Denote by \(\mathbb{H}^n\) the Heisenberg group, the set \(\mathbb{R}\times\mathbb{R}^{2n}\) equipped with the following non-commutative group operation: for all \((u,x), (v,y)\in\mathbb{H}^n\), \[ (u,x)\cdot (v,y):=(u+v+x^TBy,x+y). \] Here, \(B:=\begin{pmatrix}0&-bI_n\\ bI_n&0\end{pmatrix}\) for some \(b\neq 0\) (one typically choose \(b=1/2\)). For \(\mu_1\equiv\mu\), the normalized surface measure on \(\{0\}\times\mathbb{S}^{2n-1}\), let \(\mu_t\) denote its dilation supported on \(t\mathbb{S}^{2n-1}\). For a function \(f:\mathbb{H}^n\rightarrow\mathbb{C}\), one may formally define its spherical means as \[ f\ast\mu_t(u,x):=\int_{\mathbb{S}^{2n-1}}\!f(u-tx^TBy,x-ty)\,d\mu(y) \] and its spherical maximal function as \[ Mf(u,x):=\sup_{t>0}|f\ast\mu_t(u,x)|. \] In this paper, the authors complement known \(L^p\)-boundedness results for \(M\) on \(\mathbb{H}^n\), \(n\geq{2}\), by initiating the study of the case \(n=1\), where currently nothing is known for any \(p<\infty\). For \(2<p\leq\infty\), they show the existence of a constant \(C_p\), depending only on \(p\), such that \[ \|Mf\|_{L^p(\mathbb{H}^1)}\leq C_p\|f\|_{L^p(\mathbb{H}^1)} \] for all \(\mathbb{H}\)-radial functions \(f\) on \(\mathbb{H}^1\). A function \(f:\mathbb{H}^1\rightarrow\mathbb{C}\) is said to be \(\mathbb{H}\)-radial if \(f(u,Rx)=f(u,x)\) for all \((u,x)\in\mathbb{H}^1\) and all \(R\) belonging to the special orthogonal group, \(SO(2)\). Equivalently, \(f\) is \(\mathbb{H}\)-radial if and only if there exists some function \(f_0:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{C}\) such that \(f(u,x)=f_0(u,|x|)\) for all \((u,x)\in\mathbb{H}^1\).
The authors accomplish this by reducing the problem to studying the boundedness of a maximal function given by \(\sup_{t>0} |A_tf|\), where \(\{A_t\}\) are non-convolution averaging operators on \(\mathbb{R}^2\). While the reduction is not difficult, the associated curve distribution has vanishing rotational and cinematic curvatures, precluding the straightforward application of the standard techniques used to study the Euclidean spherical maximal function. A significant portion of this paper is spent overcoming these challenges, along the way performing an \(L^2\) analysis of two-parameter oscillatory integrals with two-sided fold singularities.
The appendices contain, among other things, a discussion of the use of repeated integration by parts often seen when studying oscillatory integrals.