#  $L^{2}$-Maximal functions on graded Lie groups

Duván Cardona  Duván Cardona: Department of Mathematics: Analysis, Logic and
Discrete Mathematics Ghent University, Belgium E-mail address
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Bourgain in his seminal paper [2] about the analysis of maximal functions
associated to convex bodies, has estimated in a sharp way the $L^{2}$-operator
norm of the maximal function associated to a kernel $K\in L^{1},$ with
differentiable Fourier transform $\widehat{K}.$ We formulate the extension to
Bourgain’s $L^{2}$-estimate in the setting of maximal functions on graded Lie
groups. Our criterion is formulated in terms of the group Fourier transform of
the kernel. We discuss the application of our main result to the
$L^{p}$-boundedness of maximal functions on graded Lie groups.

###### Key words and phrases:

Group Fourier transform. $L^{2}$-estimate. Maximal function. Graded Lie group.
Noncommutative harmonic analysis.

###### 2020 Mathematics Subject Classification:

42B25, 22E27.

The author was supported by the Research Foundation-Flanders (FWO) under the
postdoctoral grant No 1204824N. The author was supported by the FWO Odysseus 1
grant G.0H94.18N: Analysis and Partial Differential Equations and by the
Methusalem programme of the Ghent University Special Research Fund (BOF)
(Grant number 01M01021).

###### Contents

  1. 1 Introduction
    1. 1.1 $L^{2}$-estimates for maximal functions on $\mathbb{R}^{n}$
    2. 1.2 $L^{2}$-estimates for maximal functions on graded Lie groups
  2. 2 Fourier analysis on nilpotent Lie groups and Rockland operators
    1. 2.0.1 Nilpotent Lie groups
    2. 2.0.2 The unitary dual $\widehat{G}$
    3. 2.0.3 Plancherel theorem on nilpotent Lie groups
    4. 2.0.4 Rockland operators on graded Lie groups
  3. 3 Estimation of maximal functions using the Fourier transform

##  1\. Introduction

Let $G$ be a homogeneous (nilpotent) Lie group and let us consider a
measurable function $K\in L^{1}(G).$ Extending a result due to Bourgain [2] we
estimate the constant $\Gamma(K)$ that makes valid the $L^{2}$-inequality

$\|\sup_{t>0}|f\ast
K_{t}|\|_{L^{2}(G)}\lesssim\Gamma(K)\|f\|_{L^{2}(G)},\,f\in
C^{\infty}_{0}(G).$ (1.1)

The quantity $\Gamma(K)$ involves information of the group Fourier transform
of the kernel $K.$ Here, $K_{t}(x):=t^{-Q}K\circ D_{t^{-1}}(x)$ where $Q$ is
the homogeneous dimension of the group $G,$ and $\\{D_{t}\\}_{t>0},$ denotes
its family of dilations. Since our criteria are presented in terms of the
symbols of Rockland operators we consider $G$ to be a graded Lie group. Our
main result is Theorem 1.6 of Subsection 1.2. To illustrate our main result we
are going to recall some aspects related to Bourgain’s $L^{2}$-estimate [2] as
well as some of its consequences in the analysis of maximal functions on
$\mathbb{R}^{n}$.

###  1.1. $L^{2}$-estimates for maximal functions on $\mathbb{R}^{n}$

Bourgain in [2] proved that the maximal function

$M_{B}f(x)=\sup_{r>0}\frac{1}{\textnormal{Vol}(r\cdot B)}\smallint_{r\cdot
B}|f(x+y)|dy,$ (1.2)

associated to a centrally symmetric convex body in $\mathbb{R}^{n},$ satisfies
the $L^{2}$-operator norm estimate

$\|M_{B}f\|_{L^{2}(\mathbb{R}^{n})}\leq D\|f\|_{L^{2}(\mathbb{R}^{n})},$ (1.3)

for a numerical constant $D>0$ independent of $B$ and of the dimension $n$.
Bourgain’s estimate in (1.3) extended in this general setting the celebrated
results by Stein and Stein and Strömberg [17, 18] in the $L^{2}$-case. For a
discussion on the subject we refer to the introduction of [5, Pages 60-63].
Bourgain’s argument for the proof of (1.3) combines two methods, the first one
from Fourier analysis when estimating the $L^{2}$-operator norm of an
arbitrary maximal function associated to an integrable kernel with
differentiable Fourier transform and, the second one, from the geometrical
properties of convex sets as Brunn’s theorem, see e.g. Gromov and Milman [12].

The $L^{2}$-estimate for arbitrary maximal functions
$M_{K}f(x):=\sup_{t>0}|f\ast K_{t}(x)|$ due to Bourgain [2] can be stated as
follows. Here, we have used the notation $K_{t}(x)=t^{-n}K(t^{-1}x)$ for any
function in the family $\\{K_{t}\\}_{t>0},$ among other things, preserving the
$L^{1}$-norm of $K.$ Also, $\widehat{K}(\xi)=\smallint e^{-2\pi
ix\cdot\xi}f(x)dx$ denotes the Fourier transform of $K.$

######  Theorem 1.1 (Bourgain [2]).

Let $K\in L^{1}(\mathbb{R}^{n})$ be such that its Fourier transform
$\widehat{K}$ is differentiable. Define the following quantities

$\alpha_{j}=\sup_{|\xi|\sim
2^{j}}|\widehat{K}(\xi)|,\quad\beta_{j}=\sup_{|\xi|\sim
2^{j}}|\langle\nabla\widehat{K}(\xi),\xi\rangle|.$ (1.4)

Then, we have the following estimate on the maximal operator associated to
$K,$

$\|\sup_{t>0}|f\ast K_{t}|\|_{L^{2}(\mathbb{R}^{n})}\leq
C\Gamma(K)\|f\|_{L^{2}(G)},$ (1.5)

for every $f\in\mathscr{S}(\mathbb{R}^{n}),$ where

$\Gamma(K)=\sum_{j\in\mathbb{Z}}\alpha_{j}^{\frac{1}{2}}(\alpha_{j}+\beta_{j})^{\frac{1}{2}}.$
(1.6)

There are several reasons making Theorem 1.1 interesting by itself. First, the
constant $\Gamma(K)$ in (1.6) is sharp in some sense, see [2, Page 1475]. On
the other hand, it has been useful in the analysis of the $L^{2}$-theory of
maximal functions, for convex bodies and for the sphere
$\mathbb{S}^{n-1}\subset\mathbb{R}^{n}$, as we describe in the following
remarks.

######  Remark 1.2.

Let $\chi_{A}$ be the characteristic function of an arbitrary set
$A\subset\mathbb{R}^{n}.$ A direct application of Theorem 1.1 to
$K(x)=\chi_{B}-P_{L}$222Here, $\\{P_{t}\\}_{t>0}$ denotes the Poisson kernel
defined via $\widehat{P}_{t}(\xi)=e^{-t|\xi|}$, $t=L(B)$ is a suitable
constant introduced by Milman, see [2, Page 1469] for details. provides the
existence of the numerical constant $D$ in (1.3), see [2, Page 1473].

######  Remark 1.3.

Consider the full maximal function on $\mathbb{R}^{n}$

${M}^{d\sigma}_{F}f(x)=\sup_{r>0}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-ry)d\sigma(y)\right|,\,n\geq
2,$ (1.7)

where $d\sigma$ is the surface measure on the sphere $\mathbb{S}^{n-1}.$ For
$n\geq 3,$ by an appropriate splitting procedure
$d\sigma=d\sigma_{1}+d\sigma_{2},$ and denoting by $M_{1}$ and $M_{2}$ the
corresponding maximal functions for the measures $d\sigma_{1}$ and
$d\sigma_{2},$ respectively, the $L^{2}$-theory of $M_{2}$ has been
established in [1], leading to the proof of the restricted weak
$(\frac{n}{n-1},1)$ inequality for ${M}^{d\sigma}_{F}.$ Note that the
interpolation of the equivalent to this result in the setting of Lorentz
spaces (namely, the boundedness of ${M}^{d\sigma}_{F}:L^{p,1}\rightarrow
L^{p,\infty}$ with $p=\frac{n}{n-1}$) with the boundedness of
${M}^{d\sigma}_{F}$ on $L^{\infty}$ provides an alternative proof of the
$L^{p}$-boundedness result due to Stein [15], for the spherical maximal
function ${M}^{d\sigma}_{F}$ in the range $p>\frac{n}{n-1}.$

######  Remark 1.4.

Bourgain in [4] proved the $L^{p}$-boundedness of the spherical maximal
function ${M}^{d\sigma}_{F}$ in the lower dimension case $n=2,$ for all $p>2.$
As it was pointed out in [4, Page 70] the $L^{2}$-boundedeness of
${M}^{d\sigma}_{F}$ fails, but the analysis of its $L^{2}$-behaviour is
important to understand its $L^{p}$-behaviour. Moreover, if one consider the
regularised version

${M}^{d\sigma}_{\varepsilon}f(x)=\sup_{r>0}\left|\smallint\limits_{\mathbb{S}^{n-1}}f(x-ry)(d\sigma\ast
P_{\varepsilon})(y)\right|,\,n\geq 2,$ (1.8)

the $L^{2}$-operator norm of ${M}^{d\sigma}_{\varepsilon}$ can be estimated as
follows

$\|{M}^{d\sigma}_{\varepsilon}f\|_{L^{2}(\mathbb{R}^{n})}\lesssim\log\left(\frac{1}{\varepsilon}\right)\|f\|_{L^{2}(\mathbb{R}^{n})}.$
(1.9)

Bourgain proved this inequality of norms using Theorem 1.1, see [4, Page 70].

######  Remark 1.5.

Theorem 1.1 has been fundamental as well its consequences in further works
about the $L^{2}$-theory of maximal functions in the Euclidean setting. For
instance, see the references [3, 5, 6]. In the next subsection we present our
extension to Bourgain’s estimate (1.5) in Theorem 1.1 to the setting of graded
Lie groups.

###  1.2. $L^{2}$-estimates for maximal functions on graded Lie groups

The analysis of the $L^{p}$-boundnedness of maximal functions on graded Lie
groups and more general homogeneous Lie groups has been a problem of wide
interest with a long tradition, see Folland and Stein [11]. There has been
several criteria for establishing these $L^{p}$-boundendness results. On
graded Lie groups criteria in terms of the symbols of Rockland operators have
been introduced in the work [8] by the author and M. Ruzhansky for the dyadic
maximal function and in [7] for the full maximal function. The understanding
of the $L^{2}$-theory for these operators is fundamental in Littlewood-Paley’s
theory in the setting of graded Lie groups, and the analysis of this problem
is the main goal of this manuscript.

To present our main Theorem 1.6 we introduce the required notation. In what
follows, $G$ denotes a graded Lie group. This means that its Lie algebra
admits a graduation
$\mathfrak{g}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}\oplus\cdots\oplus\mathfrak{g}_{s},$
with $[\mathfrak{g}_{i},\mathfrak{g}_{j}]\subset\mathfrak{g}_{i+j}.$ When
$i+j>s,$ $\mathfrak{g}_{i+j}=\\{0\\},$ see Section 2 for details. Here,
$\mathcal{R}$ denotes an arbitrary positive Rockland operator which, by
definition, is a left-invariant positive hypoelliptic partial differential
operator on $G.$ We denote by $\nu>0$ the homogeneity degree of $\mathcal{R}.$
Here $\eta\in C^{\infty}_{0}(\mathbb{R})$ denotes a compactly supported
function on the interval $I_{\nu}=[2^{-\nu},2^{\nu}].$ For every
$j\in\mathbb{Z},$ define $\eta_{j}(t):=\eta(2^{-(j+1)}t).$ We assume that
$\eta$ generates a partition of unity, namely, that
$\sum_{j\in\mathbb{Z}}\eta_{j}(t)=1,$ $\forall t>0.$ The operator
$\eta_{j}(\mathcal{R})$ is defined by the spectral calculus of the Rockland
operator $\mathcal{R}$. Let $\widehat{G}$ be the unitary dual of $G.$ Each
operator $\eta_{j}(\mathcal{R})$ is a Fourier multiplier on $L^{2}(G)$ and
$\eta_{j}(\pi(\mathcal{R})),$ $\pi\in\widehat{G},$ denotes its symbol. Our
main theorem is the following extension to the aforementioned result due to
Bourgain.

######  Theorem 1.6.

Consider a kernel $K\in L^{1}(G)$ on a graded Lie group $G.$ Define the
quantities

$\alpha_{j}=\sup_{\pi\in\widehat{G}}\|\widehat{K}(\pi)\eta_{j}(\pi(\mathcal{R}))\|_{\textnormal{op}},\quad\beta_{j}=\sup_{s>0;\pi\in\widehat{G}}\|s\frac{d}{ds}\\{\widehat{K}(s\cdot\pi)\\}\eta_{j}((s\cdot\pi)(\mathcal{R}))\|_{\textnormal{op}}.$
(1.10)

Then, we have the following estimate on the maximal operator associated to
$K,$

$\|\sup_{t>0}|f\ast K_{t}|\|_{L^{2}(G)}\leq C\Gamma(K)\|f\|_{L^{2}(G)},$
(1.11)

for every $f\in\mathscr{S}(G),$ where

$\Gamma(K)=\sum_{j\in\mathbb{Z}}\alpha_{j}^{\frac{1}{2}}(\alpha_{j}+\beta_{j})^{\frac{1}{2}}.$
(1.12)

We note that the smoothness condition on $\eta$ can be removed. Indeed, one
can assume that $\eta$ is a continuous piecewise function generating a
partition of unity, see Remark 3.1. Theorem 1.6 applied to the Euclidean
setting, that is $(\mathbb{R}^{n},+)$ with the usual isotropic dilation, and
with $\mathcal{R}=-\Delta_{x}$ being the positive Laplacian recovers Theorem
1.1. Indeed, note that in the Euclidean setting $\pi(\mathcal{R})$ is replaced
by $|\xi|^{2}$ where $\xi\in\mathbb{R}^{n},$ is the Fourier variable. We note
that Theorem 1.6 provides substantial information on the Littlewood-Paley
components of the Fourier transform $\widehat{K}$ related to the Rockland
operator $\mathcal{R}.$ Note also, that the hypothesis $K\in L^{1}(G),$
implies that the corresponding maximal operator
$M_{K}:f\mapsto\sup_{t>0}|f\ast K_{t}|,$ is bounded on
$L^{\infty}(G)$333Indeed, note that $|f\ast K_{t}(x)|\leq\|f\ast
K_{t}\|_{L^{\infty}(G)}\leq\|f\|_{L^{\infty}(G)}\|K_{t}\|_{L^{1}(G)}=\|f\|_{L^{\infty}(G)}\|K\|_{L^{1}(G)},$
in view of Young’s inequality. From where one deduces that
$\|M_{K}f\|_{L^{\infty}(G)}\leq\|K\|_{L^{1}(G)}\|f\|_{L^{\infty}(G)}.$. The
Riesz-Thorin interpolation theorem implies the following $L^{p}$-boundedness
result.

######  Corollary 1.7.

Consider a kernel $K\in L^{1}(G)$ on a graded Lie group $G$ and assume that
$\Gamma(K)$ in (1.12) is finite. Then, we have the following $L^{p}$-estimate
on the maximal operator associated to $K,$

$\|\sup_{t>0}|f\ast K_{t}|\|_{L^{p}(G)}\leq
C_{p}\Gamma(K)^{\frac{2}{p}}\|K\|_{L^{1}(G)}^{1-\frac{2}{p}}\|f\|_{L^{p}(G)},$
(1.13)

for every $f\in\mathscr{S}(G),$ and for all $2\leq p\leq\infty.$

This paper is organised as follows. In Section 2 we present some basics about
the Fourier analysis on nilpotent Lie groups and the spectral calculus of
Rockland operators. In Section 3 we present the the proof of our main Theorem
1.6.

#### Notation and remarks

We write $A\lesssim B$ when $A$ is less than $B$ up to a constant and $A\asymp
B$ (or $A\sim B$) if $A\lesssim B$ and $B\lesssim A.$ For a bounded linear
operator $T:H\rightarrow H$ defined on a Hilbert space $(H,\|\cdot\|_{H}),$
the quantity $\|T\|_{\textnormal{op}}=\sup_{\|x\|_{H}\leq 1}\|T(x)\|_{H}$
denotes its operator norm. On a Lie group $G,$ we write $(x,y)\mapsto xy$ for
the group operation and $x\mapsto x^{-1}$ for the inversion mapping. Here
$e=e_{G}$ denotes the neutral element of $G.$ The Lie algebra of $G$ will be
denoted by $\mathfrak{g}.$ The Haar measure of a nilpotent Lie group $G$ will
be denoted by $dx.$ The corresponding $L^{p}$-spaces for the measure $dx$ will
be denoted by $L^{p}(G)=L^{p}(G,\,dx).$ The case $p=2$ is crucial in this
work. Since the group $G$ is nilpotent, in view of the identification
$G\cong\mathbb{R}^{n}$ we can define the Schwartz class
$\mathscr{S}(G)\cong\mathscr{S}(\mathbb{R}^{n}),$ and the corresponding space
of distributions $\mathscr{D}^{\prime}(G),$ see Section 2. A partial
differential operator $P:C^{\infty}(G)\rightarrow\mathscr{D}^{\prime}(G)$ is
(globally) hypoelliptic, if the equation $Pu=f\in C^{\infty}(G)$ with
$u\in\mathscr{D}^{\prime}(G)$ imply that $u\in C^{\infty}(G),$ see Hörmander
[13]. The class of hypoelliptic differential operators of interest for this
work is the family of Rockland operators, see Section 2.

##  2\. Fourier analysis on nilpotent Lie groups and Rockland operators

In this section, we present a consistent background of the Fourier analysis on
graded Lie groups and on the spectral calculus of Rockland operators. We
follow the notation related to the quantisation on graded Lie groups as in
[10]. We introduce it as follows.

####  2.0.1. Nilpotent Lie groups

Let us consider a Lie group $G$ and let $(\mathfrak{g},[\cdot,\cdot])$ be its
Lie algebra. Let us use the notation $\textnormal{ad}X(Y):=[X,Y]$ for the
adjoint action
$\textnormal{ad}:\mathfrak{g}\rightarrow\textnormal{End}(\mathfrak{g}).$ The
Lie algebra $\mathfrak{g}$ is nilpotent, if for each $X\in\mathfrak{g},$
$(\textnormal{ad}X)^{k}=0$ for some $k\in\mathbb{N}.$ A nilpotent Lie group
$G$ is a Lie group whose Lie algebra $\mathfrak{g}\cong T_{e}G$ is nilpotent.
Here we recall that the linear adjoint action on the Lie algebra
$\mathfrak{g}$ of $G$ is denoted by
$\textnormal{Ad}:G\rightarrow\textnormal{GL}(\mathfrak{g})$ and defined as
follows. Each element $g$ of $G$ induces an inner automorphism of the Lie
group $G$ by the formula $\textnormal{I}(g):x\in G\mapsto gxg^{-1}.$ Its
differential $\textnormal{Ad}(g):\mathfrak{g}\rightarrow\mathfrak{g}$ gives an
automorphism of the Lie algebra $\mathfrak{g}.$ The resulting representation
$\textnormal{Ad}:G\rightarrow\textnormal{GL}(\mathfrak{g})$ is called the
adjoint representation of $G.$ The differential of the adjoint representation
$\textnormal{Ad}:G\rightarrow\textnormal{GL}(\mathfrak{g})$ gives rise to to
the linear representation
$\textnormal{ad}:\mathfrak{g}\rightarrow\textnormal{End}(\mathfrak{g}),$
defined above as $\textnormal{ad}X(Y)=[X,Y].$

If $G$ is a connected and simply connected nilpotent Lie group then we have
that $\textnormal{exp}:\mathfrak{g}\rightarrow G$ is an analytic
diffeomorphism. Among other things this diffeomorphism allows us to identify
$G\cong\mathfrak{g}$ (and then
$\mathfrak{g}\cong\mathbb{R}^{\dim(\mathfrak{g})}$) and define the Schwartz
class $\mathscr{S}(G)\cong\mathscr{S}(\mathfrak{g})$ and the space of
distributions $\mathscr{D}^{\prime}(G)$ using the exponential mapping.

####  2.0.2. The unitary dual $\widehat{G}$

Let $G$ be a simply connected nilpotent Lie group. We say that a mapping
$\pi:G\rightarrow\textnormal{End}(H_{\pi}),$ for some separable Hilbert space
$H_{\pi},$ is a continuous, unitary and irreducible representation of $G,$ if
it satisfies the algebraic properties (i) and (iii), and the analytic property
(ii) below:

  * (i)

$\pi\in\textnormal{Hom}(G,\textnormal{U}(H_{\pi})),$ i.e.
$\pi(xy)=\pi(x)\pi(y)$ and for the adjoint of $\pi(x),$
$\pi(x)^{*}=\pi(x^{-1}),$ for every $x,y\in G.$

  * (ii)

The map $(x,v)\mapsto\pi(x)v,$ from $G\times H_{\pi}$ into $H_{\pi}$ is
continuous.

  * (iii)

For every $x\in G,$ and $W_{\pi}\subset H_{\pi},$ if $\pi(x)W_{\pi}\subset
W_{\pi},$ then $W_{\pi}=H_{\pi}$ or $W_{\pi}=\\{0\\}.$

Two unitary representations
$\pi\in\textnormal{Hom}(G,\textnormal{U}(H_{\pi}))$ and
$\eta\in\textnormal{Hom}(G,\textnormal{U}(H_{\eta}))$ are equivalent if there
exists a bounded linear mapping $T:H_{\pi}\rightarrow H_{\eta}$ such that for
any $x\in G,$ $T\pi(x)=\eta(x)T.$ The mapping $T$ is called an intertwining
operator between $\pi$ and $\eta.$ The relation $\sim$ on the set of unitary
and irreducible representations $\textnormal{Rep}(G)$ defined by:
$\pi\sim\eta$ if and only if $\pi$ and $\eta$ are equivalent representations,
is an equivalence relation. The quotient set
$\widehat{G}:={\textnormal{Rep}(G)}/{\sim},$ is called the unitary dual of
$G.$

####  2.0.3. Plancherel theorem on nilpotent Lie groups

The group Fourier transform of $f\in L^{1}(G),$ is defined by

$f\mapsto\mathscr{F}_{G}(f)\equiv\widehat{f},\,\quad\widehat{f}(\pi):=\smallint\limits_{G}f(x)\pi(x)^{*}dx:H_{\pi}\rightarrow
H_{\pi},\quad\forall\pi\in\widehat{G}.$

The Schwartz space on the dual $\widehat{G}$ is defined by the image under the
Fourier transform of the Schwartz space $\mathscr{S}(G),$ that is
$\mathscr{F}_{G}:\mathscr{S}(G)\rightarrow\mathscr{S}(\widehat{G}):=\mathscr{F}_{G}(\mathscr{S}(G)).$
If we identify a representation $\pi$ with its equivalence class,
$[\pi]=\\{\pi^{\prime}:\pi\sim\pi^{\prime}\\}$, for every $\pi\in\widehat{G},$
the Kirillov trace character $\Theta_{\pi}$ defined by (see Kirillov [14])

$[\Theta_{\pi},f]:=\textnormal{Tr}(\widehat{f}(\pi)),$

is a tempered distribution on $\mathscr{S}(G).$ The Fourier inversion formula
can be written in terms of Kirillov’s character as follows

$\forall f\in L^{1}(G)\cap
L^{2}(G),\,\,f(x)=\smallint\limits_{\widehat{G}}\textnormal{Tr}[\pi(x)\widehat{f}(\pi)]d\pi=:\mathscr{F}_{G}^{-1}[\widehat{f}\,](x),\,\,x\in
G.$

Here, $d\pi$ denotes the Plancherel measure on $\widehat{G}.$ The
$L^{2}$-space on the dual is defined by the completion of $\mathscr{S}(G)$
with respect to the norm

$\|\sigma\|_{L^{2}(\widehat{G})}:=\left(\smallint\limits_{G}\|\sigma(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}},\,\,\sigma(\pi)\in\mathscr{S}(\widehat{G}),$
(2.1)

where $\|\cdot\|_{\textnormal{HS}}$ denotes the Hilbert-Schmidt norm of
operators on every representation space. The corresponding inner product on
$L^{2}(\widehat{G})$ is given by

$(\sigma,\tau)_{L^{2}(\widehat{G})}:=\smallint\limits_{G}\textnormal{Tr}[\sigma(\pi)\tau(\pi)^{*}]d\pi,\,\,\sigma,\tau\in
L^{2}(\widehat{G}),$ (2.2)

where the notation $\tau(\pi)^{*}$ indicates the adjoint operator. Then, the
Plancherel theorem says that
$\|f\|_{L^{2}(G)}=\|\widehat{f}\|_{L^{2}(\widehat{G})}$ for all $f\in
L^{2}(G).$

An important subspace of each representation space $H_{\pi}$ is the one
determined by the set of smooth vector $H_{\pi}^{\infty},$ that is, the space
of vectors $v\in{H}_{\pi}$ such that the function $x\mapsto\pi(x)v,$
$x\in\widehat{G},$ is smooth. The space of smooth vectors $H_{\pi}^{\infty}$
is a dense sub-space of ${H}_{\pi},$ see [10, Page 42]. Finally, we record
that the group convolution $f\ast g,$ of two functions, namely, the mapping
defined via $f\ast g(x)=\smallint_{G}f(xy^{-1})g(y)dy,$ $f,g\in L^{1}(G),$
under suitable conditions satisfies the identity
$\widehat{g}(\pi)\widehat{f}(\pi)=\mathscr{F}_{G}[f\ast g](\pi).$

####  2.0.4. Rockland operators on graded Lie groups

A connected, simply connected nilpotent Lie group $G$ is graded if its Lie
algebra $\mathfrak{g}$ may be decomposed as the direct sum of subspaces
$\mathfrak{g}=\mathfrak{g}_{1}\oplus\mathfrak{g}_{2}\oplus\cdots\oplus\mathfrak{g}_{s},$
such that the following bracket conditions are satisfied:
$[\mathfrak{g}_{i},\mathfrak{g}_{j}]\subset\mathfrak{g}_{i+j},$ where
$\mathfrak{g}_{i+j}=\\{0\\}$ if $i+j\geq s+1.$ Each graded Lie group is
nilpotent and there is a class of homogeneous operators that classifies the
family of graded Lie groups among the family of nilpotent Lie groups, namely,
the family of (left-invariant hypoelliptic partial differential operators,
also called) Rockland operators. We introduce this family in what follows.
Assume the Lie algebra $\mathfrak{g}$ to be endowed with a family of dilations
$\\{D_{r}\\}_{r>0},$ compatible with its graduation, namely, for every $r>0,$
$D_{r}\in\textnormal{End}(\mathfrak{g}),$ and $D_{r}$ is a mapping of the form
$D_{r}=\textnormal{Exp}(\ln(r)A),$ for some diagonalisable linear operator
$A\equiv\textnormal{diag}[\nu_{1},\cdots,\nu_{n}]:\mathfrak{g}\rightarrow\mathfrak{g}.$
We call the eigenvalues of the matrix $A,$ $\nu_{1},\nu_{2},\cdots,\nu_{n},$
the dilations weights or weights of $G$. The homogeneous dimension of $G$ is
given by

$Q=\textnormal{Tr}(A)=\nu_{1}+\cdots+\nu_{n}.$

Note that the identification $G\cong\mathfrak{g}$ induced by the exponential
coordinates allows us to define a family of dilations on $G$ that we denote
also by $\\{D_{r}\\}_{r>0}.$ We write $D_{r}(x)=r\cdot x,$ $\forall x\in G,$
to simplify the notation. Then, we record that a continuous linear operator
$T:C^{\infty}(G)\rightarrow C^{\infty}(G)$ is homogeneous of degree
$\nu_{T}\in\mathbb{C}$ if, for every $r>0,$ the following equality holds

$T(f\circ D_{r})=r^{\nu_{T}}(Tf)\circ D_{r},\,\forall f\in
C^{\infty}(G),\,\forall r>0.$

Let us consider $\pi\in\widehat{G}.$ We denote by ${H}_{\pi}^{\infty}$ the set
of smooth vectors, also called Gårding vectors, formed by those
$v\in{H}_{\pi},$ such that the function $x\mapsto\pi(x)v,$ $x\in\widehat{G},$
is smooth. Then, a Rockland operator is a left-invariant partial differential
operator

$\mathcal{R}=\sum_{[\alpha]=\nu}a_{\alpha}X^{\alpha}:C^{\infty}(G)\rightarrow
C^{\infty}(G),\quad[\alpha]:=\sum_{i=1}^{n}\alpha_{i}\nu_{i},$

which is homogeneous of positive degree $\nu=\nu_{\mathcal{R}}$ and such that,
it satisfies the Rockland condition, namely, for every unitary irreducible
non-trivial representation $\pi\in\widehat{G},$ its symbol $\pi(\mathcal{R})$
defined via the Fourier inversion formula by

$\mathcal{R}f(x)=\smallint\limits_{\widehat{G}}\textnormal{Tr}[\pi(x)\pi(\mathcal{R})\widehat{f}(\pi)]d\pi,\,\,x\in
G,$ (2.3)

is injective on ${H}_{\pi}^{\infty}.$ Note that
$\sigma_{\mathcal{R}}(\pi)=\pi(\mathcal{R})$ coincides with the infinitesimal
representation of $\mathcal{R}$ as an element of the universal enveloping
algebra $\mathfrak{U}(\mathfrak{g})$.

If the Rockland operator $\mathcal{R}$ is symmetric, then $\mathcal{R}$ and
$\pi(\mathcal{R})$ admit self-adjoint extensions on $L^{2}(G)$ and
${H}_{\pi},$ respectively. Now if we preserve the same notation for their
self-adjoint extensions and we denote by $E_{\mathcal{R}}$ and
$E_{\pi(\mathcal{R})}$ their spectral measures, we will denote by

$\psi(\mathcal{R})=\smallint\limits_{-\infty}^{\infty}\psi(\lambda)dE_{\mathcal{R}}(\lambda),\,\,\,\textnormal{and}\,\,\,\pi(\psi(\mathcal{R}))\equiv\psi(\pi(\mathcal{R}))=\smallint\limits_{-\infty}^{\infty}\psi(\lambda)dE_{\pi(\mathcal{R})}(\lambda),$
(2.4)

the functions defined by the functional calculus. In general, we will reserve
the notation $\\{dE_{A}(\lambda)\\}_{0\leq\lambda<\infty}$ for the spectral
measure associated with a positive and self-adjoint operator $A$ on a Hilbert
space $H.$ Moreover if $f\in L^{\infty}(\mathbb{R}^{+}_{0}),$ we have that

$\|f(A)\|_{\textnormal{op}}=\left\|\smallint_{0}^{\infty}f(\lambda)dE_{A}(\lambda)\right\|_{\textnormal{op}}\leq\|f\|_{L^{\infty}(\mathbb{R}^{+}_{0})}.$
(2.5)

##  3\. Estimation of maximal functions using the Fourier transform

In this section we present the proof of Theorem 1.6. Before presenting our
main result we introduce an additional notation and recall some fundamental
facts. For every $\pi\in\widehat{G},$ let us define for any $r>0,$ and any
$x\in G,$ $(r\cdot\pi)(x):=\pi(r\cdot x).$ Then, for each test function $f\in
C^{\infty}_{0}(\mathbb{R}^{+}_{0}),$
$f(r\cdot\pi(\mathcal{R}))=f({r^{\nu}\pi(\mathcal{R})}),$ see Lemma 4.3 of
[10]. We use the notation

$K_{t}:=t^{-Q}K(t^{-1}\cdot),\quad t>0,$ (3.1)

and the following identity for the Fourier transforms $\widehat{K}_{t}$ and
$\widehat{K},$ of $K_{t}$ and $K,$ respectively,

$\widehat{K}_{t}(\pi)=\smallint\limits_{G}t^{-Q}K(t^{-1}\cdot
x)\pi(x)^{*}dx=\smallint\limits_{G}K(y)\pi(t\cdot
y)^{*}dy=\widehat{K}(t\cdot\pi),$ (3.2)

always that $K\in L^{1}(G).$ In (3.2) we have applied the changes of variables
$t^{-1}\cdot x\mapsto y,$ and we have used the new volume element
$dy=t^{-Q}dx.$ We also note that the mapping
$\frac{d}{ds}\\{\widehat{K}(s\cdot\pi)\\},$ can be defined as a densely
defined operator on each representation space $H_{\pi},$ taking the set of
smooth vectors $H_{\pi}^{\infty}$ as its domain, see e.g. [7, Section 3].

######  Remark 3.1.

Note that $\alpha_{j}=\alpha_{j}^{(\eta)}$ and $\beta_{j}^{(\eta)}$ defined by

$\alpha_{j}=\sup_{\pi\in\widehat{G}}\|\widehat{K}(\pi)\eta_{j}(\pi(\mathcal{R}))\|_{\textnormal{op}},\quad\beta_{j}=\sup_{s>0;\pi\in\widehat{G}}\|s\frac{d}{ds}\\{\widehat{K}(s\cdot\pi)\\}\eta_{j}((s\cdot\pi)(\mathcal{R}))\|_{\textnormal{op}},$
(3.3)

initially are quantities depending on $\eta.$ However, if we consider $\phi$
being a positive and continuous piecewise function on $\mathbb{R},$ supported
in a closed interval $I=[c,d],$ and generating a partition of unity
$\sum_{j\in\mathbb{Z}}\phi_{j}(t)=1,\,\phi_{j}(t)=\phi(2^{-(j+1)}t),\,t>0,$
and we define

$\alpha_{j}^{(\phi)}=\sup_{\pi\in\widehat{G}}\|\widehat{K}(\pi)\phi_{j}(\pi(\mathcal{R}))\|_{\textnormal{op}},\quad\beta_{j}^{(\phi)}=\sup_{s>0;\pi\in\widehat{G}}\|s\frac{d}{ds}\\{\widehat{K}(s\cdot\pi)\\}\phi_{j}((s\cdot\pi)(\mathcal{R}))\|_{\textnormal{op}},$
(3.4)

we have that $\alpha_{j}^{(\phi)}\asymp\alpha_{j}^{(\eta)},$
$\beta_{j}^{(\phi)}\asymp\beta_{j}^{(\eta)}.$ So, in view of these estimates
for the constants $\alpha_{j}^{(\eta)},\beta_{j}^{(\eta)},$ we simplify the
notation and we just write $\alpha_{j},\beta_{j},$ as in (1.10), for any
$j\in\mathbb{Z}$. We continue this discussion, namely, the proof of the
previous fact in Remark 3.2.

###### Proof of Theorem 1.6.

We start the proof by considering the partition of unity for the spectrum of
the Rockland operator $\mathcal{R}$ generated by $\eta\in
C^{\infty}_{0}(\mathbb{R}),$

$\forall\lambda\in(0,\infty),\,\sum_{j=-\infty}^{\infty}\eta(2^{j\nu}\lambda)=1.$
(3.5)

We have that $\textnormal{supp}(\eta)\subset[1/2^{\nu},2^{\nu}].$ As a
consequence of (3.5), the spectral calculus of Rockland operators implies that
$\sum_{j=-\infty}^{\infty}\eta(2^{j\nu}\mathcal{R})=I,$ as well as the
convergence of
$\sum_{j=-\infty}^{\infty}\eta(2^{j\nu}\mathcal{R})\delta=\delta,$ in
$\mathscr{S}^{\prime}(G)$ to the Dirac distribution $\delta.$ By defining

$\eta_{j}(\mathcal{R})=\smallint_{0}^{\infty}\eta_{j}(\lambda)dE_{\mathcal{R}}(\lambda),\,\,\eta_{j}(\lambda)=\eta(2^{-(j+1)\nu}\lambda),\,\textnormal{supp}(\eta_{j})\subset[2^{j\nu},2^{(j+2)\nu}],$
(3.6)

note that

$\widehat{\eta_{j}(\mathcal{R})\delta}(\pi)=\eta((2^{-(j+1)}\cdot\pi)(\mathcal{R})),$
(3.7)

and with $k_{j}$ defined by

$\widehat{k}_{j}(\pi)=\widehat{K}(\pi)\eta_{j}(\pi(\mathcal{R}))=\widehat{K}(\pi)\eta(2^{-(j+1)}\cdot\pi)(\mathcal{R}),$
(3.8)

we have that

$\displaystyle\|\sup_{t>0}|f\ast
k_{t}|\|_{L^{2}(G)}\leq\sum_{j\in\mathbb{Z}}\|\sup_{t>0}|f\ast(k_{j})_{t}|\|_{L^{2}(G)}.$

Let us fix $j\in\mathbb{Z}.$ We have that

$\sup_{t>0}|f\ast(k_{j})_{t}(x)|\leq\left(\sum_{\ell\in\mathbb{Z}}\sup_{2^{\ell}\leq
t<2^{\ell+1}}|f\ast(k_{j})_{t}(x)|^{2}\right)^{\frac{1}{2}}.$ (3.9)

In our further analysis we are going to estimate the $L^{2}$-norm of the right
hand side of (3.9) and later we make the summation of these norms over
$j\in\mathbb{Z}.$ In view of the Fourier inversion formula can write

$\displaystyle
f\ast(k_{j})_{t}(x)=\smallint_{\widehat{G}}\textnormal{Tr}[\pi(x)\widehat{k}_{j}(t\cdot\pi)\widehat{f}(\pi)]=\smallint_{\widehat{G}}\textnormal{Tr}[\pi(x)\widehat{K}(\pi)\eta([2^{-(j+1)}\cdot
t\cdot\pi](\mathcal{R}))\widehat{f}(\pi)].$

Fix $t>0$ such that $2^{\ell}\leq t<2^{\ell+1}.$ The spectral calculus for
Rockland operators allows us to write

$\displaystyle\eta([2^{-(j+1)}\cdot t\cdot\pi](\mathcal{R}))\widehat{f}(\pi)$
$\displaystyle=\smallint\eta(2^{-(j+1)\nu}t^{\nu}\lambda)dE_{\pi(\mathcal{R})}(\lambda)$
$\displaystyle=\smallint_{2^{-\nu}\leq 2^{-(j+1)\nu}t^{\nu}\lambda\leq
2^{\nu}}\eta(2^{-(j+1)\nu}t^{\nu}\lambda)dE_{\pi(\mathcal{R})}(\lambda)$
$\displaystyle=\smallint_{2^{(j-\ell-2)\nu}\leq\lambda\leq
2^{(j-\ell+2)\nu}}\eta(2^{-(j+1)\nu}t^{\nu}\lambda)dE_{\pi(\mathcal{R})}(\lambda).$

For $m\in\mathbb{Z},$ let us denote

$E_{m}(\pi):=E_{\pi(\mathcal{R})}[2^{(m-2)\nu},2^{(m+2)\nu}].$ (3.10)

Observe that with $m=j-\ell,$

$\displaystyle E_{m}(\pi)\eta([2^{-(j+1)}\cdot
t\cdot\pi](\mathcal{R}))\widehat{f}(\pi)$
$\displaystyle=E_{\pi(\mathcal{R})}[2^{(m-2)\nu},2^{(m+2)\nu}]\smallint_{2^{(j-\ell-2)\nu}\leq\lambda\leq
2^{(j-\ell+2)\nu}}\eta(2^{-(j+1)\nu}t^{\nu}\lambda)dE_{\pi(\mathcal{R})}(\lambda)\widehat{f}(\pi)$
$\displaystyle=\smallint_{2^{(j-\ell-2)\nu}\leq\lambda\leq
2^{(j-\ell+2)\nu}}\eta(2^{-(j+1)\nu}t^{\nu}\lambda)dE_{\pi(\mathcal{R})}(\lambda)E_{\pi(\mathcal{R})}[2^{(m-2)\nu},2^{(m+2)\nu}]\widehat{f}(\pi)$
$\displaystyle=\eta([2^{-(j+1)}\cdot
t\cdot\pi](\mathcal{R}))E_{m}(\pi)\widehat{f}(\pi).$

In consequence,

$\displaystyle
f\ast(k_{j})_{t}(x)=\smallint_{\widehat{G}}\textnormal{Tr}[\pi(x)\widehat{k}_{j}(t\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]d\pi=\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)].$

Let us fix an integer $A_{j}\geq 1,$ and for a fixed $\ell\in\mathbb{Z},$ let
$\\{t_{\tau}\\}_{\tau\leq A_{j}}$ be a $(2^{\ell}A_{j}^{-1})$-net in the
interval $I_{\ell}=[2^{\ell},2^{\ell+1}].$ Le us assume that this net also
includes the point $t\in[2^{\ell},2^{\ell+1}].$ By a
$(2^{\ell}A_{j}^{-1})$-net of $I_{\ell},$ we mean a partition
$P_{\ell}=\\{t_{\tau}\\}_{\tau\leq A_{j}}$ of $I_{\ell}$ such that the
following properties are satisfied:

  * •

the number of elements of $P_{\ell},$ namely, its cardinality is $A_{j}.$

  * •

The partition $P_{\ell}=\\{t_{\tau}\\}_{\tau\leq A_{j}}$ is ordered in such a
way that two consecutive elements $t_{\tau},t_{\tau+1},$ satisfy the distance
condition $|t_{\tau}-t_{\tau+1}|\leq 2^{\ell}A_{j}^{-1}.$

Additionally,

  * •

without loss of generality we have assume that $t=t_{\tau_{0}}$ also belongs
to the partition $P_{\ell}.$

The fundamental theorem of calculus allows us to write

$\displaystyle f\ast(k_{j})_{t}(x)$
$\displaystyle=\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]$
$\displaystyle=\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t_{\tau_{0}-1}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]+\smallint_{t_{\tau_{0}-1}}^{t_{\tau_{0}}}\frac{d}{ds}\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]ds.$

In view of triangle inequality, we have that

$\displaystyle|f\ast(k_{j})_{t}(x)|$
$\displaystyle\leq|\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t_{\tau_{0}-1}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]|+\smallint_{t_{\tau_{0}-1}}^{t_{\tau_{0}}}|\frac{d}{ds}\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]|ds$
$\displaystyle\leq|\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t_{\tau_{0}-1}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]|+\smallint_{t_{\tau_{0}-1}}^{t_{\tau_{0}}+1}|\frac{d}{ds}\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]|ds.$

In consequence, summing over the points of the partition
$\tilde{P_{\ell}}=P_{\ell}\setminus\\{t\\},$ (and enumerating these points
again in order to keep the notation $t_{\tau},$ $\tau\leq A_{j},$ for the
points in $\tilde{P_{\ell}}$) we have that

$\displaystyle|f\ast(k_{j})_{t}(x)|$
$\displaystyle\leq\left(\sum_{\tau}|\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]|^{2}+\left[\smallint_{t_{\tau}}^{t_{\tau}+1}|\frac{d}{ds}\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]|ds\right]^{2}\right)^{\frac{1}{2}}.$

Let us observe that the right hand side of the previous inequality does not
depend on $t.$ In consequence

$\displaystyle\sup_{2^{\ell}\leq t<2^{\ell+1}}|f\ast(k_{j})_{t}(x)|$
$\displaystyle\leq\left(\sum_{\tau}|\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)](x)|^{2}+\left[\smallint_{t_{\tau}}^{t_{\tau}+1}|\frac{d}{ds}\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)](x)|ds\right]^{2}\right)^{\frac{1}{2}}.$

Now, by applying both, Plancherel theorem and Minkowski integral inequality
observe that

$\displaystyle\left\|\sup_{2^{\ell}\leq
t<2^{\ell+1}}|f\ast(k_{j})_{t}(x)|\right\|_{L^{2}(G)}^{2}$
$\displaystyle\leq\smallint_{G}\sum_{\tau}|\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)](x)|^{2}$
$\displaystyle\hskip
56.9055pt+\left[\smallint_{t_{\tau}}^{t_{\tau}+1}|\frac{d}{ds}\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)](x)|ds\right]^{2}dx$
$\displaystyle=\sum_{\tau}\smallint_{G}|\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)](x)|^{2}dx$
$\displaystyle\hskip
56.9055pt+\smallint_{G}\left[\smallint_{t_{\tau}}^{t_{\tau}+1}|\frac{d}{ds}\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)](x)|ds\right]^{2}dx$
$\displaystyle\leq\sum_{\tau}\smallint_{G}|\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)](x)|^{2}dx$
$\displaystyle\hskip
56.9055pt+\left[\smallint_{t_{\tau}}^{t_{\tau}+1}\left(\smallint_{G}|\frac{d}{ds}\mathscr{F}_{G}^{-1}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)](x)|^{2}dx\right)^{\frac{1}{2}}ds\right]^{2}$
$\displaystyle=\sum_{\tau}\smallint_{\widehat{G}}\|[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)\|_{\textnormal{HS}}^{2}dx$
$\displaystyle\hskip
56.9055pt+\left[\smallint_{t_{\tau}}^{t_{\tau}+1}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}ds\right]^{2}$
$\displaystyle\leq\sum_{\tau}\smallint_{\widehat{G}}\|[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)\|_{\textnormal{HS}}^{2}d\pi$
$\displaystyle\hskip
56.9055pt+\left[\smallint_{t_{\tau}}^{t_{\tau}+1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}ds^{\prime}\right]^{2}$
$\displaystyle\leq\sum_{\tau}\smallint_{\widehat{G}}\|[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)\|_{\textnormal{HS}}^{2}d\pi$
$\displaystyle\hskip 56.9055pt+\left[2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}\right]^{2}$
$\displaystyle=A+B.$

We shall estimate the first term $A.$ In order to simplify the notation let us
define $f^{(m)}=\mathscr{F}_{G}^{-1}[E_{m}(\pi)\widehat{f}(\pi)],$ for any
$m\in\mathbb{Z}.$ Note that

$\displaystyle A$
$\displaystyle=\smallint_{\widehat{G}}\|[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)\|_{\textnormal{HS}}^{2}d\pi$
$\displaystyle\leq\sup_{\pi^{\prime}\in\widehat{G}}\|\widehat{k}_{j}(\pi^{\prime})\|_{\textnormal{op}}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}$
$\displaystyle=\sup_{\pi^{\prime}\in\widehat{G}}\|K(\pi^{\prime})\eta_{j}(\pi^{\prime}(\mathcal{R}))\|_{\textnormal{op}}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}$
$\displaystyle=\alpha_{j}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}.$

In consequence

$\displaystyle
A=\smallint_{\widehat{G}}\|[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)\|_{\textnormal{HS}}^{2}d\pi\leq\alpha_{j}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}.$

On the other hand, let us estimate the term

$\sqrt{B}:=2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}.$
(3.11)

Since

$\frac{d}{ds}[\widehat{k}_{j}(s\cdot\pi)]=\frac{d}{ds}\\{\widehat{K}(s\cdot\pi)\\}\eta_{j}((s\cdot\pi)(\mathcal{R}))+\widehat{K}(s\cdot\pi)\frac{d}{ds}\\{\eta_{j}((s\cdot\pi)(\mathcal{R}))\\}$

we have that

$\displaystyle\left(\smallint_{\widehat{G}}\|\frac{d}{ds}[\widehat{k}_{j}(s\cdot\pi)]\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}\leq\left(\smallint_{\widehat{G}}\|\frac{d}{ds}\widehat{K}(s\cdot\pi)\eta_{j}((s\cdot\pi)(\mathcal{R}))\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\hskip
56.9055pt+\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\frac{d}{ds}\eta_{j}((s\cdot\pi)(\mathcal{R}))\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}.$

Let $\kappa(t)$ be defined by $\kappa(t)=\eta(t^{\nu}).$ Note that

$\displaystyle\eta_{j}((s\cdot\pi)(\mathcal{R}))$
$\displaystyle=\smallint_{0}^{\infty}\eta(2^{-j\nu}s^{\nu}\lambda)dE_{\pi(\mathcal{R})}(\lambda)=\smallint_{0}^{\infty}\kappa(2^{-j}s\lambda^{\frac{1}{\nu}})dE_{\pi(\mathcal{R})}(\lambda)$
$\displaystyle=\smallint_{0}^{\infty}\kappa(2^{-j}st)dE_{\pi(\mathcal{R})^{\frac{1}{\nu}}}(t).$

The function $\kappa$ allows us to avoid unbounded terms when computing the
derivative $\frac{d}{ds}\eta_{j}((s\cdot\pi)(\mathcal{R})).$ To estimate it in
an efficient way let us make use of the functional calculus for Rockland
operators. Indeed,

$\displaystyle\frac{d}{ds}\eta_{j}((s\cdot\pi)(\mathcal{R}))=\frac{d}{ds}\smallint_{0}^{\infty}\kappa(2^{-j}st)dE_{\pi(\mathcal{R})^{\frac{1}{\nu}}}(t)=\smallint_{0}^{\infty}\kappa^{\prime}(2^{-j}st)2^{-j}tdE_{\pi(\mathcal{R})^{\frac{1}{\nu}}}(t)$

Now, we have that

$\displaystyle\sqrt{B}$ $\displaystyle=2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\lesssim 2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}\widehat{K}(s\cdot\pi)\eta_{j}((s\cdot\pi)(\mathcal{R}))\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle+2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\frac{d}{ds}\eta_{j}((s\cdot\pi)(\mathcal{R}))\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\lesssim 2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}\widehat{K}(s\cdot\pi)\eta_{j}((s\cdot\pi)(\mathcal{R}))\|_{\textnormal{op}}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle+2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\frac{d}{ds}\eta_{j}((s\cdot\pi)(\mathcal{R}))\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle=I+II.$

In order to estimate $I$, observe that

$\displaystyle I$ $\displaystyle=2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}\widehat{K}(s\cdot\pi)\eta_{j}((s\cdot\pi)(\mathcal{R}))\|_{\textnormal{op}}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\asymp 2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}s^{-1}\left(\smallint_{\widehat{G}}\|s\frac{d}{ds}\widehat{K}(s\cdot\pi)\eta_{j}((s\cdot\pi)(\mathcal{R}))\|_{\textnormal{op}}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\lesssim 2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}s^{-1}\left(\smallint_{\widehat{G}}\sup_{s^{\prime}>0;\pi\in\widehat{G}}\|s^{\prime}\frac{d}{ds^{\prime}}\widehat{K}(s^{\prime}\cdot\pi)\eta_{j}((s^{\prime}\cdot\pi)(\mathcal{R}))\|_{\textnormal{op}}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\asymp
2^{\ell}A_{j}^{-1}2^{-\ell}\left(\smallint_{\widehat{G}}\beta_{j}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle=A_{j}^{-1}\beta_{j}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}.$

For the analysis of the remainder term $II,$ consider the test function
$\rho(\lambda):=\kappa^{\prime}(\lambda^{\frac{1}{\nu}})\lambda^{\frac{1}{\nu}},$
$\lambda>0.$ Note that $\rho(t^{\nu}):=\kappa^{\prime}(t)t,$ for all $t>0.$
Now, we proceed as follows

$\displaystyle 2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\frac{d}{ds}\eta_{j}((s\cdot\pi)(\mathcal{R}))\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle=2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\smallint_{0}^{\infty}\kappa^{\prime}(2^{-j}st)2^{-j}tdE_{\pi(\mathcal{R})^{\frac{1}{\nu}}}(t)\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\asymp A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\smallint_{0}^{\infty}\kappa^{\prime}(2^{-j}st)2^{-j}\times
s\times
tdE_{\pi(\mathcal{R})^{\frac{1}{\nu}}}(t)\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle=A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\smallint_{0}^{\infty}\rho(2^{-j\nu}s^{\nu}t^{\nu})dE_{\pi(\mathcal{R})^{\frac{1}{\nu}}}(t)\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle=A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\smallint_{0}^{\infty}\rho(2^{-j\nu}s^{\nu}t)dE_{\pi(\mathcal{R})}(t)\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle=A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\widehat{K}(s\cdot\pi)\rho(2^{-j}\cdot
s\cdot\pi(\mathcal{R}))\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\leq
A_{j}^{-1}\left(\smallint_{\widehat{G}}\sup_{\pi^{\prime}\in\widehat{G}}\|\widehat{K}(\pi^{\prime})\rho(2^{-j}\cdot\pi^{\prime}(\mathcal{R}))\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\leq
A_{j}^{-1}\left(\smallint_{\widehat{G}}\sup_{\pi^{\prime}\in\widehat{G}}\|\widehat{K}(\pi^{\prime})\rho(2^{-j}\cdot\pi^{\prime}(\mathcal{R}))\|_{\textnormal{op}}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle\asymp\alpha_{j}A_{j}^{-1}\left(\smallint_{\widehat{G}}\|\widehat{f}^{(j-\ell)}(\pi)\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}$
$\displaystyle=\alpha_{j}A_{j}^{-1}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}.$

All the analysis above allows us to write the estimates below

$\displaystyle\left\|\sup_{2^{\ell}\leq
t<2^{\ell+1}}|f\ast(k_{j})_{t}(x)|\right\|_{L^{2}(G)}^{2}$
$\displaystyle\leq\sum_{\tau}\smallint_{\widehat{G}}\|[\widehat{k}_{j}(t_{\tau}\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)\|_{\textnormal{HS}}^{2}d\pi$
$\displaystyle+\left[2^{\ell}A_{j}^{-1}\sup_{s\sim
2^{\ell}}\left(\smallint_{\widehat{G}}\|\frac{d}{ds}[\widehat{k}_{j}(s\cdot\pi)E_{j-\ell}(\pi)\widehat{f}(\pi)]\|_{\textnormal{HS}}^{2}d\pi\right)^{\frac{1}{2}}\right]^{2}$
$\displaystyle\leq\sum_{\tau}\alpha_{j}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}+\left[A_{j}^{-1}\beta_{j}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}+\alpha_{j}A_{j}^{-1}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}\right]^{2}$
$\displaystyle\lesssim
A_{j}\alpha_{j}^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}+A_{j}A_{j}^{-2}(\beta_{j}+\alpha_{j})^{2}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}$
$\displaystyle=(A_{j}\alpha_{j}^{2}+A_{j}^{-1}(\beta_{j}+\alpha_{j})^{2})\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}.$

Therefore, we have proved that

$\displaystyle\left\|\sup_{2^{\ell}\leq
t<2^{\ell+1}}|f\ast(k_{j})_{t}(x)|\right\|_{L^{2}(G)}\lesssim\sqrt{A_{j}\alpha_{j}^{2}+A_{j}^{-1}(\beta_{j}+\alpha_{j})^{2}}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}.$

Let us optimise the constant inside of the square root. It can be done by
choosing

$A_{j}:=\left[\alpha_{j}^{-1}(\beta_{j}+\alpha_{j})\right],$

to be the integer part of $\alpha_{j}^{-1}(\beta_{j}+\alpha_{j})>1.$ It
follows that $A_{j}\sim\alpha_{j}^{-1}(\beta_{j}+\alpha_{j}),$ and
consequently

$\sqrt{A_{j}\alpha_{j}^{2}+A_{j}^{-1}(\beta_{j}+\alpha_{j})^{2}}\sim\alpha_{j}^{\frac{1}{2}}(\beta_{j}+\alpha_{j})^{\frac{1}{2}}.$

Now, we can return to the estimate of the maximal function. Application of the
estimates above gives therefore

$\displaystyle\|\sup_{t>0}|f\ast k_{t}|\|_{L^{2}(G)}$
$\displaystyle\leq\sum_{j\in\mathbb{Z}}\|\sup_{t>0}|f\ast(k_{j})_{t}|\|_{L^{2}(G)}\leq\sum_{j\in\mathbb{Z}}\left\|\left(\sum_{\ell\in\mathbb{Z}}\sup_{2^{\ell}\leq
t<2^{\ell+1}}|f\ast(k_{j})_{t}|^{2}\right)^{\frac{1}{2}}\right\|_{L^{2}(G)}$
$\displaystyle=\sum_{j\in\mathbb{Z}}\left(\smallint_{G}\sum_{\ell\in\mathbb{Z}}\sup_{2^{\ell}\leq
t<2^{\ell+1}}|f\ast(k_{j})_{t}(x)|^{2}dx\right)^{\frac{1}{2}}$
$\displaystyle=\sum_{j\in\mathbb{Z}}\left(\sum_{\ell\in\mathbb{Z}}\smallint_{G}\sup_{2^{\ell}\leq
t<2^{\ell+1}}|f\ast(k_{j})_{t}(x)|^{2}dx\right)^{\frac{1}{2}}$
$\displaystyle\lesssim\sum_{j\in\mathbb{Z}}\left(\sum_{\ell\in\mathbb{Z}}\alpha_{j}(\alpha_{j}+\beta_{j})\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}\right)^{\frac{1}{2}}$
$\displaystyle=\sum_{j\in\mathbb{Z}}\alpha_{j}^{\frac{1}{2}}(\alpha_{j}+\beta_{j})^{\frac{1}{2}}\left(\sum_{\ell\in\mathbb{Z}}\|\widehat{f}^{(j-\ell)}(\pi)\|_{L^{2}(\widehat{G})}^{2}\right)^{\frac{1}{2}}$
$\displaystyle=\sum_{j\in\mathbb{Z}}\alpha_{j}^{\frac{1}{2}}(\alpha_{j}+\beta_{j})^{\frac{1}{2}}\left(\sum_{m\in\mathbb{Z}}\|\widehat{f}^{(m)}(\pi)\|_{L^{2}(\widehat{G})}^{2}\right)^{\frac{1}{2}}$
$\displaystyle\asymp\sum_{j\in\mathbb{Z}}\alpha_{j}^{\frac{1}{2}}(\alpha_{j}+\beta_{j})^{\frac{1}{2}}\|f\|_{L^{2}(G)},$

where in the last line we have use the almost orthogonality of the sequence
$\widehat{f}^{(m)}(\pi),$ $m\in\mathbb{Z},$ on $L^{2}(\widehat{G})$ and the
Plancherel theorem. The proof is complete. ∎

######  Remark 3.2.

We continue with the discussion of Remark 3.1. Indeed, note that there exists
$M=M(\phi,\eta)\in\mathbb{N},$ independent of $j\in\mathbb{Z},$ such that

$\displaystyle\|\widehat{K}(\pi)\eta_{j}(\pi(\mathcal{R}))\|_{\textnormal{op}}\leq\sum_{k=j-M}^{j+M}\|\widehat{K}(\pi)\phi_{k}(\pi(\mathcal{R}))\eta_{j}(\pi(\mathcal{R}))\|_{\textnormal{op}}\lesssim_{M}\sum_{k=j-M}^{j+M}\alpha_{k}^{(\phi)}\|\eta_{j}(\pi(\mathcal{R}))\|_{\textnormal{op}}.$

Since
$\|\eta_{j}(\pi(\mathcal{R}))\|_{\textnormal{op}}\leq\|\eta\|_{L^{\infty}},$
and certainly
$\alpha_{j}^{(\phi)}\asymp_{M}\sum_{k=j-M}^{j+M}\alpha_{k}^{(\phi)},$ we have
that $\alpha_{j}^{(\eta)}\lesssim\alpha_{j}^{(\phi)}.$ In the same way one can
prove that $\alpha_{j}^{(\phi)}\lesssim\alpha_{j}^{(\eta)}.$ In consequence
$\alpha_{j}^{(\phi)}\asymp\alpha_{j}^{(\eta)}.$ The same argument proves that
$\beta_{j}^{(\eta)}\asymp\beta_{j}^{(\phi)}.$

#### Conflict of interests statement - Data Availability Statements

The author states that there is no conflict of interest. Data sharing does not
apply to this article as no datasets were generated or analysed during the
current study.

### Acknowledgements

The author is currently a FWO fellow supported by the Research Foundation
Flanders FWO under the postdoctoral grant No 1204824N.

## References

  * [1] Bourgain, J. Estimations de certaines fonctions maximales. C. R. Acad. Sci. Paris Sér. I Math. 301(10), 499–502, (1985).
  * [2] Bourgain, J. On high dimensional maximal functions associated to convex bodies, Amer. J. Math., 108, 1467–1476, (1986).
  * [3] Bourgain, J. On the Hardy-Littlewood maximal function for the cube. Israel J. Math. 203, 275–293, (2014).
  * [4] Bourgain, J. Averages in the plane over convex curves and maximal operators, J. Anal. Math. 47, 69–85, (1986).
  * [5] Bourgain, J., Mirek, M., Stein, E. M., Wróbel, B. On dimension-free variational inequalities for averaging operators in Rd. Geom. Funct. Anal. 28(1), 58–99, (2018).
  * [6] Carbery, A. An almost-orthogonality principle with applications to maximal functions associated to convex bodies. Bull. Amer. Math. Soc. 14(2), 269–274, (1986).
  * [7] Cardona, D. Estimates for the full maximal function on graded Lie groups, submitted, arXiv:2401.07086.
  * [8] Cardona, D., Delgado, J., Ruzhansky, M. Boundedness of the dyadic maximal function on graded Lie groups, to appear in Quart. J. Math., arXiv:2301.08964.
  * [9] Corwin, L. J., Greenleaf, F. P. Representations of nilpotent Lie groups and their applications. Part I. Basic theory and examples. Cambridge Studies in Advanced Mathematics, 18. Cambridge University Press, Cambridge, 1990. viii+269 pp. ISBN: 0-521-36034-X
  * [10] Fischer V., Ruzhansky M. Quantization on nilpotent Lie groups, Progress in Mathematics, Vol. 314, Birkhauser, 2016. xiii+557pp.
  * [11] Folland, G. Stein, E. Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, N.J., 1982.
  * [12] Gromov, M., Milman, V. Brunn theorem and a concentration of volume of convex bodies, GAFA Seminar Notes, Tel-Aviv, (1983-1984).
  * [13] Hörmander, L. The Analysis of the linear partial differential operators. Vol. III. IV, Springer-Verlag, (1985).
  * [14] Kirillov, A.A. Unitary representations of nilpotent Lie groups. Russ. Math. Surv. 17, 57–110, (1962).
  * [15] Stein, E. M. Maximal functions. I. Spherical means, Proc. Natl. Acad. Sci. USA 73, 2174–2175, (1976).
  * [16] Stein, E. M. The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. 7, 359–376, (1982).
  * [17] Stein, E. M. The development of square functions in the work of A. Zygmund. Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), 2–30, Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983.
  * [18] Stein, E. M., Strömberg, J.-O. Behavior of maximal functions in Rn for large n. Ark. Mat. 21, no. 2, 259–269, (1983).


# Very- and ultra-high frequency electric field detection using high angular
momentum Rydberg states

Roger C. Brown<EMAIL_ADDRESS>Georgia Tech Research Institute, Atlanta,
Georgia 30332, USA  NIST 325 Broadway, Boulder, CO 80305  Baran Kayim  Michael
A. Viray  Georgia Tech Research Institute, Atlanta, Georgia 30332, USA
Abigail R. Perry  Georgia Tech Research Institute, Atlanta, Georgia 30332, USA
Quantinuum, 303 S. Technology Ct., Broomfield, CO 80021  Brian C. Sawyer
Robert Wyllie<EMAIL_ADDRESS>Georgia Tech Research Institute,
Atlanta, Georgia 30332, USA

###### Abstract

We demonstrate resonant detection of rf electric fields from 240 MHz to 900
MHz (very-high-frequency (VHF) to ultra-high-frequency (UHF)) using
electromagnetically induced transparency to measure orbital angular momentum
$L=3\rightarrow L^{\prime}=4$ Rydberg transitions. These Rydberg states are
accessible with three-photon infrared optical excitation. By resonantly
detecting rf in the electrically small regime, these states enable a new class
of atomic receivers. We find good agreement between measured spectra and
predictions of quantum defect theory for principal quantum numbers $n=45$ to
$70$. Using a super-hetrodyne detection setup, we measure the noise floor at
$n=50$ to be $13\,\mathrm{\mu V/m/\sqrt{Hz}}$. Additionally, we utilize data
and a numerical model incorporating a five-level master equation solution to
estimate the fundamental sensitivity limits of our system.

††preprint: APS/123-QED

##  I Introduction

00footnotetext:
Approved for public release; distribution is unlimited. Public Affairs release
approval #AFRL-2022-1980

Rydberg atoms have recently been used to measure radio frequency (rf) electric
field amplitude [1], polarization [2], phase [3, 4], and angle of arrival [5].
The detected field amplitude is traceable to fundamental atomic structure and
has led to an artifact-free paradigm in rf field calibrations [6, 7, 8, 9].
Since the rf field sensing region is defined by laser-atom interaction volume,
new opportunities in sub-wavelength rf field visualization have emerged [10,
11]. Temporal modulation of the detected rf field has resulted in
communications demonstrations of atomic reception using: amplitude modulation,
frequency modulation (AM, FM) [12, 13, 14, 15, 16], Binary Phase Shift Keying,
and Quadrature Amplitude Modulation [17]. A previous study demonstrated that
atomic receivers can operate non-resonantly in the electrically small regime
[18]. In this case, the data switching rate was taken to be equal to the
carrier frequency (DC to 30 MHz) and not resonant with nearby Rydberg
transition frequencies ($>10$ GHz). In order for atomic receivers to be
compatible with common broadcast technologies (AM FM radio, television [19]),
they must be able to operate with a tunable carrier frequency distinct from
the data rate. For example, U.S. UHF television channels 14 through 89 111pre
2020 F.C.C. spectrum re-allocation
https://www.ntia.doc.gov/legacy/osmhome/alloctbl/ allocmhz.html occupy a
frequency band from 470 MHz to 890 MHz (with 6 MHZ allocated per channel).

Among the attractive properties of so-called Rydberg receivers is the fact
that they are not subject to the Chu limit for electrically small antennas
[21, 22, 23]. This stems from the difference in the underlying physical
mechanism of rf reception between conducting antennas and atoms. The Chu limit
states that the bandwidth ($BW$) is constricted for electrically small passive
conductor antennas, where the characteristic radius of the antenna is less
than the wavelength of the rf field,
$\ell_{\mathrm{ant}}<\lambda_{\mathrm{rf}}$. Specifically

$\frac{BW_{\mathrm{Chu}}}{f_{0}}\lesssim\frac{(2\pi\ell_{\mathrm{ant}})^{3}}{\lambda_{\mathrm{rf}}^{3}},$
(1)

where $f_{0}=c/\lambda_{\mathrm{rf}}$ is the carrier frequency and $c$ is the
speed of light in vacuum. For example, for a lossless electrically small
classical antenna with $2\pi\ell_{\mathrm{ant}}/\lambda_{\mathrm{rf}}=0.5$,
the bandwidth is limited to $BW/f_{0}\lesssim 0.1$ [24].

In a typical Rydberg atomic receiver experiment [12, 13, 14, 15, 16, 17],
$f_{0}\approx 10$ to $40$ GHz, the $BW\lesssim 30$ MHz, and the optical path
length in the atomic vapor is $0.5$ cm $<\ell_{v}<10$ cm. To create an
electrically small atomic receiver able to surpass a Chu-limited antenna in
the $10$ to $40$ GHz range, atomic receiver bandwidth will need to be
increased about 100-fold while reducing the apparatus size from centimeters to
millimeters. However, at reduced carrier frequency, $\lambda_{\mathrm{rf}}$
becomes large such that $\ell_{v}\ll\lambda_{\mathrm{rf}}$. For example,
choosing $f_{0}=300$ MHz with typical atomic parameters ($BW=10$ MHz,
$\ell_{v}\approx 3$ cm $<\lambda_{\mathrm{rf}}=1$ m) enables an electrically
small receiver.

Here, we explore $nF_{7/2}\rightarrow nG_{9/2}$ Rydberg transitions in
rubidium vapor with resonances from 240 MHz to 900 MHz for VHF to UHF rf
detection. Two demonstrated approaches to reduce $f_{0}$ are: (i) to use non-
resonant detection [25, 26, 27, 28] and (ii) to increase the principal quantum
number, $n$, of resonant detection [1]. In non-resonant detection schemes, the
extension to low rf frequency requires that the amplitude of the detected
field be large enough (V/cm to kV/cm) to mix in nearby Rydberg states and can
require the interpretation and simulation of Floquet spectra [26, 27, 28]. To
maximize sensitivity to incident rf fields, we choose sensors where Autler-
Townes (AT) [29] splitting of Rydberg states is resonantly detected with
electromagnetically induced transparency (EIT) [30, 31]. Resonant AT-EIT
detection has been shown to sense rf fields with amplitudes near $\mu$V/cm at
carrier frequencies defined by the difference in energy between Rydberg states
[1, 32, 4]. Reducing $f_{0}$ requires optical excitation to higher Rydberg
states that are increasingly susceptible to perturbation from other effects,
e.g. DC Stark induced state mixing and long range Rydberg-Rydberg and Rydberg-
ground state collisions [33, 34, 35]. To our knowledge, the lowest published
AT EIT-detected rf signal is at 724 MHz with $n>130$ [7]. Higher angular
momentum $(nF_{7/2}\rightarrow nG_{9/2})$ Rydberg transitions are more than an
order of magnitude lower in energetic separation at a given principal quantum
number compared to lower angular momentum transitions (see Fig. 3). Using
these higher angular momentum states, we demonstrate resonant AT EIT rf
sensing with $f_{0}$ from 240 MHz to 900 MHz using $n=75$ to $n=40$. To
achieve $f_{0}=240$ MHz using the more familiar
$nP_{3/2}\rightarrow(n-1)D_{5/2}$ transitions would require excitation to
$n>200$.

This paper is structured as follows: In Sec. II we present our experimental
apparatus and present measurements of rf transitions between $nF\rightarrow
nG$. In Sec. III we present a numerical model and in Sec. IV use it to assess
the sensitivity of the system estimate the fundamental noise limits of our
data. Finally, in Sec. V, we conclude with an outlook for future work.

##  II Experiment

To access Rydberg states with orbital angular momentum, $L>3$, we use the
three photon ladder system shown in Figure 1 a. This ladder system has
previously been studied in a number of contexts including fundamental atomic
structure [36, 37], rf field calibration in the 100 GHz range using both EIT
and electromagnetically induced absorption [38], time domain signal reception
at 1.2 GHz [39], and quantum optics [40]. A variety of other multi-photon
Rydberg excitation pathways have been explored [41, 42, 43, 44, 45] however,
none of these have been used to optically couple to $L\geq 3$ states 222This
is often because it is either technologically or scientifically attractive for
the second leg of the ladder to connect $P\rightarrow S$ states. This ladder
system is appealing for a number of reasons. First, it utilizes transitions
accessible by diode lasers. The two initial steps are also accessible by
frequency-doubled telecom band fiber-lasers with linewidth below the Rydberg-
state linewidth. Second, the transition dipole moments between successive
ladder states are larger than in one- or two-photon Rydberg excitation
(increasing Rabi-excitation rates for fixed optical powers). Third, the near
degeneracy in optical frequency between the first two steps in optical
excitation allows access to a broader range of atomic velocity classes in a
vapor cell [38]333Access to the largest number of atomic velocity classes
occurs when the 780 nm beam counter propagates with the 776 nm and 1260 nm
beams and with small frequency offset from $5D_{5/2}$ state, creating stronger
spectral features in room temperature atomic vapors than in other three photon
excitation schemes.

The EIT probe laser beam addresses the $5S_{1/2}$, $F=2\rightarrow 5P_{3/2}$,
$F^{\prime}=3$ transition at 780 nm. The effective Rydberg EIT coupling beam
is comprised of a two-step optical excitation with a variable intermediate
detuning from the $5D_{5/2},F=4$ state. The $5P_{3/2},F=3\rightarrow
5D_{5/2},F^{\prime}=4$ dressing transition is at 776 nm, and the
$5D_{5/2},F=4\rightarrow nF_{7/2},F^{\prime}=5$ (or $nP_{3/2},F^{\prime}=3$)
is tuned from 1260 nm to 1253 nm to access $n=45$ to $n=70$ Rydberg states.
The intermediate detuning from the $5D_{5/2}$ state can be adjusted to trade
Rydberg excitation rate and intermediate-state-lifetime broadening effects.

Figure 1: (a) Level diagram for $F$-state Rydberg excitation and rf induced
Autler-Townes splitting in 87Rb. Three electric dipole-allowed infrared
transitions connect the ground state to the Rydberg $nF_{7/2}$ (or $nP_{3/2}$)
states. Dashed virtual levels show the single photon detunings of each step
from atomic resonance. UHF to VHF fields, depicted with a green bidirectional
arrow, can be detected via Autler Townes splitting on the $nF\rightarrow nG$
Rydberg transitions while 1 GHz to 500 GHz can be detected on $nS\rightarrow
nP$, $nP\rightarrow(n-1)D$, and $nF\rightarrow(n+1)D$ transitions. (b)
Experimental schematic: the 780 nm probe beam counter propagates in a Rb vapor
cell with 776 nm and 1260 nm beams which form the effective EIT coupling beam.
PD: photodiode; DBS1,2: dichroic beam splitter; BD: beam displacer; Pset:
polarization optics consisting of a $\lambda/2$ waveplate and a polarizing
beam splitter cube for power control followed by $\lambda/2$ and $\lambda/4$
waveplates for polarization control.

The optical layout is shown in Figure 1 b. Our rf sensing volume consists of a
75 mm long, 19 mm diameter cylindrical atomic vapor cell with counter-
propagating probe and dressing and coupling laser beams. The room temperature
quartz vapor cell is filled with isotopically pure ($98~{}\%$) 87Rb, and has
wedged fused silica windows at a $11^{\circ}$ angle with respect to the laser
beam propagation direction. After interacting with the atomic medium, the 780
nm probe beam is split from the 776 nm and 1260 nm beams and monitored with a
photodiode.

The 780 nm, 776 nm, and 1260 nm beam waists ($1/e^{2}$ radius) are measured to
be 587(60) $\mu$m, 598(60) $\mu$m, and 592(60) $\mu$m with typical powers of
141(9) $\mu$W, 13.7(8) mW, and 233(14) mW, respectively. We estimate the upper
bounds for optical excitation Rabi rates using the stretched-state dipole
matrix elements to be $2\pi\times$17(3) MHz for the 780 nm transition,
$2\pi\times$52(6) MHz for the 776 nm transition, and $2\pi\times$13(2) MHz
{$2\pi\times$7(1) MHz} for the 1260 nm transition to $n=45$ {$n=70$}[34]. The
780 nm and 776 nm lasers are frequency stabilized to reference vapor cells
using one and two color polarization-rotation spectroscopy [48, 49],
respectively. The probe laser is locked 10 MHz below the optical cycling
$5S_{1/2},F=2\rightarrow 5P_{3/2},F=3$ transition. The intermediate coupling
laser is locked 30 MHz below the $5P_{3/2}F=3\rightarrow 5D_{5/2}F=4$
transition. The 1260 nm laser can either be locked to a wavemeter or scanned
across the three photon resonance.

The 780 nm probe intensity is measured by a photodiode during a scan of the
1260 nm laser frequency across EIT resonance. To suppress probe-laser
intensity noise, we can optionally use a reference probe beam without counter-
propagating lasers and a differential photodetector [4, 50]. To calibrate the
1260 nm laser frequency scan, we simultaneously record the transmission of a
fiber-based Michelson interferometer with a free spectral range of 30(3) MHz,
allowing frequency scan non-linearity to be removed in post-processing. The
scan center frequency is recorded on a wavemeter with $\pm 50$ MHz accuracy.
This wavemeter accuracy is sufficient to identify the principal quantum number
of each Rydberg transition; it is not used to measure rf transition
frequencies.

The source of rf radiation in our setup is a linear 17 cm monopole antenna
connected to an rf synthesizer and placed roughly 20 cm away from the vapor
cell. The cell and antenna are mounted 15 cm above an aluminum bread board on
a dielectric post. We expect non-trivial contributions from the breadboard to
the overall radiation pattern of the antenna since the separation is smaller
than the 33 cm to 1.5 m rf wavelengths investigated here.

Figure 2:  (a) AT-EIT Spectrum of the $50F\rightarrow 50G$ transition with an
applied rf field of 655 MHz (red). A 5-level simulated spectrum is underlaid
with the data (blue). (b) Autler-Townes splitting of the $50F\rightarrow 50G$
EIT signal as a function of applied UHF electric field. Point markers (red)
are the measured uncertainties, with a linear fit to guide the eye (blue).
This plot confirms the expected linear scaling between rf electric field and
Autler-Townes splitting.

When the resonant $nF\rightarrow nG$ rf radiation coupling strength exceeds
the EIT linewidth, the EIT line splits into two Autler Townes peaks. Figure 2
a shows the AT splitting of the $50F\rightarrow 50G$ transition as a function
of 1260 nm laser detuning, in good agreement with a five-level optical Bloch
equation simulation of the spectrum (see section IV for further discussion).
The observed EIT linewidth is primarily due to power broadening associated
with the dressing and coupling Rabi rates and detuning from the intermediate
$D_{5/2}$ state. In Figure 2 b we plot the measured Autler-Townes peak
frequency splitting of the $50F\rightarrow 50G$ transition as a function of
the applied UHF electric field. This shows the expected linear scaling,
applicable for use as an rf power standard [1, 7, 9]. The rf electric field
can be calculated as:

$|\vec{E}|=\frac{\hbar}{\mu_{R}}\Omega_{\mathrm{RF}}$ (2)

using measured AT splitting, $\Omega_{\mathrm{RF}}$, and the known transition
dipole moment $\mu_{R}$. The slope of Figure 2 b. and the Rydberg transition
dipole moment (for the $|m_{J}|=1/2$ $\pi$-transition of $50F\rightarrow 50G$,
$\mu_{R}=1858ea_{0}$ [34]) can be used to calibrate the apparatus calibration
factor from signal generator power $P_{s}$ to electric field at the atomic
sample, $\xi=|E|P_{s}^{-1/2}$. This will be used in section IV to evaluate the
sensitivity of our system.

Figure 3: The upper three (blue-dashed, red, and black) curves show calculated
rf carrier frequencies [34] of commonly used lower angular momentum Rydberg
states accessible by two photon optical excitation. The middle (green) curve,
located primarily in the UHF band, is the calculated rf carrier frequency
associated with $nF_{7/2}\rightarrow nG_{9/2}$ transitions; the overlaid
(green) point markers with uncertainty show corresponding measured carrier
frequencies in the $n=45$ to $n=70$ range. The lowest (yellow) trace shows the
calculated rf carrier frequency associated with $nG_{9/2}\rightarrow
nH_{11/2}$ transitions.

Figure 3 illustrates the $>10$x reduction in carrier frequency (increase in
wavelength) at a given $n$ that can be attained by using transitions between
higher ($L\geq 3$) angular momentum states. The experimentally measured
resonant transition frequencies and uncertainties are shown in green. The
measurements are in good agreement with carrier frequencies computed from the
85Rb $G$-series quantum defect model [51, 52, 53, 54] after mass scaling
444The isotope shift between values in 85Rb and the data measured here in 87Rb
is accounted for by mass scaling the infinite nuclear mass Rydberg constant
$\mathrm{R}_{\infty}=(e^{4}m_{e})/({16\pi^{2}\epsilon_{0}^{2}\hbar^{2}})$
(here $e$ is the electron charge, $m_{e}$ is the mass of the electron) by the
factor $\mathrm{R_{n}}=\mathrm{R}_{\infty}\times M_{n}/(M_{n}+m_{e})$)
($M_{n}$ is the mass of the atomic nucleus) not by adjusting the quantum
defects.. These quantum defect-derived carrier frequencies are shown as solid
curves, with the $nF_{7/2}\rightarrow nG_{9/2}$ transitions shown in green.
The upper curves on this graph show resonant transition frequencies as a
function of $n$ for transitions with lower ($L\leq 3$) orbital angular
momenta: $nS_{1/2}\rightarrow nP_{3/2}$, $nP_{3/2}\rightarrow(n-1)D_{5/2}$,
and $nF_{7/2}\rightarrow(n+1)D_{5/2}$. These transitions were chosen because
they have the largest dipole matrix elements and thus are the most sensitive
for communications applications.

To determine the resonant Rydberg transition frequency in Fig 3, we record the
amplitude and separation of the Autler-Townes peaks as in Fig 2 a. 555For the
data shown in Fig 3, the 780 nm, 776 nm, and 1260 nm beam waists ($1/e^{2}$)
are measured to be 55(6) $\mu$m, 55(6) $\mu$m, and 65(7) $\mu$m with typical
powers of 4(.3) $\mu$W, 130(8) $\mu$W, and 40(3) mW, respectively. On
resonance, the amplitude of the peaks is equal and the separation between the
two peaks is minimized. Off of resonance, the peak amplitudes become
imbalanced and the peak separation increases. The peak separations as a
function of rf frequency reveal an approximately quadratic minimum about the
rf carrier frequency. Spectra were recorded in a 30 MHz region approximately
centered about the $nF\rightarrow nG$ transition frequency, with a frequency
step size of 1 MHz. The peak positions are found by fitting independent
Gaussian functions to each Autler-Townes peak. The peak separations are then
fit to a quadratic to determine Rydberg resonance, with combined fit
uncertainties of $<24$ MHz on average. We repeat this measurement for several
principal quantum numbers in the $n=45$ to $n=70$ range.

For this measurement at $n=70$, the vapor cell is more than an order of
magnitude smaller than the rf wavelength ($\ell_{vapor}/\lambda_{rf}\approx
0.05$). Carrier frequencies may be further reduced using high angular momentum
Rydberg states, and future experimental work will will investigate the lower-
frequency limits of this approach. For example, the $nG_{9/2}\rightarrow
nH_{11/2}$ transitions, shown at the bottom of Figure 3, are a factor of 4.6
lower in frequency at a given principal quantum number than the
$nF_{7/2}\rightarrow nG_{9/2}$ transitions. These transitions can be accessed
with additional rf fields. However, when the energetic spacing between
transitions approaches the Rabi coupling rate (or potentially data bandwidth
in communications applications) between the Rydberg states of interest, the
incident rf will couple nearly resonantly to other higher $L$ dipole allowed
transitions. This will result in complex spectra not easily interpreted for
use in calibration or communication.

##  III Numerical Model

We develop a numerical model to benchmark our experimental data and compute
fundamental sensitivity limits. We use a master equation formalism to simulate
the light-atom interaction, as in previous work [38, 57, 58]. We numerically
compute the steady-state density matrix for the thermal 87Rb sample. Our
numerical model includes the experimental laser beam and rf intensities,
propagation directions, polarizations, and frequency detunings from relevant
Rb electronic transitions. We account for a number of state decay ($T_{1}$)
and dephasing ($T_{2}$) processes via Lindblad operators [59]. These
transition broadening processes include finite laser linewidth, Rb state
decay, transit broadening, and electronic-state-dependent collisional
broadening (see Fig. 4). We model the system using the five electronic states:
$5S_{1/2}$, $5P_{3/2}$, $5D_{5/2}$, $nF_{7/2}$, and $nG_{9/2}$ while
neglecting the full 45-level Zeeman structure for computational efficiency.
The steady state is calculated for a range of atom velocities chosen from the
Maxwell-Boltzmann distribution.

To account for collision broadening effects, we calculate Van der Waals
($C_{6}$) coefficients [60] for both Rydberg-Rydberg ($nF_{7/2}$-$nF_{7/2}$
and $nF_{7/2}$-$nG_{9/2}$) and $5S_{1/2}$-Rydberg atomic collisions. We then
use the eikonal approximation with the optical theorem [61] to calculate the
total rate $\Gamma_{SR}$ ($\Gamma_{RR}$) for $5S_{1/2}$-Rydberg (Rydberg-
Rydberg) collisions as shown in Fig. 4. We include $5S_{1/2}$-Rydberg
collisional broadening using a phenomenological jump operator of the form
$\Gamma_{SR}(|5S_{1/2}\rangle\langle R|+|R\rangle\langle 5S_{1/2}|)$, where
$|R\rangle$ is the relevant Rydberg state.

Figure 4: Dephasing rates versus principal quantum number. Due to competing
Rydberg state lifetime and collisional broadening effects, the atom-limited
dephasing rate (atom-limited coherence time) is minimized (maximized) at
intermediate principal quantum number $40<n<80$. This illustrates the
advantage of decreasing rf carrier frequency by increasing $L$ rather than
increasing $n$. The traces in the legend share the same vertical ordering with
the $n=20$ data.

We calculate a transit broadening rate coefficient as the inverse of the
average laser beam transit time using a vapor cell temperature $T=300$ K and
the beam waists and generate a corresponding Lindblad operator by summing the
outer products of each state vector with the ground state to indicate a
$T_{1}$ decay. Similarly, we include Lindblad operators for the laser
linewidth broadening and the various atomic state lifetimes. The magnitude of
each dephasing rate is represented in Fig. 4, in addition to a “total atom”
dephasing rate, which is the sum of all of the non-laser-dependent atomic
broadening effects (collisions and Rydberg state lifetime). Remaining
broadening not included in this model may be due to electric field
inhomogeneity in the applied rf or nonuniform distribution of metallic
rubidium within the vapor cell along the axis of beam propagation. Figure 4
shows two atomic structure limitations to coherence time. At low principal
quantum number short atomic lifetimes dominate, while at high principal
quantum number Rydberg-Rydberg collisions dominate. The atom-limited dephasing
rate is therefore minimized at intermediate principal quantum number
$40<n<80$. This optimal atom-limited sensitivity again supports the choice to
use $L\geq 3$ states to achieve lower rf carrier frequencies rather than
simply going to higher $n$ states.

To compare with absorption measurements, we compute the detuning-dependent
optical absorption coefficient ($\alpha$) experienced by the 780 nm probe
laser beam as [38]

$\alpha=\frac{2k_{p}n_{V}\epsilon\mu_{p}}{\epsilon_{0}E_{p}},$ (3)

where

$\epsilon=\int_{-\infty}^{\infty}{P(v_{x})\text{Im}(\rho_{01})dv_{x}}$ (4)

is the atomic excitation fraction obtained via integration of the velocity-
dependent quantity $\text{Im}(\rho_{01})$ weighted by the Maxwell-Boltzmann
velocity distribution along the beam propagation direction,
$P(v_{x})=\sqrt{\frac{m}{2\pi kT}}\exp({-\frac{mv_{x}^{2}}{2kT}})$. In Eq.
(3), $k_{p}$ is the probe laser k-vector magnitude, $E_{p}$ is the probe laser
electric field, $n_{V}$ is the volumetric number density of 87Rb atoms in the
cell, $\mu_{p}$ is the electric dipole moment of the $5S_{1/2}\rightarrow
5P_{3/2}$ probe transition, and $\epsilon_{0}$ is the permittivity of vacuum.
Finally, we compute the fractional probe laser beam transmission as
$\exp(-\alpha\ell_{v})$.

##  IV Sensitivity Assessment

We characterize the sensitivity of our system using the $50F\rightarrow 50G$
transition. Experimentally we implement super-heterodyne detection [3, 4]
using two rf fields: a signal field
$E_{\mathrm{{s}}}\cos{[(\omega_{\mathrm{{LO}}}+\delta)t]}$ and a local
oscillator
$E_{\mathrm{{LO}}}\cos{(\omega_{\mathrm{{LO}}}t-\phi_{\mathrm{{LO}}})}$. We
first make a series of calibration measurements to obtain $\xi$ (Fig. 2b)
relating the signal generator power setting $P_{s}$to the electric field
through Eq. 2 and $\xi=|E|P_{s}^{-1/2}$. We choose
$\Omega_{\mathrm{{LO}}}\approx\Gamma$ to bias the sensor to maximum
sensitivity [60] and apply a calibrated signal field $E_{\mathrm{{s}}}\ll
E_{\mathrm{{LO}}}$. When $\delta\ll\omega_{LO}$ and $\delta$ is within the
atomic response bandwidth, the time-dependent atomic response is
$~{}E_{s}/2\cos(\delta t+\phi_{\mathrm{{LO}}})$ [62] The beatnote amplitude
then serves to calibrate the receiver response $\chi$ between the photodiode
voltage and signal field $E_{\mathrm{{s}}}$. We measure an amplitude spectral
density noise floor of $E_{n}=13(2)\,\mathrm{\mu V/m/\sqrt{Hz}}$ for the
$50F-50G$ transition. The LO parameters are $E_{\mathrm{{LO}}}=0.37$ V/m
(corresponding to $\Omega_{\mathrm{{LO}}}/2\pi=9.5$ MHz) at
$\omega_{LO}/2\pi=655$ MHz. The signal field parameters are
$E_{\mathrm{s}}=9.3$ mV/m with $\delta/2\pi=50$ kHz. Both fields are applied
via the same rf-horn. Figure 5 compares this calibrated amplitude spectral
density with those of the probe only (without the dressing or coupling lasers)
and the electronic noise floor without probe. We find the noise dominated by
probe detection and well above the photon shot noise floor, indicating room
for further improvement.

Figure 5: Amplitude spectral density noise measurements at $n=50$,
$f_{\mathrm{rf}}=655$ MHz utilizing a super-heterodyne technique and the
parameters described in the text. The traces in the legend share the same
vertical ordering with the data. The upper (red) trace shows the noise with a
9.3 mV/m calibration field on, with a $13\,\mathrm{\mu V/m/\sqrt{Hz}}$ white-
noise floor near the calibration frequency offset of 50 kHz from the LO field.
The calibrated noise with only probe light (black trace), without any light
(light gray trace) and photon shot noise (psn) (black dashed trace) are shown
for reference. The estimated quantum projection noise is
$34\,\mathrm{nV/m/\sqrt{Hz}}$, well below the range displayed.

The fundamental sensitivity limits of Rydberg vapor quantum sensors are
determined by the quantum projection noise and the photon shot noise [63].
Below, we utilize our model and measurements to estimate the applicable limits
to our system in the UHF frequency range investigated here.

The quantum projection noise limited sensitivity to an rf electric field
amplitude $E_{\mathrm{qpn}}$ is

$E_{\mathrm{qpn}}=\frac{\hbar}{\mu_{R}}\frac{1}{\sqrt{NT_{2}t}},$ (5)

for the coherence time $T_{2}$ and $t$ is the total measurement time. The
number of atoms participating in the measurement is $N=\epsilon n_{V}V$ for an
interaction volume defined by optical beam geometry $V=\pi
w_{0}^{2}\ell_{\mathrm{eff}}$ for beam waists $w_{0}$ and effective
interaction lenth $\ell_{\mathrm{eff}}$. Typical parameters in our simulations
yield $10^{-3}<\epsilon<10^{-2}$, significantly less than an estimation based
on all available atoms in the laser beam column [64, 63]. At principal $n=50$
at room temperature and the experimental parameters defined above, we estimate
$\epsilon=10.7\times 10^{-3}$ and participating atom number $N\sim 10^{6}$. We
use the bare EIT full width at half maximum $\Gamma=2\pi\times 9.7$ MHz to
estimate $T_{2}=\Gamma^{-1}$ [4] and estimate $\Delta
E_{\mathrm{qpn}}=38\,\mathrm{nV/m/\sqrt{Hz}}$.

The photon shot noise for a single photodiode intensity measurement is $\Delta
I_{\mathrm{psn}}=\sqrt{2e(\eta e\Phi_{p})\Delta f}$, where $e$ is the electron
charge, $\eta$ is the photodiode quantum efficiency, $\Phi_{p}$ is the probe
photon flux incident upon the photodiode, and $\Delta f$ is the measurement
bandwidth [32]. Using the transimpedance amplifier gain $G_{v}$ and the
receiver response $G_{v}\chi=$ (V/(V/m)) in volts per applied
$E_{\mathrm{{sig}}}$, we can express this as a field sensitivity limit

$\Delta E_{\mathrm{psn}}=G_{v}\chi\Delta I_{\mathrm{psn}}.$ (6)

The photon shot noise in Fig. 5 is $\Delta E_{psn}=1.6\,\mathrm{\mu
V/m/\sqrt{Hz}}$, a factor of 40 higher than $E_{qpn}$. Technical improvements
that will be pursued in future work to reach the photon shot noise limit
include intensity stabilization and the addition of a laser to repump ground
state population [50] and potentially comb-based optical probing [65].

##  V Outlook

We have demonstrated AT-EIT rf field sensitivity on $nF\rightarrow nG$ Rydberg
transitions that enable resonant electrically small UHF receivers. Future work
includes recording temporally modulated fields to characterize sensitivity and
bandwidth at UHF and lower frequencies. This can be done by using portable
laser systems and operating in a controlled rf-environment, since
$\lambda_{\mathrm{rf}}$ is comparable to or larger than many optical elements.
Furthermore, this approach is compatible with using auxiliary rf fields to
Stark-tune the desired rf frequency [66, 57] offering continuous tuning
between atomic transitions.

The three photon all-infrared optical excitation approach also offers a number
of benefits that may be further explored. The three optical beams may be
aligned in a planar orientation to achieve Doppler-free and recoil-free
excitation [67, 68, 69] potentially enabling Rydberg lifetime-limited narrow
spectral features useful in precise rf field calibrations. Infrared optical
excitation may also enable simplified all-dielectric vapor cell sensor heads
with more uniformly applied EIT coupling fields compared to infrared-blue
excitation [70]. This should reduce transit broadening and increase
sensitivity in a deployable package.

##  VI Acknowledgements

We thank V. Gerginov, R. Westafer, C. D. Herold, and M. Lombardi for comments
and careful reading of the manuscript. We acknowledge funding from the Air
Force Research Laboratory (AFRL) and the GTRI HIVES program.

