# Transferring Fourier Multipliers from $\mathbb{R}^{n}$ to Compact Lie Groups

David G. Maher111<EMAIL_ADDRESS>
###### Abstract

This paper uses the wrapping map of Dooley and Wildberger to prove $L^{p}$
boundedness of multipliers on compact Lie groups by transferring the estimate
from $\mathbb{R}^{n}$. This improves the bounds in several cases, and
simplifies the proofs of others.

Contents
abcde 1\. Introduction
abcde 2\. Notation and Preliminaries
abcde 3\. The Wrapping Map and Multipliers
abcde 4\. Main Results: $L^{p}-L^{q}$ bounds
abcde 5\. An Application: Convergence of Fourier Series
abcde 6\. Further directions
abcde 7\. References

##  1 Introduction

Let $G$ be a compact connected Lie group and $\mathfrak{g}$ its associate Lie
algebra.

There are many papers dealing with the $L^{p}$ boundedness of multipliers on
compact Lie groups, including [10], [11], [12]. These papers have made use of
one or both of the following techniques: the first is to try and transfer the
analysis from the curved manifold to $\mathbb{R}^{n}$. The main drawback here
is that this can only occur within a neighbourhood of where the exponential
map is injective.

The second technique is to transfer the analysis directly to a maximal torus
$T\subseteq G$ to appeal to existing (abelian) Fourier analysis. This has
involved studying the integrability of negative powers of the Weyl function to
transfer estimates on $G$ to estimates on $T$. Since this function vanishes at
the origin, this creates many technical problems.

The wrapping map $\Phi$ introduced in [3] is a homomorphism of central
measures or distributions under (abelian) convolution on $\mathfrak{g}$
(viewed as $\mathbb{R}^{n}$) to their corresponding counterparts on $G$,
utilising the whole of the lift of the exponential map, not just its
restriction to a fundamental domain. That is,

$\Phi(\mu*_{\mathfrak{g}}\nu)=\Phi(\mu)*_{G}\Phi(\nu)$

As any Fourier multiplier can be written as a convolution with the associated
kernel operator, the wrapping map provides a natural mechanism to transfer
results concerning Fourier multipliers from $\mathfrak{g}\cong\mathbb{R}^{n}$
to $G$.

##  2 Notation and Preliminaries

Let $G$ be a compact connected Lie group, $\mathfrak{g}$ its Lie algebra, $T$
a maximal torus and $\mathfrak{t}$ the Lie algebra of $T$. Let $n$ be the
dimension of $G$, and $l$ the dimension of $T$ (also known as the rank of
$G$). Let $B(\cdot,\cdot)$ be the Killing form on $\mathfrak{g}$, with
$\mathfrak{g}^{*}$ and $\mathfrak{t}^{*}$ the respective duals of
$\mathfrak{g}$ and $\mathfrak{t}$ with respect to the Killing form.

We denote by $\Sigma^{+}$ the set of positive roots
$\\{\alpha_{1},\dots,\alpha_{k}\\}$, with $k=\bigl{|}\Sigma^{+}\bigr{|}$, and
thus we have $n=l+2k$. Let $W$ denote the Weyl group, $\mathfrak{t}^{*}_{+}$
the positive Weyl chamber, and let
$\rho=\tfrac{1}{2}\sum_{\alpha\in\Sigma^{+}}\alpha$. Let
$\Lambda\subseteq\mathfrak{t}^{*}$ denote the set of integral weights, and
$\Lambda^{+}=\Lambda\cap\mathfrak{t}^{*}_{+}$ the set of positive integral
weights.

Every irreducible representation $\pi\in\widehat{G}$ is associated to a unique
highest weight $\lambda\in\lambda^{+}$. If $\chi_{\pi}=\chi_{\lambda}$ is the
character of this representation, the Kirillov’s character formula is given by

$j(X)\chi_{\lambda}(\exp
X)=\int_{\mathcal{O}_{\lambda+\rho}}e^{i\beta(X)}d\mu_{\lambda+\rho}(\beta),\;\;\;\text{for
all}\;\;\;X\in\mathfrak{g}$

where $\mathcal{O}_{\lambda+\rho}$ is the co-adjoint orbit through
$\lambda+\rho\in\mathfrak{t}^{*}_{+}$, and $\mu_{\lambda+\rho}$ is the
Liouville measure on $\mathcal{O}_{\lambda+\rho}$ with total mass
$d_{\lambda}=d_{\pi}=\dim\pi$, and where $j(X)$ is is the analytic square root
of the Jacobian of $\exp$ with $j(0)=1$, given by

$j(X)=\prod_{\alpha\in\Sigma^{+}}\frac{\sin\alpha(X)/2}{\alpha(X)/2}$

That is, for $f\in C_{c}^{\infty}(G)$:

$\int_{\mathfrak{g}}f(\exp X)|j(X)|^{2}dX=\int_{G}f(g)dg:$

where we have normalised the Haar measure $dg$ on $G$ to have total mass 1,
with subsequent normalisation of the Lebesgue measure $dX$ on $\mathfrak{g}$.
Similarly, we normalise the Haar measure and $dt$ on $T$ and the Lebesgue
measure $dH$ on $\mathfrak{t}$ so that we have for $f\in C_{c}^{\infty}(T)$:

$\int_{T}f(\exp H)dH=\int_{\mathfrak{t}}f(t)dt$

Integrals over $G$ of central functions may be computed by integrals on $T$ by
the Weyl integral formula:

$\int_{G}f(g)dg=\frac{1}{|W|}\int_{T}f(t)|\triangle(t)|^{2}dt,\;\text{ where
}\;\triangle(\exp H)=\prod_{\alpha\in\Sigma_{+}}2\sin\alpha(H)/2$

Correspondingly, we also have a similar integral formula for an Ad-invariant
function of compact support $\varphi$ on $\mathfrak{g}$:

$\int_{\mathfrak{g}}\varphi(X)dX=\int_{t_{+}}\prod_{\alpha\in\Sigma_{+}}\alpha(H)^{2}\varphi(H)dH.$

We also denote by $L^{p}(\mathfrak{g})$, $L^{p}(\mathfrak{t})$, $L^{p}(G)$ and
$L^{p}(T)$ the set of $p$-integrable functions on $\mathfrak{g}$,
$\mathfrak{t}$, $G$ and $T$, respectively, and $\|\cdot\|_{p}$ the $p$-norm
when the space concerned is unambiguous. We also denote by
$L^{p}_{I}(\mathfrak{g})$ and $L^{p}_{I}(G)$ the respective sets of
$G$-invariant $L^{p}$ functions.

Let $\mu^{\wedge}$ denote the Fourier transform of $\mu$, with the convention

$\mu^{\wedge}(\xi)=\int_{\mathbb{R}^{n}}\mu(\mathbf{x})e^{i\mathbf{x}\cdot\xi}d\mathbf{x}.$

when $\mu\in L^{1}(\mathbb{R}^{n})$. Correspondingly, Let $\pi\in\widehat{G}$
have highest weight $\lambda\in\Lambda^{+}$. The Fourier transform of a
central function, $\nu$, is a multiple $c_{\pi}I_{\pi}$ of the identity, where

$c_{\pi}=\frac{1}{d_{\pi}}\int_{G}\nu(g)\chi_{\lambda}(g)dg$

when $\mu\in L^{1}_{I}(G)$, such that we may write the Fourier series

$\mu(g)=\sum_{\lambda\in\Lambda^{+}}c_{\lambda}d_{\lambda}\chi_{\lambda}(g).$

We also employ the notation $X\lesssim Y$ to refer to the situation $X\leq
CY$, where $C$ is an unspecified constant.

The Wrapping Map

Assume that $\nu$ is a distribution of compact support on $\mathfrak{g}$, or
$j\nu\in L^{1}(\mathfrak{g})$. We define the wrapping map, $\Phi$ by

$\langle\Phi(\nu),f\rangle=\langle\nu,j\tilde{f}\rangle$ (2.1)

where $f\in C^{\infty}(G)$, $\tilde{f}=f\circ\exp$. We call $\Phi(\nu)$ the
wrap of $\nu$. The principal result is the wrapping formula ([4], Thm. 2)
given by

$\Phi(\mu*_{\mathfrak{g}}\nu)=\Phi(\mu)*_{G}\Phi(\nu)$ (2.2)

What (2.2) shows us is that problems of convolution of central measures or
distributions on a (non-abelian) compact Lie group can be “transferred” to
Euclidean convolution of Ad-invariant distributions on $\mathfrak{g}$. Thus,
since a Fourier multiplier operator may be regarded as a convolution operator,
a natural question arises as to how we may “wrap” a Fourier multiplier
operator. To explicitly show this connection, we must consider the formulation
for $\Phi$.

######  Proposition 1.

We have the following formulations for $\Phi$ for $G$-invariant distributions
of compact support, firstly given as summations over the integer lattice:

$\Phi(\mu)(\exp\,H)=\sum_{\gamma\in\Gamma}\frac{\mu}{j}(H+\gamma)\;\;\;\forall
H\in\mathfrak{t}.$ (2.3)

$\Phi(j\mu)(\exp\,H)=\sum_{\gamma\in\Gamma}\mu(H+\gamma)\;\;\;\forall
H\in\mathfrak{t}.$ (2.4)

and as the Fourier series:

$\Phi(\mu)(\exp
H)=\sum_{\lambda\in\Lambda^{+}}d_{\lambda}\mu^{\wedge}(\lambda+\rho)\chi_{\lambda}(\exp
H)$ (2.5)

$\Phi(j\mu)(\exp
H)=\sum_{\lambda\in\Lambda^{+}}d_{\lambda}\int_{G}\mu^{\wedge}(\lambda+\rho-\mathrm{ad}(g)(\rho))\mathrm{\,d}g\,\chi_{\lambda}(\exp
H)$ (2.6)

where the values are given on $H\in\mathfrak{t}$

(2.4) is calculated in [4], and (2.3) follows naturally. In [4] it is shown
that the Fourier transform of $\Phi(\mu)$ at $\pi$ is given by

$c_{\pi}=(\Phi(\mu))^{\wedge}(\lambda+\rho)=\mu^{\wedge}(\lambda+\rho)$

by applying Kirillov’s character formula, that is:

$c_{\pi}=\frac{1}{d_{\pi}}\langle\Phi(\mu),\chi_{\lambda}\rangle=\frac{1}{d_{\pi}}\langle\mu,j\tilde{\chi}_{\lambda}\rangle=\langle\mu,\int_{G}e^{ig(\lambda+\rho)(\cdot)}dg\rangle$

By the $G$-invariance of $\mu$ this is

$c_{\pi}=\langle\mu,e^{i(\lambda+\rho)(\cdot)}\rangle=\mu^{\wedge}(\lambda+\rho)$
(2.7)

which yields (2.5). Note that (2.2) also follows from (2.7), since

$\displaystyle(\Phi(\mu*\nu))^{\wedge}(\pi)$
$\displaystyle=(\mu*\nu)^{\wedge}(\lambda+\rho)I_{\pi}$
$\displaystyle=\mu^{\wedge}(\lambda+\rho)\nu^{\wedge}(\lambda+\rho)I_{\pi}$
$\displaystyle=(\Phi(\mu))^{\wedge}(\lambda+\rho)(\Phi(\nu))^{\wedge}(\lambda+\rho)I_{\pi}$
$\displaystyle=(\Phi(\mu)*\Phi(\nu))^{\wedge}(\pi)$

We now have an analogue of (2.7) for Ad-invariant distributions with compact
support of the form $j\mu$:

######  Lemma 1.

Let $\mu$ be an Ad-invariant distribution of compact support on
$\mathfrak{g}$. Then the Fourier transform of $\Phi(\mu)$ at $\pi$ is a
multiple $c_{\pi}I_{\pi}$ of the identity, where

$c_{\pi}=(\Phi(\mu))^{\wedge}(\lambda+\rho)=\int_{G}\mu^{\wedge}(\lambda+\rho-\mathrm{ad}(g)(\rho))\mathrm{\,d}g$

Proof: From (2.7) we have that as a multiple of the identity:

$\Phi(\mu)^{\wedge}(\pi_{\lambda})=\mu^{\wedge}(\lambda+\rho)$

Thus,

$\displaystyle\Phi(j\mu)^{\wedge}(\pi_{\lambda})$
$\displaystyle=(j\mu)^{\wedge}(\lambda+\rho)$
$\displaystyle=(j^{\wedge}*\mu^{\wedge})(\lambda+\rho)$
$\displaystyle=\int_{G}\mu^{\wedge}(\lambda+\rho-\mathrm{ad}(g)(\rho))\mathrm{\,d}g\phantom{abcde}\square$

Reconciling (2.3) with (2.5), or (2.4) with (2.6), gives the Poisson summation
formula for compact Lie groups.

