the Heisenberg group

TL;DR

The article develops dimension-free, vector-valued bounds for maximal functions on the Heisenberg group $\mathbb{H}^n$, focusing on Korányi ball averages and the associated Hardy–Littlewood maximal operator. Central to the approach is a Müller–Seeger decomposition of horizontal spherical means, which yields frequency-localized components whose $L^p$-bounds are established and shown to be dimension-free; these are combined with a one-dimensional vertical maximal operator to obtain dimension-free bounds for the scalar and vector-valued Hardy–Littlewood maximal operators. A key stepping stone is the L^p-boundedness of the vector-valued Nevo–Thangavelu spherical maximal function $M_S$, proved for sequences $(f_j)$ with $(\sum_j |f_j|^q)^{1/q} \in L^p$ in a specific range, using the decomposition to control each piece and interpolation. The results are extended to general UMD Banach lattices, using heat semigroup methods and complex interpolation to yield dimension-free, $Z$-valued bounds. Together, these findings advance dimension-free harmonic analysis on non-commutative groups and broaden the vector-valued theory in spaces of homogeneous type and on the Heisenberg group with UMD targets.

Abstract

In this article, we establish dimension-free Fefferman-Stein inequalities for the Hardy-Littlewood maximal function associated with averages over Korányi balls in the Heisenberg group. We also generalize the result to more general UMD lattices. As a key stepping stone, we establish the $L^p$- boundedness of the vector-valued Nevo-Thangavelu spherical maximal function, which plays a crucial role in our proofs of the main theorems.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• Proposition 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof