---
abstract: |
    Let $X$ be a non-compact symmetric space of dimension $n$. We prove that if $f\in L^{p}(X)$, $1\leq p\leq 2$, then the Riesz means $S_{R}^{z}\left(
    f\right) $ converge to $f$ almost everywhere as $R\rightarrow \infty $, whenever $\operatorname{Re}z>\left( n-\frac{1}{2}\right) \left( \frac{2}{p}-1\right) $.
author:
- 'A. Fotiadis, M. Marias, and E. Papageorgiou'
title:
- Riesz means on symmetric spaces
- Riesz means on symmetric spaces
---

Introduction and statement of the results
=========================================

The Riesz means $S_{R}^{z}\left( f\right) $, $R>0$, $\operatorname{Re}z\geq 0$, of a function $f$ defined on the cube $\left[ 0,1\right] ^{n}$, are given by $$S_{R}^{z}\left( f\right) \left( \theta \right) =\sum_{\Vert k\Vert <R}\left[
\left( 1-\frac{\left\Vert k\right\Vert ^{2}}{R^{2}}\right) \right] _{+}^{z}%
\hat{f}\left( k\right) e^{2\pi i\left( k,\theta \right) }\text{, \ }k\in
\mathbb{Z}^{n}\text{, }\theta \in \left[ 0,1\right] ^{n},$$where $\hat{f}$ is the Fourier transform of $f$. If $z=0$, then they are just the partial sums $S_{R}\left( f\right) \left( \theta \right)
=\sum_{\left\Vert k\right\Vert <R}\hat{f}\left( k\right) e^{2\pi i\left(
k,\theta \right) }$ of the multiple Fourier series of $f$. Riesz means are used in order to understand the strange behavior of $S_{R}\left( f\right) $ as $R\longrightarrow \infty $. In fact, if $f\in L^{p}\left( \left[ 0,1%
\right] \right) $, $p\in \left( 1,\infty \right) $, then M. Riesz proved in 1910 that $$\left\Vert S_{R}^z\left( f\right) -f\right\Vert _{p}\longrightarrow 0,\text{
as }R\longrightarrow \infty ,  \label{ri}$$[@ZYG], while for $n\geq 2$, C. Fefferman [@fef Theorem 1] proved in 1972 that (\[ri\]) is valid iff $p=2$.

In his seminal work [@STEIN27], E.M. Stein proved in 1958 that for all $n\geq 1$ and $f\in L^{p}\left( \left[ 0,1\right] ^{n}\right) $, $p\in \left( 1,2%
\right] $, then $$\left\Vert S_{R}^{z}\left( f\right) -f\right\Vert _{p}\longrightarrow 0,%
\text{ as }R\longrightarrow \infty ,  \label{st}$$whenever $\operatorname{Re}z$ is larger than the *critical index* $%
z_0(n,p)=\left( \frac{n-1}{2}\right) \left( \frac{2}{p}-1\right) $.

Since then, many authors have investigated the almost everywhere convergence of Riesz means. They have already been extensively studied in the case of $%
\mathbb{R}^{n}$ ([@CHRIST7; @CHRIST8; @STEIN27; @STEIN29] as well as in the book [@DAVCHANG]). In the case of elliptic differential operators on compact manifolds they are treated in ([Berard,ChristSogge,GIUTRA,HORM,SEEGER,Sogge]{}). The case of Lie groups of polynomial volume growth and of Riemannian manifolds of nonnegative curvature is studied in [@ALEXOLO; @MAR] and the case of compact semisimple Lie groups in [@CLERC2].

The rank one noncompact symmetric spaces are treated in 1991 by Giulini and Mauceri in [@GIUMA] and in 1997 the case of $SL(3,\mathbb{H})/Sp(3),$ which has rank 2, is studied in [@zhu]. Here we treat the general case of noncompact symmetric spaces of *all* ranks.

To state our results, we need to introduce some notation. Let $G$ be a semi-simple, noncompact, connected Lie group with finite center and let $K$ be a maximal compact subgroup of $G$. We consider the symmetric space of noncompact type $X=G/K$. Denote by $\mathfrak{g}$ and $\mathfrak{k}$ the Lie algebras of $G$ and $K$, respectively. We have the Cartan decomposition $%
\mathfrak{g}=\mathfrak{p}\oplus \mathfrak{k}$. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$ and $\mathfrak{a}^{\ast }$ its dual. If $\operatorname{dim}\mathfrak{a}=l$, we say that $X$ has rank $l$.

Denote by $\rho $ the half sum of positive roots, counted with their multiplicities. Fix $R\geq \Vert \rho \Vert ^{2}$ and $z\in \mathbb{C}$ with $\operatorname{Re}z\geq 0$, and consider the bounded function $$s_{R}^{z}(\lambda )=\left( 1-\frac{\Vert \rho \Vert ^{2}+\Vert \lambda \Vert
^{2}}{R}\right) _{+}^{z},\;\lambda \in \mathfrak{a^{\ast }}.  \label{mult}$$

Denote by $\kappa _{R}^{z}$ its inverse spherical Fourier transform in the sense of distributions and consider the so-called Riesz means operator $%
S_{R}^{z}$: $$S_{R}^{z}(f)(x)=\int_{G}\kappa _{R}^{z}(y^{-1}x)f(y)dy=(\kappa _{R}^{z}\ast
f)(x),\quad f\in C_{0}(X).  \label{operatorX}$$

For every pair $p,q$ such that $1\leq p,q\leq \infty $, denote by $%
(L^{p}+L^{q})(X)$ the Banach space of all functions $f$ on $X$ which admit a decomposition $f=g+h$ with $g\in L^{p}$ and $h\in L^{q}$. The norm of $f\in
(L^{p}+L^{q})(X)$ is given by $$\Vert f\Vert _{\left( p,q\right) }=\inf \left\{ \Vert f\Vert _{p}+\Vert
g\Vert _{q}:\text{ for all decompositions }f=g+h\right\} .$$For $q\geq 1$, denote by $q^{\prime }$ its conjugate. In the present work we prove the following results.

\[th0\]Let $z\in \mathbb{C}$ with $\operatorname{Re}z\geq n-\frac{1}{2}$ and consider $q>2$. Then, for every $p$ such that $1\leq p\leq q^{\prime }$, and for every $r\in \lbrack qp^{\prime }/(p^{\prime }-q),\infty ]$, $S_{R}^{z}$ is uniformly bounded from $L^{p}(X)$ to $(L^{p}+L^{r})(X)$.

Next we deal with the maximal operator $S_{\ast }^{z}$ associated with Riesz means: $$S_{\ast }^{z}(f)(x)=\sup_{R>\Vert \rho \Vert ^{2}}|S_{R}^{z}(f)(x)|,\;f\in
L^{p}({X}),\;1\leq p\leq 2.$$Set $$Z_0(n,p)=\left( n-\frac{1}{2}\right) \left( \frac{2}{p}-1\right) .$$We have the following result.

\[Th1\] Let $1\leq p\leq 2$ and consider $q>2$. If $\operatorname{Re}z>Z_0(n,p)$, then for every $s\geq pq/(2-p+pq-q)$, there is a constant $c\left( z\right) >0$, such that for every $f\in L^{p}(X)$, $$\Vert S_{\ast }^{z}f\Vert _{\left( p,s\right) }\leq c(z)\Vert f\Vert _{p}.$$

As a corollary of the Theorem \[Th1\], we obtain the almost everywhere convergence of Riesz means.

\[Th2\] Let $1\leq p\leq 2$. If $\operatorname{Re}z>Z_0(n,p)$, then for $f\in
L^{p}(X)$, $$\lim\limits_{R\rightarrow +\infty }S_{R}^{z}f(x)=f(x),\;\text{a.e.. }
\label{aec}$$

Note that in the setting of $\mathbb{R}^{n}$, [@STEIN27], as well as in case of the rank one symmetric spaces, [@GIUMA], (\[aec\]) is valid for $\operatorname{Re}z$ larger than the critical index $z_0(n,p)$ which is smaller than $Z_0(n,p)$. This is a consequence of limitations in our analysis and the lack of an explicit formula for the inverse Abel transform in the general case of a symmetric space. On the other hand, it is interesting that the index $Z_0(n,p)$ only depends on the Euclidean dimension of $X$.

Note also that the multiplier $s_{R}^{z}(\lambda )$ does not extend holomorphically to any tube domain over $\mathfrak{a}^{\ast }$. So, by [@CLERC Theorem 1], the Riesz means operators are not bounded on $L^{p}(X)$ if $p\neq 2$ and consequently the norm summability problem on $L^{p}(X)$, $%
p\neq 2$, is ill posed.

To prove Theorem \[th0\], we split the Riesz means operator in two convolution operators: $S_{R}^{z}=S_{R}^{z,0}+S_{R}^{z,\infty }$. The *local part* $S_{R}^{z,0}$ has a compactly supported kernel around the origin, while the kernel of the *part at infinity* $%
S_{R}^{z,\infty }$ is supported away from the origin. To treat the local part, we follow the approach of [@ALEXOLO; @PAPAGEO]. More precisely, we express the kernel of $S_{R}^{z,0}$ via the heat kernel $p_{t}$ of $X$, and we make use of its estimates, which combined with the fact that the wave operator $\cos t\sqrt{-\Delta }$ of $X$ propagates with finite speed, allow us to prove that $S_{R}^{z,0}$ is continuous on $L^{p}(X)$ for all $p\geq 1$. To treat the part at infinity of the operator, we proceed as in [LOMAanna]{}, and obtain estimates of its kernel by using the support preserving property of the *Abel transform*. We combine known ingredients to study Riesz means for noncompact symmetric spaces of arbitrary rank, an area that remained inactive since the seminal work [@GIUMA] in 1991.

This paper is organized as follows. In Section 2 we present the necessary ingredients for our proofs. In Section 3 we deal with the local part and the part at infinity, of the Riesz mean operator and we prove Theorem \[th0\]. In Section 4 we prove Theorem \[Th1\] and we deduce Theorem \[Th2\].

Preliminaries
=============

In this section we recall some basic facts about symmetric spaces. For details see for example [@AN; @FOMAMA; @HEL; @LOMAjga].

Symmetric spaces
----------------

Let $G$ be a semisimple Lie group, connected, noncompact, with finite center and let $K$ be a maximal compact subgroup of $G$. We denote by $X$ the noncompact symmetric space $G/K$. In the sequel we assume that $\operatorname{dim}%
X=n $. Denote by $\mathfrak{g}$ and $\mathfrak{k}$ the Lie algebras of $G$ and $K $. Let also $\mathfrak{p}$ be the subspace of $\mathfrak{g}$ which is orthogonal to $\mathfrak{k}$ with respect to the Killing form. The Killing form induces a $K$-invariant scalar product on $\mathfrak{p}$ and hence a $G$-invariant metric on $X$. Denote by $\Delta $ the Laplace-Beltrami operator on $X$, by $d(.,.)$ the Riemannian distance and by $dx$ the associated Riemannian measure on $X$. Denote by $\left\vert B\left( x,r\right)
\right\vert $ the volume of the ball $B\left( x,r\right) $, $x\in X$, $r>0$, and recall that there is a $c>0$, such that $\left\vert B\left( x,r\right)
\right\vert \leq cr^{n}$ for all $r\leq 1$.

Fix $\mathfrak{a} $ a maximal abelian subspace of $\mathfrak{p}$ and denote by $\mathfrak{a}^{\ast }$ the real dual of $\mathfrak{a}$. If $\operatorname{dim}%
\mathfrak{a}=l$, we say that $X$ has rank $l$. We also say that $\alpha \in
\mathfrak{a}^{\ast }$ is a root vector, if the space $$\mathfrak{g}^{\alpha }=\left\{ X\in \mathfrak{g}:[H,X]=\alpha (H)X,\text{
for all }H\in \mathfrak{a}\right\} \neq \left\{ 0\right\} .$$

Let $A$ be the analytic subgroup of $G$ with Lie algebra $\mathfrak{a}.$ Let $\mathfrak{a}_{+}\subset \mathfrak{a}$ be a positive Weyl chamber and let $%
\overline{\mathfrak{a}_{+}}$ be its closure. Set $A^{+}=\exp \mathfrak{a}%
_{+} $. Its closure in $G$ is $\overline{A_{+}}=\exp \overline{\mathfrak{a}
_{+}}$. We have the Cartan decomposition $$G=K\overline{A_{+}}K=K\exp \overline{\mathfrak{a}_{+}}K.  \label{kak}$$ Then, each element $x \in G $ is written uniquely as $x =k_1(\exp H)k_2$. We set $$\label{normG}
|x|=|H|, \; H \in \overline{\mathfrak{a}_{+}},$$ the norm on $G$ [@ANO p.2]. Denote by $x_0 = eK$ a base point of $X$. If $x,y\in X$, there are isometries $g,h \in G $ such that $x = gx_0$ and $y
= hx_0$. Because of the Cartan decomposition (\[kak\]), there are $k,
k^{\prime}\in K$ and a unique $H \in \overline{\mathfrak{a}_{+}}$ with $%
g^{-1}h=k \exp H k^{\prime}$ . It follows that $$\ d(x, y)=|H|,$$ where $d(x,y)$ is the geodesic distance on $X$ [@Weber].

Normalize the Haar measure $dk$ of $K$ such that $\int_{K}dk=1$. Then, from the Cartan decomposition, it follows that $$\label{haarmeas}
\int_{G}f(g)dg=\int_{K}dk_{1}\int_{{\mathfrak{a}_{+}}}{\delta (H)}%
dH\int_{K}f(k_{1}\exp (H)k_{2})dk_{2},$$ where the modular function $\delta (H)$ satisfies the estimate $$\label{modular}
\delta(H)\leq ce^{2\rho (H)} .$$ We identify functions on $X = G/K$ with functions on $G$ which are $K$-invariant on the right, and hence bi-$K$-invariant functions on $G$ are identified with functions on $X$, which are $K$-invariant on the left. Note that if $f$ is $K$-bi-invariant, then by (\[haarmeas\]), $$\int_{G}f\left( g\right) dg=\int_{X}f\left( x\right) dx=c\int_{\mathfrak{a}%
_+}f(\exp H)\delta (H)dH.  \label{rightinv}$$

The spherical Fourier transform
-------------------------------

Denote by $S(K\backslash G/K)$ the Schwartz space of $K$-bi-invariant functions on $G$. For $f\in S(K\backslash G/K),$ the spherical Fourier transform $\mathcal{H}$ is defined by $$\ \mathcal{H}f(\lambda )=\int_{G}f(x)\varphi _{\lambda }(x)\;dx,\quad
\lambda \in \mathfrak{a^{\ast }},$$ where $\varphi _{\lambda }$ is the elementary spherical function of index $%
\lambda$ on $G$. Note that from [@HEL] we have the following estimate $$\label{phi0}
\varphi_0(\exp H) \leq c(1+|H|)^de^{-\rho(H)},$$ where $d$ is the cardinality of the set of positive indivisible roots.

Let $S(\mathfrak{a^{\ast }})$ be the usual Schwartz space on $\mathfrak{\
a^{\ast}}$. Denote by $W$ the Weyl group associated to the root system of $(%
\mathfrak{g},\mathfrak{a})$ and denote by $S(\mathfrak{a^{\ast }})^{W}$ the subspace of $W$-invariant functions in $S(\mathfrak{a^{\ast }})$. Then, by a celebrated theorem of Harish-Chandra, $\mathcal{H}$ is an isomorphism between $S(K\backslash G/K)$ and $S(\mathfrak{a^{\ast }})^W$ and its inverse is given by $$\ (\mathcal{H}^{-1}f)(x)=c\int_{\mathfrak{a\ast }}f(\lambda )\varphi
_{-\lambda }(x)\frac{d\lambda }{|\mathbf{c}(\lambda )|^{2}},\quad x\in
G,\quad f\in S(\mathfrak{a^{\ast }})^{W},$$ where $\mathbf{c}(\lambda )$ is the Harish-Chandra function.

The heat kernel on $X$
----------------------

Set $$w_t(\lambda )=e^{-t(\Vert \lambda \Vert ^{2}+\Vert \rho \Vert
    ^{2})},\quad t>0,\;\lambda \in \mathfrak{a^{\ast }},$$ Then the heat kernel $p_{t}(x)$ of $X$ is given by $(\mathcal{H}^{-1}w_t)(x)$ [@ANJ].

The heat kernel $p_t$ on symmetric spaces has been extensively studied, see for example [@ANJ; @ANO]. Sharp estimates of the heat kernel have been obtained by Davies and Mandouvalos in [@DAVMAN] for the case of real hyperbolic space, while Anker and Ji [@ANJ] and later Anker and Ostellari [@ANO], generalized the results of [@DAVMAN] to all symmetric spaces of noncompact type.

Denote by $\Sigma _{0}^{+}$ the set of positive indivisible roots $\alpha $ of $(\mathfrak{g}, \mathfrak{a})$ and by $m_{\alpha }$ the dimension of the root space $\mathfrak{g}^{\alpha }$. In [@ANO Main Theorem] it is proved the following sharp estimate: $$\begin{aligned}
p_{t}(\exp {H})& \leq ct^{-n/2}\left( \underset{\alpha \in \Sigma _{0}^{+}}{    \prod }(1+\langle \alpha ,H\rangle )(1+t+\langle \alpha ,H\rangle )^{\frac{     m_{\alpha }+m_{2\alpha }}{2}-1}\right) \times \notag \\
& \times e^{-\left\Vert \rho \right\Vert ^{2}t-\langle \rho ,H\rangle
    -|H|^{2}/4t},\quad t>0, \;H\in \overline{\mathfrak{a}   _{+}},  \label{heat}\end{aligned}$$where $n=$dim$X$.

From (\[heat\]), we deduce the following crude estimate $$p_{t}(\exp {H})\leq ct^{-n/2}e^{-|H| ^{2}/4t}, \quad
t>0, \;H\in \overline{\mathfrak{a}_{+}},  \label{ostellari}$$which is sufficient for our purposes.

As it is shown in [@GRIG Lemma 3.2], the estimate (\[ostellari\]) of the heat kernel implies that $$\int_{d(x,x_{0})>a}p_{t}^{2}(x)dx\leq ct^{-n/2}e^{-a^{2}/Dt},
\label{grigoryan}$$for some constant $D>0$ sufficiently large.

Proof of Theorem 1
==================

Let $\kappa _{R}^{z}$ be the kernel of the Riesz means operator. We start with a decomposition of $\kappa _{R}^{z}$: $$\kappa _{R}^{z}=\zeta \kappa _{R}^{z}+(1-\zeta )\kappa _{R}^{z}:=\kappa
    _{R}^{z,0}+\kappa _{R}^{z,\infty },  \label{decomp}$$ where $\zeta \in C^{\infty }(K\backslash G/K)$ is a cut-off function such that $$\zeta (x)=%
    \begin{cases}
    1, & \text{if }|x|\leq 1/2, \\
    0, & \text{if }|x|\geq 1.%
    \end{cases}
    \label{zita}$$ Denote by ${S}_{R}^{z,0}$ (resp. ${S}_{R}^{z,\infty }$) the convolution operator on $X$ with kernel $\kappa _{R}^{z,0}$ (resp. $\kappa
    _{R}^{z,\infty })$.

The local part
--------------

We shall prove the following proposition.

\[T0X\] Assume that $\operatorname{Re}z >n/2$. Then the operator ${S}_{R}^{z,0}$ is bounded on $L^{p}(X)$, $1\leq p\leq \infty$, and $\|S_R^{z,
0}\|_{p\rightarrow p}\leq c(z)$, for some constant $c(z)>0$.

The proof is lengthy and it will be given in several steps. First, we shall express the kernel $\kappa _{R}^{z}$ in terms of the heat kernel $%
p_{t} $ of $X$. Then, we shall use the heat kernel estimates (\[ostellari\]) to prove that $\kappa _{R}^{z}$ is integrable in the unit ball $B(0,1)$ of $X$. This implies that $S_{R}^{z,0}$ is bounded on $L^{\infty }(X)$. We then prove that $S_{R}^{z,0}$ is bounded on $L^2(X)$, and an interpolation argument between $L^{\infty}(X)$ and $L^2(X)$ allows us to conclude.

To express the kernel $\kappa _{R}^{z}$ in terms of $p_{t}$, we follow [ALEXOLO]{} and we write $$s_{R}^{z}(\lambda )=s_{R}^{z}(\Vert \lambda \Vert ) =\left( 1-\frac{\Vert
\lambda \Vert ^{2}+\Vert \rho \Vert ^{2}}{R}\right) _{+}^{z}.$$ \[Srz\] Set $r=\sqrt{R}$, $\xi =\Vert \lambda \Vert $ and consider the function $$h_{r}^{z}(\lambda )=h_{r}^{z}(\xi ):=\left( 1-\left( \frac{\sqrt{\xi ^{2}+\Vert \rho
\Vert ^{2}}}{r} \right)^2\right) _{+}^{z}e^{(\sqrt{\xi ^{2}+\Vert \rho \Vert ^{2}}/{%
r})^{2}}.  \label{hr,z}$$ Then, from (\[Srz\]) and (\[hr,z\]) we have $$s_r^z(\lambda)=h_R^z(\lambda)e^{-(\|\lambda\|^2+\| \rho \|^{2})/r^2},$$ and thus $$s_{R}^{z}(-\Delta )=h_{r}^{z}(\sqrt{-\Delta })e^{-1/{r^{2}}(-\Delta )}.
\label{heatexpr}$$

Next, we recall the construction of the partition of unity of [@ALEXOLO p.213] we shall use for the splitting of the operator $s_{R}^{z}(-\Delta )$. For that we set $\psi (\xi )=e^{-1/\xi ^{2}}$, $\xi \geq 0$, and $\psi _{1}(\xi )=\psi (\xi )\psi (1-\xi )$. Then $\psi
_{1}\in C^{\infty }\left( [0,1]\right) $. Set also $\phi (\xi )=\psi
_{1}(\xi +\frac{5}{4})$, and $$\phi _{j}(\xi )=\phi (2^{j}(\xi -1)),\text{ }j\in \mathbb{N}.$$Then $\phi _{j}(\xi )$ is a $C^{\infty }$ function with support in $%
I_{j}=[1-5/2^{j+2},1-1/2^{j+2}]$. The functions $$\chi _{j}(\xi )=\frac{\phi _{j}(\xi )}{\sum_{i\geq 0}\phi _{i}(\xi )},\text{
}$$form the required partition of unity.

Set $$\chi _{j,r}(\xi )=\chi _{j}\left( (\xi /r)^{2}\right) ,\text{ }$$and$$h_{j,r}(\xi ):=h_{R}^{z}(\xi )\chi _{j,r}(\xi ).$$

Consider the operator $$T_{j,r}:=s_{j,r}(-\Delta )=h_{j,r}(\sqrt{-\Delta })e^{-1/r^{2}(-\Delta )}.
\label{Tj,r}$$Note that by (\[Tj,r\]) and (\[heatexpr\]), $$\sum_{j\in \mathbb{N}}T_{j,r}=\sum_{j\in \mathbb{N}}h_{r,j}(\sqrt{-\Delta })e^{-1/r^{2}(-\Delta
)}=h_{r}(\sqrt{-\Delta })e^{-1/r^{2}(-\Delta
)}=s_{R}^{z}(-\Delta ).  \label{sum}$$Denote by $\kappa _{j,r}$ the kernel of the operator $T_{j,r}$. Then, ([Tj,r]{}) implies that $$\kappa _{j,r}(x)=T_{j,r}\delta _{0}\left( x\right) =h_{j,r}(\sqrt{-\Delta }%
)e^{-1/r^{2}(-\Delta )}\delta _{0}\left( x\right) =h_{j,r}(\sqrt{-\Delta }%
)p_{1/r^{2}}(x).  \label{kj,r}$$Consequently, (\[sum\]) and (\[kj,r\]) imply that $$\kappa _{R}^{z}=\sum\limits_{j\in \mathbb{N}}\kappa _{j,r}.  \label{sumk}$$So, to estimate the kernel $\kappa _{R}^{z}$, it suffices to estimate the kernels $\kappa _{j,r}$, which by (\[kj,r\]) are expressed in terms of the heat kernel $p_{t}$ of $X$ and the functions $h_{j,r}$. For that, we shall first recall from [@ALEXOLO p.214] some properties of the functions $h_{j,r}$ we shall use in the sequel.

There is a $c>0$ such that $$|\operatorname{supp}h_{j,r}|\leq cr2^{-j},  \label{alexo6}$$[@ALEXOLO p.214]. Note that the functions $\chi _{j}$, as well as $h_{j,r}$ are radial and thus invariant by the Weyl group [@AN p.612].

Note also that for every $k\in \mathbb{N}$, there is a $c_{k}>0$, such that $$\Vert \chi _{j,r}^{(k)}\Vert _{\infty }\leq c_{k}r^{-k}2^{kj},\quad \Vert
h_{j,r}^{(k)}\Vert _{\infty }\leq c_{k}r^{-k}2^{-(\operatorname{Re}z-k)j}.
\label{alexo7}$$As it is mentioned in [@ALEXOLO p.214], the estimates (\[alexo6\]) and (\[alexo7\]) imply that for every $k\in \mathbb{N}$, there is a $c_{k}>0$ such that $$\int_{|t|\geq s}|\hat{h}_{j,r}(t)|dt\leq c_{k}s^{-k}r^{-k}2^{(k-\operatorname{Re}%
z)j},\;s>0,  \label{hhat}$$where $\hat{h}_{j,r}$ is the euclidean Fourier transform of $h_{j,r}$.

\[L1ball\] Let $\kappa _{R}^{z}$ be the kernel of the Riesz mean operator $S_{R}^{z}$. Then, there is $c>0$, independent of $R$, such that for $%
\operatorname{Re}z>n/2$, $$\Vert \kappa _{R}^{z}\Vert _{L^{1}(B(0,1))}\leq c.$$

For the proof we shall consider different cases. Recall that $R\geq \Vert
\rho \Vert ^{2}$*.*

*Case 1:* $\Vert \rho \Vert ^{2}\leq R\leq \Vert \rho \Vert ^{2}+1$*.*

Combining (\[ostellari\]) and the heat semigroup property, we get that $$\begin{aligned}
\Vert p_{t}\Vert _{L^{2}(X)}& =\left( \int_{X}p_{t}(x,y)p_{t}(y,x)dy\right)
^{1/2}  \label{L2heat} \\
& \leq p_{2t}(x,x)^{1/2}\leq ct^{-n/4}.  \notag\end{aligned}$$Thus, using (\[alexo7\]), (\[kj,r\]) and (\[L2heat\]), we have $$\begin{aligned}
\Vert \kappa _{j,r}\Vert _{L^{1}(B(0,1))}& \leq |B(0,1)|^{1/2}\Vert \kappa
_{j,r}\Vert _{L^{2}(X)}  \notag \\
& \leq c\Vert h_{j,r}(\sqrt{-\Delta })\Vert _{L^{2}\rightarrow L^{2}}\Vert
p_{1/r^{2}}\Vert _{L^{2}(X)}  \notag \\
& \leq c\Vert h_{j,r}\Vert _{\infty }(1/r^{2})^{-n/4}  \label{case1} \\
& \leq c2^{-j\operatorname{Re}z},  \notag\end{aligned}$$where $c=c(\Vert \rho \Vert )$. So, $$\Vert \kappa _{R}^{z}\Vert _{L^{1}(B(0,1))}\leq \sum_{j\in \mathbb{N}}\Vert
\kappa _{j,r}\Vert _{L^{1}(B(0,1))}\leq c\sum_{j\in \mathbb{N}}2^{-j\operatorname{Re}%
z}\leq c,$$ since $\operatorname{Re}z>0$.

*Case 2:* $R\geq \Vert \rho \Vert ^{2}+1.$

Recall that $r=\sqrt{R}$. So, the ball $B(0,1/r)$ is contained in the unit ball. Next, let $i\geq 0$ be such that $2^{i-1}<r\leq 2^{i}$ and consider the annulus $A_{p}=\{x\in X:2^{p}\leq |x|\leq 2^{p+1}$, with $p \geq -i$. We write $$B(0,1)\subset B(0,1/r)\bigcup_{p=-i}^{0}A_{p}.$$

Applying (\[alexo7\]), (\[kj,r\]) and (\[L2heat\]), and proceeding as in Case 1, one can show that $$\Vert \kappa _{j,r}\Vert _{L^{1}(B(0,1/r))}\leq c2^{-j\operatorname{Re}z}.
\label{B1/r}$$

So, to finish the proof of the lemma it remains to prove estimates of the kernels $\kappa _{j,r}$ on the annulus $A_{p}$. For that, we shall use the fact that the kernel $G_{t}(x,y)$, $x,y\in X$, of the wave operator $\cos t%
\sqrt{-\Delta }$, propagates with finite speed [@TAYLOR], that is $$\operatorname{supp}(G_{t})\subset \{(x,y):\;d(x,y)\leq |t|\}.  \label{finitespeed}$$Since $h_{j,r}$ is even, then by the Fourier inversion formula $$\begin{aligned}
\kappa _{j,r}(x)& =[h_{j,r}(\sqrt{-\Delta })p_{r^{-2}}(\cdot )](x)  \notag
\label{k_j,r} \\
& =(2\pi )^{-1/2}\int_{-\infty }^{+\infty }\hat{h}_{j,r}(t)[\cos t\sqrt{%
-\Delta }p_{r^{-2}}(\cdot )]( x)dt.  \notag\end{aligned}$$So, if $x\in A_{p}$, then $$\begin{aligned}
\kappa _{j,r}( x)& =(2\pi )^{-1/2}\int_{-\infty }^{+\infty }\hat{h}%
_{j,r}(t)[\cos t\sqrt{-\Delta }p_{r^{-2}}(\cdot )\mathbf{1}_{\{| y|\leq
2^{p-1}\}}]( x)dt  \notag \\
& +(2\pi )^{-1/2}\int_{-\infty }^{+\infty }\hat{h}_{j,r}(t)[\cos t\sqrt{%
-\Delta }p_{r^{-2}}(\cdot )\mathbf{1}_{\{| y|>2^{p-1}\}}]( x)dt.  \notag \\
& =(2\pi )^{-1/2}\int_{|t|\geq 2^{p-1}}\hat{h}_{j,r}(t)[\cos t\sqrt{-\Delta }%
p_{r^{-2}}(\cdot )\mathbf{1}_{_{\{| y|\leq 2^{p-1}\}}}]( x)dt  \notag \\
& +(2\pi )^{-1/2}\int_{-\infty }^{+\infty }\hat{h}_{j,r}(t)[\cos t\sqrt{%
-\Delta }p_{r^{-2}}(\cdot )\mathbf{1}_{\{| y|>2^{p-1}\}}]( x)dt,\end{aligned}$$where in the last equality we have used the finite propagation speed of the wave operator: if $|y|\leq 2^{p-1}$ and $|x|\geq 2^p$, then (\[finitespeed\]) implies that $|t|\geq 2^{p-1}$.

Using (\[k\_j,r\]), the fact that $\Vert \cos t\sqrt{-\Delta }\Vert
_{2\rightarrow 2}\leq 1$, and the inequality $\Vert \widehat{h}_{j,r}\Vert
_{1}\leq \Vert h_{j,r}\Vert _{\infty }$, applying Cauchy-Schwarz we get that $$\begin{aligned}
\Vert \kappa _{j,r}\Vert _{L^{1}(A_{p})}& \leq c|A_{p}|^{1/2}\int_{|t|\geq
2^{p-1}}|\hat{h}_{j,r}(t)|\Vert p_{r^{-2}}\Vert _{2}dt  \label{L1norm} \\
& +c|A_{p}|^{1/2}\Vert h_{j,r}\Vert _{\infty }\Vert p_{r^{-2}}\mathbf{1}%
_{\{|x|>2^{p-1}\}}\Vert _{2}:=I_{1}+I_{2}.  \notag\end{aligned}$$

From (\[alexo7\]), (\[grigoryan\]) and the fact that $2^{i-1}<r\leq
2^{i} $, it follows that $$\begin{aligned}
I_{2}& \leq c2^{p/2}2^{-j\operatorname{Re}z}(r^{-2})^{-n/4}e^{-2^{p-1}/2Dr^{-2}} \\
& \leq c2^{-j\operatorname{Re}z}2^{p/2}r^{n/2}e^{-2^{p}r^{2}/4D} \\
& \leq c2^{-j\operatorname{Re}z}2^{(p+i)n/2}e^{-D_1 2^{p+i}}.\end{aligned}$$Using the elementary estimate $$e^{-D_1x}x^{n/2}\leq c_kx^{-k},\text{ for all }x>1,\;k\in \mathbb{N},$$we obtain $$I_{2}\leq 2^{-j\operatorname{Re}z}2^{-k(p+i)}.  \label{I1}$$Also, from (\[L2heat\]) we have that $$I_{1}\leq c2^{p/2}(r^{-2})^{-n/4}\int_{|t|\geq 2^{p-1}}|\hat{h}_{j,r}(t)|dt.$$Then, applying (\[hhat\]) for $k>n/2$, we obtain $$\begin{aligned}
I_{1}& \leq c_{n}2^{(p+i)n/2}2^{-pk}r^{-k}2^{(k-\operatorname{Re}z)j}  \notag
\label{I2} \\
& \leq c2^{-(p+i)(k-n/2)}2^{-j(\operatorname{Re}z-n/2)}.\end{aligned}$$Finally, using (\[I1\]) and (\[I2\]), (\[L1norm\]) implies that $$\Vert \kappa _{j,r}\Vert _{L^{1}(A_{p})}\leq c2^{-(p+i)(k-n/2)}2^{-j(\operatorname{Re%
}z-n/2)}.  \label{L1Ap}$$*End of proof of Lemma \[L1ball\].* It follows from (\[B1/r\]) and (\[L1Ap\]) that $$\begin{aligned}
\Vert \kappa _{j,r}\Vert _{L^{1}(B(0,1))}& \leq c2^{-j\operatorname{Re}%
z}+c\sum_{p=-i}^{0}2^{-(p+i)(k-n/2)}2^{-j(\operatorname{Re}z-n/2)} \\
& \leq c2^{-j(\operatorname{Re}z-n/2)}.  \notag\end{aligned}$$So, for $\operatorname{Re}z>n/2$, $$\begin{aligned}
\Vert \kappa _{R}^{z}\Vert _{L^{1}(B(0,1))} &\leq c\sum_{j\geq 0}\Vert
\kappa _{j,r}\Vert _{L^{1}(B(0,1))} \\
&\leq c\sum_{j\geq 0}2^{-j(\operatorname{Re}z-n/2)}\leq c.\end{aligned}$$

$S_{R}^{z,0}$ is bounded on $L^{2}(X)$.

Set $$\kappa _{j,r}^{0}=\zeta \kappa _{j,r},\;T_{j,r}^{0}=\ast \kappa _{j}^{0}%
\text{ and }s_{j,r}^{0}=\mathcal{H}(\kappa _{j,r}^{0}),  \label{cutoff}$$where $\zeta $ is the cut-off function given in (\[zita\]).

By Plancherel theorem and using (\[cutoff\]), we get that $$\begin{aligned}
\Vert T_{j,r}^{0}\Vert _{L^{2}\rightarrow L^{2}}& \leq \Vert
s_{j,r}^{0}\Vert _{L^{\infty }(\mathfrak{a^{\ast }})}=\Vert \mathcal{H}%
(\kappa _{j,r}^{0})\Vert _{L^{\infty }(\mathfrak{a^{\ast }})}  \notag \\
& =\Vert \mathcal{H}(\zeta \kappa _{j,r})\Vert _{L^{\infty }(\mathfrak{%
a^{\ast }})}=\Vert \mathcal{H}(\zeta )\ast \mathcal{H}(\kappa _{j,r})\Vert
_{L^{\infty }(\mathfrak{a^{\ast }})}  \label{tj02} \\
& \leq \Vert \mathcal{H}(\zeta )\Vert _{L^{1}(\mathfrak{a^{\ast }})}\Vert
s_{j,r}\Vert _{L^{\infty }(\mathfrak{a^{\ast }})}.  \notag\end{aligned}$$But $\zeta \in S(K\backslash G/K)$. Therefore, its spherical Fourier transform $\mathcal{H}(\zeta )$, belongs in $S(\mathfrak{a^{\ast }}%
)^{W}\subset L^{1}(\mathfrak{a^{\ast }})$, (see Section 2). So, $$\Vert \mathcal{H}(\zeta )\Vert _{L^{1}(\mathfrak{a^{\ast }})}\leq c(\zeta
)<\infty .$$From (\[tj02\]), (\[Tj,r\]) and (\[alexo7\]) it follows that $$\begin{aligned}
\Vert T_{j,r}^{0}\Vert_{L^{2}\rightarrow L^{2}}&\leq c(\zeta)\Vert
s_{j,r}\Vert _{L^{\infty }(\mathfrak{a^{\ast }})}\leq c(\zeta) \| h_{j,r}(\sqrt{\cdot })e^{-1/r^{2}(\cdot)}\|_{L^{\infty }(\mathfrak{a^{\ast }})} \notag \\&\leq c(\zeta
) \| h_{j,r}(\sqrt{\cdot })\|_{L^{\infty }(\mathfrak{a^{\ast }})}\leq
c(\zeta )2^{-j\operatorname{Re}z}.  \label{Tjr0L2}\end{aligned}$$ Further, by (\[Tjr0L2\]) and the fact that $%
S_{R}^{z,0}=\sum_{j\geq 0}T_{j,r}^{0}$, it follows that $$\Vert S_{R}^{z,0}\Vert _{L^{2}\rightarrow L^{2}} \leq \sum_{j\geq 0}\Vert
T_{j,r}^{0}\Vert _{L^{2}\rightarrow L^{2}} \leq c\sum_{j\geq 0}2^{-j\operatorname{Re}%
z}\leq c<\infty .  \label{t2}$$

*End of the proof of Proposition \[T0X\]*: Since $\kappa
_{R}^{z}=\sum\limits_{j\geq 0}\kappa _{j,r}$, by Lemma \[L1ball\], we have $$\Vert \kappa _{R}^{z,0}\Vert _{L^{1}(X)}=\Vert \zeta \kappa _{R}^{z}\Vert
_{L^{1}(X)}\leq c\Vert \kappa _{R}^{z}\Vert _{L^{1}(B(0,1))}<c.$$This implies that $$\Vert S_{R}^{z,0}\Vert _{L^{\infty }\rightarrow L^{\infty }}\leq c(z).
\label{tapeiro}$$

By interpolation and duality, it follows from (\[tapeiro\]) and (\[t2\]), that for all $p\in \lbrack 1,\infty ]$, $\Vert S_{R}^{z,0}\Vert _{p\rightarrow p}\leq
c(z)$, with $\operatorname{Re}z>n/2$.

The part at infinity
--------------------

For the part at infinity $S_R^{z, \infty}$ of the operator, we proceed as in [@LOMAanna] to obtain estimates of its kernel $\kappa_{R}^{z, \infty}$. Let $l=\operatorname{rank}(X)$.

To begin with, recall that $\kappa _{R}^{z}=\mathcal{H}^{-1}s_{R}^{z}$. Recall also the following result from [@LOMAanna p.650], based on the Abel transform conservation property.

For $x=k_1(\exp H)k_2\in G$, with $|x|>1$ and $k\in \mathbb{N}$ with $k>\frac{n}{2}-\frac{l}{4}$, we have that $$|\kappa _{R}^{z}(x)|\leq c\varphi _{0}(x)\left( \int\limits_{|H|>|x|-\frac{1%
}{2}}\left( \sum_{|\alpha |\leq 2k}|\partial _{H}^{\alpha }(\mathcal{F}%
^{-1}s_{R}^{z})(H)|\right) ^{2}\right) ^{1/2}.  \label{kernelLOMA}$$

Thus, to estimate the kernel for $|x|>1$, it suffices to obtain estimates for the derivatives of the euclidean inverse Fourier transform of $%
s_{R}^{z}(\lambda )$. Denote by $\mathcal{J}_{\nu }(t)=t^{-\nu }J_{\nu }(t)$, $t>0$, where $J_{\nu }$ is the Bessel function of order $\nu $. Then, it holds $$(\mathcal{F}^{-1}s_{R}^{z})(\exp H)=c(n,z)R^{-z}(R-\Vert \rho \Vert
^{2})^{z+l/2}\mathcal{J}_{z+l/2}\left( \sqrt{R-\Vert \rho \Vert ^{2}}%
|H|\right) ,  \label{Fourier}$$ [@DAVCHANG; @GIUMA], and we shall need the following auxiliary lemma.

For every multi-index $\alpha $, it holds that $$|\partial _{H}^{\alpha }\mathcal{J}_{z+l/2}(\sqrt{R-\Vert \rho \Vert ^{2}}%
|H|)|\leq c(R-\Vert \rho \Vert ^{2})^{\frac{|\alpha |}{2}-(\frac{\operatorname{Re}z}{%
2}+\frac{l+1}{4})}|H|^{-(\operatorname{Re}z+\frac{l+1}{2})}.  \label{Bessel deriv}$$

Using the identity $\mathcal{J}_{\nu }^{\prime }(t)=-t\mathcal{J}_{\nu
+1}(t) $, it is straightforward to get that $$\mathcal{J}_{\nu }^{(a)}(t)=(-1)^{a}t^{a}\mathcal{J}_{\nu
+a}(t)+\sum_{j=1}^{[a/2]}c_{j}^{a}t^{a-2j}\mathcal{J}_{\nu +a-j}(t),\;a\in
\mathbb{N},  \label{nth derivative}$$for some constants $c_{j}^{a}$, where $[a]$ denotes the integer part of $a$. Applying the inequality $$\ |\mathcal{J}_{\mu }(t)|\leq c_{\mu }t^{-(\operatorname{Re}\mu +1/2)},\text{ for
all }t>0,$$[@GIUMA], it follows that $$|\partial _{H}^{\alpha }\mathcal{J}_{\nu }(\sqrt{R-\Vert \rho \Vert ^{2}}%
|H|)|\leq c(R-\Vert \rho \Vert ^{2})^{\frac{|\alpha |}{2}-(\frac{\operatorname{Re}%
\nu }{2}+\frac{1}{4})}|H|^{-(\operatorname{Re}\nu +\frac{1}{2})}$$and (\[Bessel deriv\]) follows by taking $\nu =z+l/2$.

\[infinity\] If $R\geq \|\rho\|^2+1$, then $$\label{kappainf}
|\kappa_R^z(x)|\leq c\varphi_{0 }(x)R^{-\frac{1}{2}(\operatorname{Re}z-n+\frac{1}{2}%
)}|x|^{-\operatorname{Re}z-\frac{1}{2}}, \quad |x|>1.$$

From (\[Bessel deriv\]), we get that $$\begin{aligned}
I^{2}& :=\int\limits_{|H|>|x|-\frac{1}{2}}\left( \sum_{|\alpha |\leq
2k}\left\vert \partial _{H}^{a}\mathcal{J}_{z+l/2}\left( \sqrt{R-\Vert \rho
\Vert ^{2}}|H|\right) \right\vert \right) ^{2}dH  \notag  \label{estimate} \\
& \leq c\left( \sum_{|\alpha |\leq 2k}(R-\Vert \rho \Vert ^{2})^{a/2}\right)
^{2}\times\notag \\ &\times \int\limits_{|H|>|x|-\frac{1}{2}}\left( (R-\Vert \rho \Vert ^{2})^{-(%
\frac{\operatorname{Re}z}{2}+\frac{l+1}{4})}|H|^{-(\operatorname{Re}z+\frac{l+1}{2})}\right)
^{2}dH  \notag \\
& \leq c(R-\Vert \rho \Vert ^{2})^{-2(\frac{\operatorname{Re}z}{2}+\frac{l+1}{4}%
)+2k}\int\limits_{u>|x|-\frac{1}{2}}u^{-(l+1)-2\operatorname{Re}z}u^{l-1}du  \notag
\\
& \leq c(R-\Vert \rho \Vert ^{2})^{-2(\frac{\operatorname{Re}z}{2}+\frac{l+1}{4}%
)+2k}\left( |x|-\frac{1}{2}\right) ^{-2\operatorname{Re}z-1}.\end{aligned}$$ For $R\geq \Vert \rho \Vert ^{2}+1$, since $k>\frac{n}{2}-\frac{l}{4}$, we have that $$I\leq c(R-\Vert \rho \Vert ^{2})^{-(\frac{\operatorname{Re}z+l-n}{2}+\frac{1}{4}%
)}\left( |x|-\frac{1}{2}\right) ^{-\operatorname{Re}z-\frac{1}{2}}.
\label{estimate1}$$Using (\[estimate1\]) and (\[Fourier\]), from (\[kernelLOMA\]) we obtain that $$\begin{aligned}
|\kappa _{R}^{z}(x)|& \leq c\varphi _{0}(x)R^{-\operatorname{Re}z}(R-\Vert \rho
\Vert ^{2})^{\operatorname{Re}z+\frac{l}{2}}\times \notag \\ &\times (R-\Vert \rho \Vert ^{2})^{-(\frac{\operatorname{%
            Re}z+l-n}{2}+\frac{1}{4})}\left( |x|-\frac{1}{2}\right) ^{-\operatorname{Re}z-\frac{1%
}{2}} \\
& \leq c\varphi _{0}(x)R^{-\frac{1}{2}(\operatorname{Re}z-n+\frac{1}{2})}|x|^{-\operatorname{%
Re}z-\frac{1}{2}},\quad |x|>1.\end{aligned}$$

Using the estimate (\[estimate1\]) and proceeding as above, one can prove the following result.

\[extracase\]If $\Vert \rho \Vert ^{2}\leq R\leq \Vert \rho \Vert
^{2}+1, $ then $$|\kappa _{R}^{z}(x)|\leq c\varphi _{0}(x)|x|^{-\operatorname{Re}z-\frac{1}{2}%
},\;|x|>1.$$

Finally, we shall prove the following result, which, combined with Proposition \[T0X\], finishes the proof of Theorem 1.

\[11\] \[TinfX\] Let $\operatorname{Re}z\geq n-\frac{1}{2}$ and consider $q>2$. Then for every $p$ such that $1\leq p\leq q^{\prime }$, $S_{R}^{z,\infty }$ is continuous from $L^{p}(X)$ to $L^{r}(X)$ for every $r\in \lbrack qp^{\prime
}/(p^{\prime }-q),\infty ]$, and $\Vert S_{R}^{z,\infty }\Vert
_{p\rightarrow r}\leq c(z)$ for all $R\geq \left\Vert \rho \right\Vert^2 $.

Recall that $\kappa_{R}^{z, \infty}(x)=\kappa_{R}^z(x)$ for every $|x|>1$. Using the estimates of $\kappa _{R}^{z }$ from Lemmata \[infinity\] and \[extracase\], as well as the estimate (\[phi0\]), it follows that $%
\kappa _{R}^{z,\infty }$ is in $L^{q}(X)$ for every $q>2$. Thus, by Young’s inequality, the operator $f\rightarrow |f|\ast \kappa _{R}^{z,\infty }$ maps $L^{p}(X)$, $p\in \lbrack 1,q^{\prime }]$, continuously into $L^{r}(X)$, for every $r\in \lbrack qp^{\prime }/(p^{\prime }-q),\infty ]$.

Further, for $z\geq n-\frac{1}{2}$, in Lemmata \[infinity\] and [extracase]{} the estimates of the kernel $\kappa _{R}^{z,\infty }$ are uniform with respect to $R$. This implies that the norm $\Vert S_{R}^{z,\infty }\Vert
_{p\rightarrow r}$ is bounded by a constant, uniform with respect to $R$.

Proof of Theorem 2 and Theorem 3
================================

In this section we give the proof of Theorem \[Th1\], which deals with the $L^{p}$-continuity of the maximal operator $S_{\ast }^{z}$ associated with the Riesz means. This allows us to deduce the almost everywhere convergence of Riesz means $S_{R}^{z}(f)$ to $f$, as $R\rightarrow +\infty $.

Recall first that $$S_{\ast }^{z}(f)=\sup_{R>\Vert \rho \Vert ^{2}}|S_{R}^{z}(f)|,\;f\in L^{p}({X%
}).  \label{maximal}$$ The following proposition holds true, [@GIUMA Lemma 4.1].

\[S\*L2\] Let $\operatorname{Re}z>0$. Then, $S_*^{z}$ is continuous on $L^2(X)$.

Recall the following decomposition of the kernel $\kappa_{R}^z$ of the operator $S_R^z$: $$\label{decomp1}
\kappa_{R}^{z}=\zeta \kappa_{R}^{z}+(1-\zeta
)\kappa_{R}^z:=\kappa_{R}^{z,0}+\kappa_{R}^{z,\infty },$$where $\zeta \in C^{\infty }(K\backslash G/K)$ is a cut-off function such that $$\zeta (x)= 
\begin{cases}
1, & \text{if }|x|\leq 1/2, \\ 
0, & \text{if }|x|\geq 1.%
\end{cases}
\label{zita1}$$ Denote by ${S}_R^{z,0}$ (resp. ${S}_R^{z,\infty}$) the convolution operators on $X$ with kernel $\kappa _R^{z,0}$ (resp. $\kappa_R^{z,\infty})$. Then, $$\
S_*^zf\leq\sup\limits_{R\geq \|\rho\|^2}|S_R^{z,0}f|+\sup\limits_{R\geq \|\rho\|^2}|S_R^{z, \infty}f|.$$ The following holds true for the part at infinity $S_{\ast}^{z, \infty}$ of the operator $S_{\ast}^{z}$.

\[propinf\] \[TinfXR\] Let $ \operatorname{Re}z \geq n-\frac{1}{2}$. Then, for every $q>2$ and $p\in[1, q^{\prime}]$, $S_*^{z,\infty}$ is continuous from $L^p(X)$ to $L^r(X)$ for every $r\in[qp^{\prime}/(p^{\prime} - q), \infty]$.

The proof relies on the uniform kernel estimates for $\kappa_{R}^{z, \infty}$ implied by Lemmata 9 and 10. It is similar to the proof of Proposition 11, thus omitted.

We shall now prove the following result concerning the local part $S_{\ast}^{z, 0}$ of the Riesz means maximal operator.

\[proplocal\] Let $\operatorname{Re}z\geq n-\frac{1}{2}$. Then, $S_*^{z,0}$ is continuous on $L^p(X)$, for every $p \in (1, \infty)$, and it maps $L^1(X)$ continuously into $L^{1,w}(X)$.

Denote by $e^{t\Delta}, \;t>0$, the heat operator on $X$. Then, $e^{t\Delta}=\ast p_t$, where $p_t$ is the heat kernel on $X$. Recall that $p_t$ is given as the inverse spherical Fourier transform of $$w_t(\lambda)=e^{-t(\|\lambda\|^2+\|\rho\|^2)}, \; \lambda\in \mathfrak{\ a^{\ast}}.$$Consider the radial multiplier $$\label{Mdef1}
M(R^{-1}\lambda):=s_R^z(\lambda)-w_{R^{-1}}(\lambda), \; R\geq \|\rho\|^2.$$

Denote by $K_R(x)$ the kernel of the operator $M(-R^{-1}\Delta)$ and set $K_R^0(x):=\zeta(x)K_R(x)$. Similarly, set $s_R^{z,0}=\mathcal{H}(\zeta \kappa_{R}^{z})=\mathcal{H}( \kappa_{R}^{z,0})$ and $w_{R^{-1}}^0=\mathcal{H}(\zeta p_{R^{-1}})=\mathcal{H}( p^0_{R^{-1}})$. Then, using (\[Mdef1\]), we have that $$\label{mult1}
\mathcal{H}(\kappa_{R}^0):=M^0(-R^{-1}\cdot)=s_R^{z,0}-w_{R^{-1}}^0,$$ From (\[mult1\]) we have that $$\label{1,0}
S_*^{z,0}f=\sup_{R\geq\|\rho\|^2}|s_R^{z,0}(-\Delta)f|\leq \sup_{R\geq\|\rho\|^2}|M^0(-R^{-1} \Delta)f|+\sup_{R\geq\|\rho\|^2}|f \ast p^0_{R^{-1}} |.$$

Consider the operator $(-\Delta)^{i\gamma}$, $\gamma \in \mathbb{R}$, which in the spherical Fourier transform variables is given by $$\
\mathcal{H}((-\Delta)^{i\gamma}f)=(\|\lambda\|^2+\|\rho\|^2)^{i\gamma}\mathcal{H}(f), \; \lambda\in \mathfrak{\ a^{\ast}}.$$ Denote by $$\kappa^{\gamma}=\mathcal{H}^{-1}((\|\lambda\|^2+|\rho\|^2)^{i\gamma})$$ the kernel of $(-\Delta)^{i\gamma}$. As in [@ALEXOLO; @GIUMA], using the Mellin transform $\mathcal{M}(\gamma)$ of the radial function $M(\lambda)$, one can express the operator $M(-R^{-1}\Delta)$ as follows: $$\label{Mellin1}
M(-R^{-1}\Delta)=\int_{-\infty}^{+\infty}\mathcal{M}(\gamma)R^{-i\gamma}(-\Delta)^{i\gamma}d\gamma,$$ where $$\label{calM}
|\mathcal{M}(\gamma)|\leq c(1+|\gamma|)^{-(\operatorname{Re}z+1)},$$ [@GIUMA]. Using (\[Mellin1\]), the kernel $K_R$ of $M(-R^{-1}\Delta)$ is given by $$K_R=\int_{-\infty}^{+\infty}\mathcal{M}(\gamma)R^{-i\gamma} \kappa^{\gamma}d\gamma,$$ and thus $$\begin{aligned}
    K_R^0(x)&=\zeta(x) K_R(x)=\int_{-\infty}^{+\infty}\mathcal{M}(\gamma)R^{-i\gamma} \zeta(x)\kappa^{\gamma}(x)d\gamma\\
    &=\int_{-\infty}^{+\infty}\mathcal{M}(\gamma)R^{-i\gamma} \kappa^{\gamma,0}(x)d\gamma .
    \end{aligned}$$ It follows that $$\
    M^0(-R^{-1}\Delta)=\int_{-\infty}^{+\infty}\mathcal{M}(\gamma)R^{-i\gamma} (-\Delta)^{i\gamma,0}d\gamma .$$ Hence, $$\label{Mellin}
\sup_{R>\|\rho\|^2} |M^0(-R^{-1}\Delta)f|\leq \int_{-\infty}^{+\infty}|\mathcal{M}(\gamma)||(-\Delta)^{i\gamma,0}f|d\gamma$$

\[Anker\] The operator $(-\Delta)^{i\gamma,0}$ is bounded on $L^p$, $p\in(1, \infty)$, with $$\label{Lpnorm}
    \|(-\Delta)^{i\gamma,0}\|_{L^p\rightarrow L^p}\leq c_p(1+|\gamma|)^{[n/2]+1}.$$ Moreover, the operator $(-\Delta)^{i\gamma,0}$ is also $L^1\rightarrow L^{1,w}$ bounded, with $$\label{L1wnorm}
    \|(-\Delta)^{i\gamma,0}\|_{L^1\rightarrow L^{1,w}}\leq c(1+|\gamma|)^{[n/2]+1}.$$

To prove the lemma, we shall proceed as in [@AN]. More precisely, by using a smooth, radial partition of unity (and thus invariant by the Weyl group), we decompose the multiplier $m^{\gamma}(\lambda)=(\|\lambda\|^2+\|\rho\|^2)^{i\gamma}$ as follows $$m^{\gamma}(\lambda)=\sum_{k=0}^{+\infty}m^{\gamma}_k(2^{-k}\lambda),$$ where $\operatorname{supp}m^{\gamma}_0 \subset \{\|\lambda\|\leq 2 \}$ and $\operatorname{supp}m^{\gamma}_k \subset \{1/2\leq \|\lambda\|\leq 2 \}$ for $k\geq 1$. Then, for every $p\in (1, +\infty)$, we have $$\label{Lpimag}
    \|(-\Delta)^{i\gamma,0}\|_{p\rightarrow p}\leq c_p \sup_{k\geq 0}\|m^{\gamma}_k\|_{H_2^{\sigma/2}},$$ with $\sigma >n$ and $H_2^{\sigma/2}$ the usual Sobolev space, [@AN Corollary 17, ii]. Note that the same upper bound also holds for the $L^1\rightarrow L^{1, w}$ norm of $(-\Delta)^{i\gamma,0}$, [@AN]. A straightforward computation yields $$\label{mknorm}
    \|m^{\gamma}_k\|_{H_2^{\sigma/2}}\leq c(1+|\gamma|)^{\sigma /2},$$ for $\sigma /2$ an integer, and Lemma \[Anker\] follows from (\[Lpimag\]).

We shall complete the proof for the $L^p$ boundedness of $S_{*}^{z, 0}$, $p\in(1, \infty)$; the $L^1\rightarrow L^{1,w}$ result is similar, thus omitted. Recall that (\[1,0\]) states that $$S_{\ast}^{z,0}f\leq \sup_{R\geq \|\rho\|^2} |M^0(-R^{-1}\Delta)f|+\sup_{R\geq \|\rho\|^2}|f\ast p^0_{R^{-1}}|.$$ Note that since $p_t(x)\geq 0$, for every $x\in X$, we have $p_t^0(x)\leq p_t(x)$. Thus, $$\label{L1w}
    |(f\ast p_t^0)(x)|\leq (|f|\ast p_t)(x).$$ Also, it is known (see for example [@AN2 Corollary 3.2]) that the heat maximal operator $\sup\limits_{t>0}|e^{t\Delta}f|$ is $L^p$-bounded and also $L^1\rightarrow L^{1,w}$ bounded. This implies that the operator $\sup\limits_{R\geq\|\rho\|^2}|\ast p_{R^{-1}}^0|$ is also $L^p$-bounded and $L^1\rightarrow L^{1,w}$ bounded. Thus, from (\[1,0\]), it follows that to prove the $L^p$-boundedness of the operator $S_{\ast}^{z,0}$, it suffices to prove the $L^p$-boundedness of the operator $\sup\limits_{R\geq\|\rho\|^2}|M^0(-R^{-1}\Delta)|$, and similarly for the $L^1\rightarrow L^{1,w}$ boundedness.

From (\[Mellin\]) and (\[mknorm\]), we have that $$\begin{aligned}
    \|\sup_{R\geq \|\rho\|^2} |M^0(-R^{-1}\Delta)|\|_p &\leq \int_{-\infty}^{+\infty}|\mathcal{M}(\gamma)|\|(-\Delta)^{i\gamma,0}\|_{p\rightarrow p}|\|f\|_pd\gamma\\
    &\leq c|\|f\|_p \int_{-\infty}^{+\infty}(1+|\gamma|)^{-(\operatorname{Re}z+1)})(1+|\gamma|)^{[n/2]+1}d\gamma \\
    &\leq c|\|f\|_p \int_{-\infty}^{+\infty}(1+|\gamma|)^{-(\operatorname{Re}z-[n/2])}d\gamma\leq c\|f\|_p,
    \end{aligned}$$ whenever $\operatorname{Re}z\geq n-\frac{1}{2}$. This completes the proof of Proposition \[proplocal\].

*Proof of Theorem 2.* The proof of Theorem 2 follows from Stein’s complex interpolation, between the $L^p$ result for $p$ close to $1$ and the $L^2$ result (Propositions 12, 13 and 14).

*Proof of Theorem 3.* As it is already mentioned in the Introduction, from Theorem [Th1]{} and Propositions 13 and 14, we deduce the almost everywhere convergence of Riesz means: if $1\leq
p\leq 2$ and $\operatorname{Re}z>\left( n-\frac{1}{2}\right) \left( \frac{2}{p}%
-1\right) $, then $$\lim\limits_{R\rightarrow +\infty }S_{R}^{z}(f)(x)=f(x),\;\text{a.e., for }%
f\in L^{p}(X).$$

[99]{} G. Alexopoulos, N. Lohoué, Riesz means on Lie groups and Riemannian manifolds of nonnegative curvature, *Bull. Soc. Math. France*, **122** (1994), no. 2, 209–223. J.-Ph. Anker, $L^{p}$ Fourier multipliers on Riemannian symmetric spaces of noncompact type, *Ann. of Math.*, **132** (1990), 597–628. J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. *Duke Math. J.*, **65** (1992), no. 2, 257–297.

J.-Ph. Anker, L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, *Geom. Funct. Anal.*, **9** (1999), no. 6, 1035–1091.

J.-Ph. Anker, P. Ostellari, The heat kernel on noncompact symmetric spaces, *Amer. Math. Soc. Transl. Ser. 2*, vol. **210** (2003), 27–46.

P. Berard, Riesz means on Riemannian manifolds, *Proc. Sympos. Pure Math.*, **36** (1980), 1–12.

M. Christ, Weak type $(1,1)$ bounds for rough operators, *Ann. of Math.* (2), **128** (1988), 19–42.

M. Christ, Weak type endpoint bounds for Bochner-Riesz operators, *Rev. Mat. Iberoamericana*, **3** (1987), 25–31.

M. Christ and C. Sogge, Weak type $L^1$ convergence of eigenfunction expansions for pseudodifferential operators, *Invent. Math.*, **94** (1988), 421–453.

J.L. Clerc, Sommes de Riesz et multiplicateurs sur un groupe de Lie compact, *Ann. Inst. Fourier*, **24** (1974), 149–172.

J.L. Clerc, E.M. Stein, $L^{p}$ multipliers for noncompact symmetric spaces, *Proc. Nat. Acad. Sci. U.S.A.*, **71** (1974), 3911–3912.

M. G. Cowling, Harmonic analysis on semigroups, *Ann. of Math.*, **117** (1983), 267–283.

K. Davis, Y. Chang, *Lectures on Bochner-Riesz Means* (London Math. Soc. Lecture Note Series), Cambridge, Cambridge University Press, 1987.

E.B. Davies, N. Mandouvalos, Heat kernel bounds on hyperbolic space and Kleinian groups, *Proc. London Math. Soc.*, (3) **57** (1988), no. 1, 182–208.

C. Fefferman, The multiplier problem for the ball, *Annals of Mathematics*, **94**, no. 2 (1971), 330–336.

A. Fotiadis, N. Mandouvalos, M. Marias, Schrödinger equations on locally symmetric spaces, *Math. Ann.*, **371** (2018), no. 3-4, 1351–1374.

S. Giulini, G. Mauceri, Almost everywhere convergence of Riesz means on certain noncompact symmetric spaces, *Annali di Matematica pura ed applicata* (1991) 159–357.

S. Giulini and G. Travaglini, Estimates for Riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds, *J. Func. Anal.*, **96** (1991), 1–30.

A. Grigor’yan, Gaussian upper bounds for the heat kernel and for its derivatives on a Riemannian manifold, in *Classical and Modern Potential Theory and Applications*, NATO ASI Series, **430**, Springer, Dordrecht.

S. Helgason, *Groups and geometric analysis*, Academic Press, New York, 1984.

C. Herz, The theory of $p$-spaces with an application to convolution operators, *Trans. Amer. Math. Soc.*, **154** (1971), 69–82.

L. H[ö]{}rmander, *On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators*, Some Recent Advances in the Basic Sciences, 155–202, Yeshiva University, New York, 1966.

N. Lohoué, M. Marias, Invariants géometriques des espaces localement symétriques et théorèms de multiplicateurs, *Math. Ann.*, **343** (2009), 639–667.

N. Lohoué, M. Marias, Multipliers on locally symmetric spaces, *J. Geom. Anal.*, **24** (2014), 627–648.

M. Marias, $L^{p}$-boundedness of oscillating spectral multipliers on Riemannian manifolds, *Ann. Math. Blaise Pascal*, **10** (2003), 133–160.

I. P. Natanson, *Constructive Function Theory, Vol. I: Uniform Approximation*, Ungar, New York, 1964.

E. Papageorgiou, Oscillating multipliers on symmetric and locally symetric spaces, https://arxiv.org/abs/1811.03313.

A. Seeger, Endpoint estimates for multiplier transformations on compact manifolds, *Indiana Univ. Math. J.*, **40** (1991), no. 2, 471–533.

E. M. Stein, Localization and summability of multiple Fourier series, Acta Math., **100** (1958), 93–147.

E. M. Stein, C. Weiss, *Introduction to Fourier Analysis on Euclidean Spaces*, Princeton University Press, Princeton, 1971.

E. M. Stein, *Topics in Harmonic Analysis Related to the Littlewood-Paley Theory*, (AM-63), Volume 63, Princeton University Press, Princeton, 1971.

C. Sogge, On the convergence of Riesz means on compact manifolds, *Ann. of Math.* (2), **126** (1987), 439–447.

M. Taylor, $L^p$-estimates on functions of the Laplace operator, *Duke Math. J.,* **58** (1989), no. 3, 773–793.

A. Weber, Heat kernel bounds, Poincaré series, and $L^2$ spectrum for locally symmetric spaces (thesis).

F. Zhu, Almost everywhere convergence of Riesz means on noncompact symmetric space $SL(3, \mathbb{H})/Sp(3)$, *Acta Math. Sinica*, New Series, **13** (1997), no.4, 545–552.

A. Zygmund, *Trigonometric series*, Cambridge, Cambridge University Press, 1935.