In the next section we will consider these two function spaces in the
treatment of multipliers.

##  3 The Wrapping Map and Multipliers

Consider the Fourier multiplier operator, $T$, on $\L^{p}$ by

$\widehat{Tf}(\xi)=m(\xi)\hat{f}(\xi)$

where $m(\xi)$ is the symbol of the operator $T$. By the standard kernel
theorems, every Fourier multiplier $T$ maybe expressed as a convolution
operator with kernel $K$, that is: $Tf=K*f$.

On $\mathfrak{g}\cong\mathbb{R}^{n}$, a multiplier $T_{\psi}$ takes the form:

$(T_{\psi}f)(x)=\frac{1}{(2\pi)^{n}}\int_{\mathfrak{g}}\psi(\xi)f(\xi)e^{-iB(\xi,x)}d\xi$

with kernel

$K_{\psi}(x)=\int_{\mathfrak{g}}\psi(\xi)e^{iB(\xi,x)}d\xi$

that is

$\widehat{K_{\psi}}(\xi)=\psi(\xi)$

On $G$, a multiplier $T_{\Psi}$ takes the form:

$(T_{\Psi}f)(g)=\sum_{\lambda\in\Lambda^{+}}\Psi(\lambda)f(\lambda)\chi_{\lambda}(g)$

with kernel

$K_{\Psi}(g)=\sum_{\lambda\in\Lambda^{+}}\Psi(\lambda)\chi_{\lambda}(g)$

that is

$\widehat{K_{\Psi}}(\lambda)=\Psi(\lambda)$

Wrapping the multiplier $T_{\psi}$ from $\mathfrak{g}\cong\mathbb{R}^{n}$ to
$G$ we have

$\Phi(T_{\psi}f)=\Phi(K_{\psi}*f)=\Phi(K_{\psi})*\Phi(f)$

To compute the kernel $\Phi(K_{\psi})$, we see that its Fourier coefficients
are

$\Phi^{\wedge}(K_{\psi})(\pi_{\lambda})=\Psi(\lambda)=K_{\psi}^{\wedge}(\lambda)$

Thus, we need to compute $\Psi(\lambda)=K_{\psi}^{\wedge}(\lambda)$. In light
of (2.7), we see that

$\Psi(\lambda)=\psi(\lambda+\rho)$ (3.1)

Alternatively, if we wish to consider $\Phi(j\cdot K_{\psi})$, we see by Lemma
1 that

$\Psi(\lambda)=\int_{G}\psi(\lambda+\rho-\mathrm{ad}(g)(\rho))\mathrm{\,d}g$
(3.2)

These two expressions 3.1 and 3.2 for the multiplier $\Psi$ on $G$ constitute
to two forms considered in [11], where they are referred to as “(*)” and
“(**)”, respectively.

Thus, to obtain bounds for multipliers on $L^{p}_{I}(G)$, we now need to
consider bound of the form $\|\Phi(\nu)\|_{q}\lesssim\|\nu\|_{p}$ and
$\|\Phi(j\cdot\nu)\|_{q}\lesssim\|\nu\|_{p}$

##  4 Main Results: $L^{p}-L^{q}$ bounds

In this section we consider the following problem: Suppose $\nu\in
L^{p}(\mathfrak{g})$. For what $q$ is the wrap of $\nu$ in $L^{q}(G)$? That
is, for what $p$ and $q$ do we have

$\|\Phi(\nu)\|_{q}\lesssim\|\nu\|_{p}$

Furthermore, for what $p$ and $q$ do we have

$\|\Phi(j\nu)\|_{q}\lesssim\|\nu\|_{p}$

These will then be applied to the $L^{p}-L^{q}$ bounds of multipliers.

We firstly have the following key result:

######  Lemma 2.

$j\in\L^{p}(\mathfrak{g})$ for $p>2n/(n-l)$.

Proof: Firstly, note that $j$ is bounded at $0\in\mathfrak{g}$. Denote a
neighbourhood $B$ of $0\in\mathfrak{g}$, with
$\int_{B}|j(X)|^{p}\mathrm{\,d}X=C$ being a finite quantity.

$\displaystyle\|j(X)\|^{p}_{p}$
$\displaystyle=\int_{\mathfrak{g}}|j(X)|^{p}\mathrm{\,d}X$
$\displaystyle=\int_{\mathfrak{g}\setminus B}|j(X)|^{p}\mathrm{\,d}X+C$
$\displaystyle\leq\int_{\mathfrak{g}\setminus
B}\;\Bigl{|}\prod_{\alpha\in\Sigma_{+}}\alpha(X/2)\Bigr{|}^{-p}\mathrm{\,d}X+C$
$\displaystyle=\int_{\mathfrak{t}_{+}\setminus
B}\;\Bigl{|}\prod_{\alpha\in\Sigma_{+}}\alpha(H/2)\Bigr{|}^{2-p}\mathrm{\,d}H+C$

The function $\prod_{\alpha\in\Sigma_{+}}\alpha(H/2)$ is a homogeneous
polynomial of degree $k$ on a space of dimension $l$, having positive co-
efficients, and thus will be bounded provided $k(2-p)<l$. That is,
$p>2n/(n-l)$. $\phantom{abcde}\square$

As a consequence of this bound on $j$, we have the following bounds on $p$ and
$q$ for which $\|jf\|_{q}\lesssim\|f\|_{p}$:

######  Lemma 3.

We have

abc a) $\|jf\|_{p}\leq\|f\|_{p}$ for $1<p<\infty$.

abc b) $\|jf\|_{p}\leq\|f\|_{\infty}$ for $p>2n/(n-l)$.

abc c) $\|jf\|_{1}\leq\|f\|_{q}$ for $q<2n/(n+l)$

Proof: These follow from the Hausdorf-Young, Young, and Hölder inequalities,
and are thus best possible.

Proof of a): Let $p$ and $q$ be conjugate exponents. We have

$\displaystyle\|jf\|_{p}$ $\displaystyle\leq\|\widehat{jf}\|_{q}$ (by
Hausdorf-Young) $\displaystyle=\|\hat{j}*\hat{f}\|_{q}$
$\displaystyle\leq\|\hat{j}\|_{1}\|\hat{f}\|_{q}$ (by Young)
$\displaystyle=\|\mu_{\rho}\|_{1}\|f\|_{p}$ $\displaystyle=\|f\|_{p}$

since the mass of the Liouville measure $\mu_{\rho}$ is equal to
$\dim\pi_{0}=1$.

Proof of b): Let $p$ and $q$ be conjugate exponents. We have

$\displaystyle\|jf\|_{p}$ $\displaystyle\leq\|\widehat{jf}\|_{q}$ (by
Hausdorf-Young) $\displaystyle=\|\hat{j}*\hat{f}\|_{q}$
$\displaystyle\leq\|\hat{j}\|_{q}\|\hat{f}\|_{1}$ (by Young)
$\displaystyle=\|j\|_{p}\|f\|_{\infty}$

which is bounded so long as $p>2n/(n-l)$, since we have
$j\in\L^{p}(\mathfrak{g})$ by Lemma 2.

Proof of c): Let $p$ and $q$ be conjugate exponents. By Hölder we have:

$\|jf\|_{1}\leq\|j\|_{p}\|f\|_{q}$

By Lemma 2, $j\in\L^{p}(\mathfrak{g})$ for $p>2n/(n-l)$. Hence, the result
follows if $f\in\L^{q}(\mathfrak{g})$ for $q<2n/(n+l)$.
$\phantom{abcde}\square$

Regarding $L^{p}$ bounds for $\Phi$, we have the following from [4], which is
almost obvious from the definition: If $\nu\in L^{1}(\mathfrak{g})$, then we
have:

$\|\Phi(\nu)\|_{1}\leq\|\nu\|_{1}$ (4.1) $\|\Phi(j\nu)\|_{1}\leq\|\nu\|_{1}$
(4.2)

We now prove a more general $L^{p}$ bound:

######  Theorem 1.

We have the bounds

$\|\Phi(j\mu)\|_{p}\lesssim\|\mu\|_{p},,\;\;1\leq p\leq\infty$ (4.3)

$\|\Phi(\mu)\|_{p}\lesssim\|\mu\|_{p}\,,\;\;\;p\geq 2$ (4.4)

Proof: Let $\mathfrak{t}_{\Gamma}\subseteq\mathfrak{t}$ be a fundamental
domain for $\Gamma$ in $\mathfrak{t}$. We have:

$\displaystyle\|\Phi(j\mu)\|_{p}^{p}$
$\displaystyle=\int_{G}|\Phi(j\mu)(g)|^{p}\mathrm{\,d}g$
$\displaystyle=\frac{1}{|W|}\int_{T}|\Phi(j\mu)(t)|^{p}|\triangle(t)|^{2}\mathrm{\,d}t$
$\displaystyle=\frac{1}{|W|}\int_{\mathfrak{t}_{\Gamma}}|\sum_{\gamma\in\Gamma}\mu(H+\gamma)|^{p}|\triangle(\exp
H)|^{2}\mathrm{\,d}H$
$\displaystyle\leq\frac{1}{|W|}\int_{\mathfrak{t}}|\mu(H)|^{p}|\triangle(\exp
H)|^{2}\mathrm{\,d}H$
$\displaystyle=\int_{\mathfrak{t}^{+}}|\mu(H)|^{p}|j(H)|^{2}\Bigl{|}\prod_{\alpha\in\Sigma_{+}}\alpha(H/2)\Bigr{|}^{2}\mathrm{\,d}H$
$\displaystyle=\int_{\mathfrak{g}}|\mu(X)|^{p}|j(X)|^{2}\mathrm{\,d}X$
$\displaystyle\leq\int_{\mathfrak{g}}|\mu(X)|^{p}\mathrm{\,d}X=\|\mu(X)\|_{p}^{p}$

Similarly for $\Phi(\mu)$, we arrive at

$\displaystyle\|\Phi(\mu)\|_{p}^{p}$
$\displaystyle=\int_{\mathfrak{g}}|\tfrac{\mu}{j}(X)|^{p}|j(X)|^{2}\mathrm{\,d}X$
$\displaystyle=\int_{\mathfrak{g}}|\mu(X)|^{p}|j(X)|^{2-p}\mathrm{\,d}X$
$\displaystyle\leq\|\mu\|_{p}^{p}\||j(X)|^{2-p}\|_{\infty}$
$\displaystyle\leq\|\mu\|_{p}^{p}$

as long as $p\geq 2$. $\phantom{abcde}\square$

Remark: Compare this to [11] Lemma 3, which asserts these bounds as long as

$\sum_{\gamma\in\Gamma}|j(H+2\pi\gamma)|^{np^{\prime}-2}$

is bounded, which is only explicitly addressed for the case of $SU(2)$. Here,
$n=1$ is the case for $\Phi(\mu)$, and $n=2$ is the case for $\Phi(j\mu)$.
However, there appears to be an error in this calculation.

We can use this to prove further $L^{p}-L^{q}$ for $\Phi(j\mu)$ only.

######  Theorem 2.

We have the $L^{q}-L^{1}$ bound

$\|\Phi(j\mu)\|_{1}\lesssim\|\mu\|_{q}\,,\;\;\;q<2n/(n+l)$ (4.5)

and the $L^{\infty}-L^{q}$ bound

$\|\Phi(j\mu)\|_{q}\lesssim\|\mu\|_{\infty}\,,\;\;\;q>2n/(n-l)$ (4.6)

Proof: Let $\nu=j\mu$. From, (4.1) we have that

$\|\Phi(\nu)\|_{1}\lesssim\|\nu\|_{1}$

and so by setting $\nu=j\mu$, we have

$\|\Phi(j\mu)\|_{1}\lesssim\|j\mu\|_{1}$

By Lemma 3 c), it is only possible to have

$\|\Phi(j\mu)\|_{1}\lesssim\|j\mu\|_{1}\leq\|\mu\|_{q}$

for $q<2n/(n+l)$. Similarly, By lemma 3 b), it is only possible to have

$\|\Phi(j\mu)\|_{q}\lesssim\|j\mu\|_{q}\leq\|\mu\|_{\infty}$

for $q>2n/(n-l)$. $\phantom{abcde}\square$

We concluded this section with an additional proof of the $L^{2}$ bound for
$\Phi(j\nu)$ only. This uses the fact that the Fourier transform of $j$ leads
to a region around each point $\lambda\in\Lambda^{+}$ to tessellate over
$\mathfrak{t}^{*}$. We then apply Parseval.

######  Theorem 3.

$\bigl{\|}\Phi(j\nu)\bigr{\|}_{2}\leq\bigl{\|}\nu\bigl{\|}_{2}$

Proof: From Lemma 1 we have that

$d_{\lambda}\Phi(j\nu)^{\wedge}(\pi_{\lambda})=\int_{G}\nu^{\wedge}(\lambda+\rho-\mathrm{ad}(g)(\rho))\mathrm{\,d}g$

Let $Q$ be the convex hull of $\\{w\rho\;|\;w\in W\\}$ and $E$ a bounded
function on $Q$. We have,

$\displaystyle\bigl{\|}\Phi(j\nu)\bigr{\|}^{2}_{2}$
$\displaystyle=\sum_{\lambda\in\Lambda^{+}}\biggl{|}\int_{G}\nu^{\wedge}(\lambda+\rho-\mathrm{ad}(g)(\rho))\mathrm{\,d}g\biggr{|}^{2}$
$\displaystyle\leq\sum_{\lambda\in\Lambda^{+}}\biggl{|}\int_{Q}\biggl{(}\prod_{\alpha\in\Sigma^{+}}\frac{\partial}{\partial\alpha}\biggr{)}\nu^{\wedge}(\lambda+\rho-\xi)E(\xi)\mathrm{\,d}\xi\biggr{|}^{2}$
$\displaystyle\lesssim\int_{\mathfrak{t}^{*}}\,\biggl{|}\biggl{(}\prod_{\alpha\in\Sigma^{+}}\frac{\partial}{\partial\alpha}\biggr{)}\nu(\xi)\biggr{|}^{2}\mathrm{\,d}\xi$
$\displaystyle=\int_{\mathfrak{t}}\,\biggl{|}\prod_{\alpha\in\Sigma^{+}}\alpha(H)\nu(H)\biggr{|}^{2}\mathrm{\,d}H$
$\displaystyle\lesssim\int_{\mathfrak{g}}\,|\nu(X)|^{2}\mathrm{\,d}X$
$\displaystyle=\bigl{\|}\nu\bigl{\|}_{2}^{2}\phantom{abcde}\square$

##  5 An Application: Convergence of Fourier series

We firstly require some notation, which we adopt from [10]. Let $R$ be a Weyl-
invariant, closed, convex polyhedral subset of $\mathfrak{t}$ which contains
the origin. Let $tR=\\{tX|X\in R\\}$. As $t$ ranges over $[1,\infty)$,
$(tR)\cap\mathfrak{t}^{*}_{+}$ generates only a countable number of distinct
sets, which is denoted by $\\{R_{N}\\}$. Define the Weyl-invariant polygonal
operator on $\mathfrak{g}\cong\mathbb{R}^{n}$ by:

$K_{t}\mu(x)=\int_{tR}\exp(-isx)(\hat{\mu}(s))ds$

We define the $N$th partial sum of the Fourier series of a function $f\in
L^{p}(G),\;1\leq p\leq\infty$, as

$S_{N}f(g)=\sum_{\lambda\in R_{N}}d_{\lambda}\chi_{\lambda}*f(g)$

Then we have

######  Theorem 4.

Suppose $R$ is a regular polyhedron, then if $f\in L^{p}_{I}(G),\;p>2n/(n+l)$,
then $S_{N}f$ converges to $f$ almost everywhere.

Proof: Since $\Phi(K_{t})=S_{N}$, and from [6] the Fourier transform converges
for any polygonal region containing the origin for $1<p<\infty$, and therefore
from section 4 $\Phi(K_{t})$ is bounded for $1\leq
p\leq\infty.\phantom{abcde}\square$

##  6 Further Directions

The wrapping map devised in [3] has already been employed as a transference
method in analysis. These could be further extended using the estimates (or
the idea of transferring from the tangent space to a curved space) in this
paper. For example:

  * •

In [4] the concept of a ‘modulator’ was introduced. In particular, this allows
one to consider the class of modulators that ‘wraps’ to a (generalised)
character. Thus, one approach to considering the norms of characters on $G$
would be to consider the norm of the modulator on $\mathfrak{g}$.

  * •

In [7] the wrapping map was used to ‘wrap’ not only heat kernels but also
Brownian motion from $\mathfrak{g}\cong\mathbb{R}^{n}$ to $G$. The estimates
in this paper could be used to compute $L^{p}$ norms for solutions for these
and other partial differential equations on compact Lie groups, as well as
Brownian motions or other stochastic processes.

  * •

In [8] the wrapping map was extended to complex Lie groups and compact
symmetric spaces. $L^{p}$ bounds on these spaces could again be computed by
considering the $L^{p}$ bounds on their respective Lie algebra and tangent
space.

We will consider these in future work.

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