## References

  * Sedlacek  _et al._ [2012] J. A. Sedlacek, A. Schwettmann, H. Kübler, R. Löw, T. Pfau, and J. P. Shaffer, Microwave electrometry with Rydberg atoms in a vapour cell using bright atomic resonances, Nature Physics 8, 819 (2012).
  * Sedlacek  _et al._ [2013] J. A. Sedlacek, A. Schwettmann, H. Kübler, and J. P. Shaffer, Atom-based vector microwave electrometry using rubidium rydberg atoms in a vapor cell, Phys. Rev. Lett. 111, 063001 (2013).
  * Simons  _et al._ [2019] M. T. Simons, A. H. Haddab, J. A. Gordon, and C. L. Holloway, A Rydberg atom-based mixer: Measuring the phase of a radio frequency wave, Applied Physics Letters 114, 114101 (2019).
  * Jing  _et al._ [2020] M. Jing, Y. Hu, J. Ma, H. Zhang, L. Zhang, L. Xiao, and S. Jia, Atomic superheterodyne receiver based on microwave-dressed Rydberg spectroscopy, Nature Physics 16, 911 (2020).
  * Robinson  _et al._ [2021] A. K. Robinson, N. Prajapati, D. Senic, M. T. Simons, and C. L. Holloway, Determining the angle-of-arrival of a radio-frequency source with a rydberg atom-based sensor, Applied Physics Letters 118, 114001 (2021).
  * Donley  _et al._ [2003] E. A. Donley, T. Crowley, T. P. Heavner, and B. Riddle, A quantum-based microwave power measurement performed with a miniature atomic fountain, in  _IEEE International Frequency Control Symposium and PDA Exhibition Jointly with the 17th European Frequency and Time Forum, 2003. Proceedings of the 2003_ (IEEE, 2003) pp. 135–137.
  * Holloway  _et al._ [2017a] C. L. Holloway, M. T. Simons, J. A. Gordon, P. F. Wilson, C. M. Cooke, D. A. Anderson, and G. Raithel, Atom-Based RF Electric Field Metrology: From Self-Calibrated Measurements to Subwavelength and Near-Field Imaging, IEEE Transactions on Electromagnetic Compatibility 59, 717 (2017a).
  * Holloway  _et al._ [2017b] C. L. Holloway, M. T. Simons, J. A. Gordon, A. Dienstfrey, D. A. Anderson, and G. Raithel, Electric field metrology for si traceability: Systematic measurement uncertainties in electromagnetically induced transparency in atomic vapor, Journal of Applied Physics 121, 233106 (2017b).
  * Chen  _et al._ [2022] S. Chen, D. J. Reed, A. R. MacKellar, L. A. Downes, N. F. A. Almuhawish, M. J. Jamieson, C. S. Adams, and K. J. Weatherill, Terahertz electrometry via infrared spectroscopy of atomic vapor, Optica 9, 485 (2022).
  * Fan  _et al._ [2014] H. Q. Fan, S. Kumar, R. Daschner, H. Kübler, and J. P. Shaffer, Subwavelength microwave electric-field imaging using rydberg atoms inside atomic vapor cells, Opt. Lett. 39, 3030 (2014).
  * Holloway  _et al._ [2018] C. L. Holloway, M. T. Simons, M. D. Kautz, A. H. Haddab, J. A. Gordon, and T. P. Crowley, A quantum-based power standard: Using rydberg atoms for a si-traceable radio-frequency power measurement technique in rectangular waveguides, Applied Physics Letters 113, 094101 (2018).
  * Meyer  _et al._ [2018] D. H. Meyer, K. C. Cox, F. K. Fatemi, and P. D. Kunz, Digital communication with Rydberg atoms and amplitude-modulated microwave fields, Applied Physics Letters 112, 211108 (2018).
  * Song  _et al._ [2019] Z. Song, H. Liu, X. Liu, W. Zhang, H. Zou, J. Zhang, and J. Qu, Rydberg-atom-based digital communication using a continuously tunable radio-frequency carrier, Optics Express 27, 8848 (2019).
  * Anderson  _et al._ [2021] D. A. Anderson, R. E. Sapiro, and G. Raithel, An atomic receiver for am and fm radio communication, IEEE Transactions on Antennas and Propagation 69, 2455 (2021).
  * Deb and Kjærgaard [2018] A. B. Deb and N. Kjærgaard, Radio-over-fiber using an optical antenna based on Rydberg states of atoms, Applied Physics Letters 112, 211106 (2018).
  * Anderson  _et al._ [2020] D. A. Anderson, R. E. Sapiro, and G. Raithel, Rydberg atoms for radio-frequency communications and sensing: Atomic receivers for pulsed rf field and phase detection, IEEE Aerospace and Electronic Systems Magazine 35, 48 (2020).
  * Holloway  _et al._ [2019] C. L. Holloway, M. T. Simons, J. A. Gordon, and D. Novotny, Detecting and receiving phase-modulated signals with a rydberg atom-based receiver, IEEE Antennas and Wireless Propagation Letters 18, 1853 (2019).
  * Cox  _et al._ [2018] K. C. Cox, D. H. Meyer, F. K. Fatemi, and P. D. Kunz, Quantum-limited atomic receiver in the electrically small regime, Phys. Rev. Lett. 121, 110502 (2018).
  * Prajapati  _et al._ [2022] N. Prajapati, A. P. Rotunno, S. Berweger, M. T. Simons, A. B. Artusio-Glimpse, S. D. Voran, and C. L. Holloway, Tv and video game streaming with a quantum receiver: A study on a rydberg atom-based receiver’s bandwidth and reception clarity, AVS Quantum Science 4, 035001 (2022).
  * Note [1] Pre 2020 F.C.C. spectrum re-allocation https://www.ntia.doc.gov/legacy/osmhome/alloctbl/ allocmhz.html.
  * Chu [1948] L. J. Chu, Physical limitations of omni‐directional antennas, Journal of Applied Physics 19, 1163 (1948).
  * Wheeler [1947] H. Wheeler, Fundamental limitations of small antennas, Proceedings of the IRE 35, 1479 (1947).
  * Harrington [1960] R. F. Harrington, Effect of antenna size on gain, bandwidth, and efficiency, Journal of Research of the National Bureau of Standards-D. Radio Propagation 64 (1960).
  * Sievenpiper  _et al._ [2012] D. F. Sievenpiper, D. C. Dawson, M. M. Jacob, T. Kanar, S. Kim, J. Long, and R. G. Quarfoth, Experimental Validation of Performance Limits and Design Guidelines for Small Antennas, IEEE Transactions on Antennas and Propagation 60, 8 (2012), conference Name: IEEE Transactions on Antennas and Propagation.
  * Bason  _et al._ [2010] M. G. Bason, M. Tanasittikosol, A. Sargsyan, A. K. Mohapatra, D. Sarkisyan, R. M. Potvliege, and C. S. Adams, Enhanced electric field sensitivity of rf-dressed rydberg dark states, New Journal of Physics 12, 065015 (2010).
  * Miller  _et al._ [2016] S. A. Miller, D. A. Anderson, and G. Raithel, Radio-frequency-modulated Rydberg states in a vapor cell, New Journal of Physics 18, 053017 (2016).
  * Jiao  _et al._ [2017] Y. Jiao, L. Hao, X. Han, S. Bai, G. Raithel, J. Zhao, and S. Jia, Atom-based radio-frequency field calibration and polarization measurement using cesium $n{D}_{J}$ floquet states, Phys. Rev. Applied 8, 014028 (2017).
  * Paradis  _et al._ [2019] E. Paradis, G. Raithel, and D. A. Anderson, Atomic measurements of high-intensity VHF-band radio-frequency fields with a Rydberg vapor-cell detector, Physical Review A 100, 013420 (2019).
  * Autler and Townes [1955] S. H. Autler and C. H. Townes, Stark effect in rapidly varying fields, Phys. Rev. 100, 703 (1955).
  * Mohapatra  _et al._ [2007] A. K. Mohapatra, T. R. Jackson, and C. S. Adams, Coherent optical detection of highly excited rydberg states using electromagnetically induced transparency, Phys. Rev. Lett. 98, 113003 (2007).
  * Boller  _et al._ [1991] K.-J. Boller, A. Imamoğlu, and S. E. Harris, Observation of electromagnetically induced transparency, Phys. Rev. Lett. 66, 2593 (1991).
  * Kumar  _et al._ [2017] S. Kumar, H. Fan, H. Kübler, A. J. Jahangiri, and J. P. Shaffer, Rydberg-atom based radio-frequency electrometry using frequency modulation spectroscopy in room temperature vapor cells, Optics Express 25, 8625 (2017), publisher: Optical Society of America.
  * Gallagher [1994] T. F. Gallagher,  _Rydberg Atoms_ (Cambridge University Press, Cambridge, 1994).
  * Šibalić  _et al._ [2017] N. Šibalić, J. Pritchard, C. Adams, and K. Weatherill, Arc: An open-source library for calculating properties of alkali rydberg atoms, Computer Physics Communications 220, 319 (2017).
  * Robertson  _et al._ [2021] E. Robertson, N. Šibalić, R. Potvliege, and M. Jones, Arc 3.0: An expanded python toolbox for atomic physics calculations, Computer Physics Communications 261, 107814 (2021).
  * Thoumany  _et al._ [2009] P. Thoumany, T. Germann, T. Hänsch, G. Stania, L. Urbonas, and T. Becker, Spectroscopy of rubidium Rydberg states with three diode lasers, Journal of Modern Optics 56, 2055 (2009).
  * Johnson  _et al._ [2010] L. A. M. Johnson, H. O. Majeed, B. Sanguinetti, T. Becker, and B. T. H. Varcoe, Absolute frequency measurements of ${}^{\textrm{85}}$ Rb n F ${}_{\textrm{7/2}}$ Rydberg states using purely optical detection, New Journal of Physics 12, 063028 (2010).
  * Thaicharoen  _et al._ [2019] N. Thaicharoen, K. R. Moore, D. A. Anderson, R. C. Powel, E. Peterson, and G. Raithel, Electromagnetically induced transparency, absorption, and microwave-field sensing in a rb vapor cell with a three-color all-infrared laser system, Phys. Rev. A 100, 063427 (2019).
  * You  _et al._ [2022] S. H. You, M. H. Cai, S. S. Zhang, Z. S. Xu, and H. P. Liu, Microwave-field sensing via electromagnetically induced absorption of rb irradiated by three-color infrared lasers, Opt. Express 30, 16619 (2022).
  * Lahad  _et al._ [2019] O. Lahad, R. Finkelstein, O. Davidson, O. Michel, E. Poem, and O. Firstenberg, Recovering the Homogeneous Absorption of Inhomogeneous Media, Physical Review Letters 123, 173203 (2019).
  * Carr  _et al._ [2012a] C. Carr, M. Tanasittikosol, A. Sargsyan, D. Sarkisyan, C. S. Adams, and K. J. Weatherill, Three-photon electromagnetically induced transparency using rydberg states, Opt. Lett. 37, 3858 (2012a).
  * Zhang  _et al._ [2022] L.-H. Zhang, Z.-K. Liu, B. Liu, Z.-Y. Zhang, G.-C. Guo, D.-S. Ding, and B.-S. Shi, Rydberg microwave-frequency-comb spectrometer, Phys. Rev. Appl. 18, 014033 (2022).
  * Kondo  _et al._ [2015] J. M. Kondo, N. Šibalić, A. Guttridge, C. G. Wade, N. R. D. Melo, C. S. Adams, and K. J. Weatherill, Observation of interference effects via four-photon excitation of highly excited rydberg states in thermal cesium vapor, Opt. Lett. 40, 5570 (2015).
  * Shaffer and Kübler [2018] J. Shaffer and H. Kübler, A read-out enhancement for microwave electric field sensing with rydberg atoms, in  _Quantum Technologies 2018_ , Vol. 10674 (SPIE, 2018) pp. 39–49.
  * Bai  _et al._ [2022] J. Bai, Y. Jiao, Y. He, R. Song, J. Zhao, and S. Jia, Autler-townes splitting of three-photon excitation of cesium cold rydberg gases, Opt. Express 30, 16748 (2022).
  * Note [2] This is often because it is either technologically or scientifically attractive for the second leg of the ladder to connect $P\rightarrow S$ states.
  * Note [3] Access to the largest number of atomic velocity classes occurs when the 780 nm beam counter propagates with the 776 nm and 1260 nm beams and with small frequency offset from $5D_{5/2}$ state, creating stronger spectral features in room temperature atomic vapors than in other three photon excitation schemes.
  * Wieman and Hänsch [1976] C. Wieman and T. W. Hänsch, Doppler-Free Laser Polarization Spectroscopy, Physical Review Letters 36, 1170 (1976).
  * Carr  _et al._ [2012b] C. Carr, C. S. Adams, and K. J. Weatherill, Polarization spectroscopy of an excited state transition, Optics Letters 37, 118 (2012b).
  * Prajapati  _et al._ [2021] N. Prajapati, A. K. Robinson, S. Berweger, M. T. Simons, A. B. Artusio-Glimpse, and C. L. Holloway, Enhancement of electromagnetically induced transparency based rydberg-atom electrometry through population repumping, Applied Physics Letters 119, 214001 (2021).
  * Han  _et al._ [2006] J. Han, Y. Jamil, D. V. L. Norum, P. J. Tanner, and T. F. Gallagher, Rb $nf$ quantum defects from millimeter-wave spectroscopy of cold ${}^{85}\mathrm{Rb}$ rydberg atoms, Phys. Rev. A 74, 054502 (2006).
  * Afrousheh  _et al._ [2006] K. Afrousheh, P. Bohlouli-Zanjani, J. A. Petrus, and J. D. D. Martin, Determination of the ${}^{85}\mathrm{Rb}$ $ng$-series quantum defect by electric-field-induced resonant energy transfer between cold rydberg atoms, Phys. Rev. A 74, 062712 (2006).
  * Johnson [2011] L. A. M. Johnson,  _Precision laser spectroscopy of rubidium with a Frequency Comb_ (University of Leeds, 2011).
  * Moore  _et al._ [2020] K. Moore, A. Duspayev, R. Cardman, and G. Raithel, Measurement of the rb $g$-series quantum defect using two-photon microwave spectroscopy, Phys. Rev. A 102, 062817 (2020).
  * Note [4] The isotope shift between values in 85Rb and the data measured here in 87Rb is accounted for by mass scaling the infinite nuclear mass Rydberg constant $\mathrm{R}_{\infty}=(e^{4}m_{e})/({16\pi^{2}\epsilon_{0}^{2}\hbar^{2}})$ (here $e$ is the electron charge, $m_{e}$ is the mass of the electron) by the factor $\mathrm{R_{n}}=\mathrm{R}_{\infty}\times M_{n}/(M_{n}+m_{e})$) ($M_{n}$ is the mass of the atomic nucleus) not by adjusting the quantum defects.
  * Note [5] For the data shown in Fig 3, the 780 nm, 776 nm, and 1260 nm beam waists ($1/e^{2}$) are measured to be 55(6) $\mu$m, 55(6) $\mu$m, and 65(7) $\mu$m with typical powers of 4(.3) $\mu$W, 130(8) $\mu$W, and 40(3) mW, respectively.
  * Simons  _et al._ [2021] M. T. Simons, A. B. Artusio-Glimpse, C. L. Holloway, E. Imhof, S. R. Jefferts, R. Wyllie, B. C. Sawyer, and T. G. Walker, Continuous radio-frequency electric-field detection through adjacent rydberg resonance tuning, Phys. Rev. A 104, 032824 (2021).
  * Sandhya and Sharma [1997] S. N. Sandhya and K. K. Sharma, Atomic coherence effects in four-level systems: Doppler-free absorption within an electromagnetically-induced-transparency window, Phys. Rev. A 55, 2155 (1997).
  * Manzano [2020] D. Manzano, A short introduction to the lindblad master equation, AIP Advances 10, 025106 (2020).
  * Fan  _et al._ [2015] H. Q. Fan, S. Kumar, S. Sedlacek, H. Kübler, S. Karimkashi, and J. P. Shaffer, Atom based rf electric field sensing, J. Phys. B: At. Mol. Opt. Phys. 48, 7 (2015).
  * Sakurai [1995] J. J. Sakurai,  _Modern quantum mechanics, revised edition_ , edited by S. F. Tuan (American Association of Physics Teachers, 1995).
  * Meyer  _et al._ [2021a] D. H. Meyer, P. D. Kunz, and K. C. Cox, Waveguide-coupled rydberg spectrum analyzer from 0 to 20 ghz, Phys. Rev. Appl. 15, 014053 (2021a).
  * Meyer  _et al._ [2021b] D. H. Meyer, C. O’Brien, D. P. Fahey, K. C. Cox, and P. D. Kunz, Optimal atomic quantum sensing using electromagnetically-induced-transparency readout, Physical Review A 104, 043103 (2021b).
  * Meyer  _et al._ [2020] D. H. Meyer, Z. A. Castillo, K. C. Cox, and P. D. Kunz, Assessment of Rydberg atoms for wideband electric field sensing, Journal of Physics B: Atomic, Molecular and Optical Physics 53, 034001 (2020).
  * Dixon  _et al._ [2023] K. Dixon, K. Nickerson, D. W. Booth, and J. P. Shaffer, Rydberg-atom-based electrometry using a self-heterodyne frequency-comb readout and preparation scheme, Phys. Rev. Appl. 19, 034078 (2023).
  * Vogt  _et al._ [2018] T. Vogt, C. Gross, T. F. Gallagher, and W. Li, Microwave-assisted Rydberg electromagnetically induced transparency, Optics Letters 43, 1822 (2018).
  * Grynberg  _et al._ [1976] G. Grynberg, F. Biraben, M. Bassini, and B. Cagnac, Three-Photon Doppler-Free Spectroscopy: Experimental Evidence, Physical Review Letters 37, 283 (1976).
  * Ryabtsev  _et al._ [2011] I. I. Ryabtsev, I. I. Beterov, D. B. Tretyakov, V. M. Entin, and E. A. Yakshina, Doppler- and recoil-free laser excitation of rydberg states via three-photon transitions, Phys. Rev. A 84, 053409 (2011).
  * Šibalić  _et al._ [2016] N. Šibalić, J. M. Kondo, C. S. Adams, and K. J. Weatherill, Dressed-state electromagnetically induced transparency for light storage in uniform-phase spin waves, Phys. Rev. A 94, 033840 (2016).
  * Simons  _et al._ [2018] M. T. Simons, J. A. Gordon, and C. L. Holloway, Fiber-coupled vapor cell for a portable Rydberg atom-based radio frequency electric field sensor, Applied Optics 57, 6456 (2018).