# Thought Graph: Generating Thought Process for Biological Reasoning

Chi-Yang Hsu  University of Texas at AustinAustinTexasUSA<EMAIL_ADDRESS>,
Kyle Cox  University of Texas at AustinAustinTexasUSA<EMAIL_ADDRESS>,
Jiawei Xu  University of Texas at AustinAustinTexasUSA<EMAIL_ADDRESS>,
Zhen Tan  Arizona State UniversityTempeArizonaUSA<EMAIL_ADDRESS>,  Tianhua
Zhai  University of PennsylvaniaPhiladelphiaPennsylvaniaUSA
<EMAIL_ADDRESS>,  Mengzhou Hu  University of California
San DiegoLa JollaCaliforniaUSA<EMAIL_ADDRESS>,  Dexter Pratt  University
of California San DiegoLa JollaCaliforniaUSA<EMAIL_ADDRESS>,
Tianlong Chen  The University of North Carolina at Chapel HillChapel HillNorth
CarolinaUSA<EMAIL_ADDRESS>,  Ziniu Hu  California Institute of
Technology PasadenaCaliforniaUSA<EMAIL_ADDRESS>and  Ying Ding  University
of Texas at AustinAustinTexasUSA<EMAIL_ADDRESS>
(2024)

###### Abstract.

We present the Thought Graph as a novel framework to support complex reasoning
and use gene set analysis as an example to uncover semantic relationships
between biological processes. Our framework stands out for its ability to
provide a deeper understanding of gene sets, significantly surpassing GSEA by
40.28% and LLM baselines by 5.38% based on cosine similarity to human
annotations. Our analysis further provides insights into future directions of
biological processes naming, and implications for bioinformatics and precision
medicine. Here’s our Github Code.

large language model, natural language processing, semantic web biological
process, gene ontology, bioinformatics

††journalyear: 2024††copyright: acmlicensed††conference: Companion Proceedings
of the ACM Web Conference 2024; May 13–17, 2024; Singapore,
Singapore.††booktitle: Companion Proceedings of the ACM Web Conference 2024
(WWW ’24 Companion), May 13–17, 2024, Singapore, Singapore††isbn:
979-8-4007-0172-6/24/05††doi: 10.1145/3589335.3651572††ccs: Applied computing
Bioinformatics††ccs: Computing methodologies Natural language processing
Figure 1. The flowchart presents the application of the Thought Graph to the
Gene Ontology (GO) database. First, Thought Graph uses a gene set and initial
prompt to generate three Biological Processes (BPs). Then, a voter evaluates
and selects the best BP (dark green) and second best BP (light green), which
are more accurately descriptive of the gene set. Each chosen BP, along with a
subsequent prompt, is utilized to generate two additional, more specific BPs.
This procedure is conducted recursively until Thought Graph has reached five
layers. Finally, a voter chooses the final answer from the last layer.

##  1\. Introduction

The systematic study of human disease necessitates an in-depth understanding
of the links between diseases, drugs, phenotypes, genes, and biological
processes (Chandak et al., 2023). Analyzing gene sets that share common
biological functions, locations, or regulatory mechanisms can reveal patterns
in gene behavior across health and disease states, contributing to the
advancement of precision medicine for cancer treatment (Subramanian et al.,
2005). Yet, the task of identifying biological processes from gene sets is
fraught with challenges. Individual genes often display weak signals, and when
strong signals are present, they rarely converge on a singular biological
theme (Subramanian et al., 2005). This complexity is compounded when different
research groups studying the same biological systems arrive at vastly
divergent conclusions.

In response to these challenges, our paper introduces the Thought Graph
framework that aims to address two critical aspects: firstly, it adopted a
Tree-of-Thought (ToT) (Yao et al., 2023) architecture to facilitate thought
expansion with Large Language Model (LLM), ensuring inclusive yet precise
coverage of biological processes across varying specificity levels. Thought
expansion is strategically directed with the assistance of a voter LLM, which
guides the decision-making for future steps. This design aims to mitigate the
potential discrepancies in human annotations encountered by researchers, yet
ensure the quality of the generated processes. Second, our framework
prioritizes the integration of domain-specific external knowledge bases to
understand the semantics of connections within the Thought Graph.
Consequently, it creates semantic relationships like “is-a” and “part-of”
among various thought steps. This strategy not only facilitates complex
decision-making processes but also ensures a more nuanced and interconnected
understanding of biological systems, facilitating data interoperability and
knowledge integration. Our novel contributions can be summarized as follows:

  1. (1)

We propose Thought Graph as a complex reasoning framework that generates
diverse yet precise entities to tackle potential annotations discrepancies in
biological processes.

  2. (2)

Thought Graph can generate thought graphs with edge semantics by recalling
external knowledge (e.g., Gene Ontology) to build rich semantics among thought
steps.

  3. (3)

We have successfully applied Thought Graph in biological process generation
with significant improvement compared to SOTA methods, surpassing GSEA by
40.28% and LLM baselines by 5.38% in cosine similarity score, and identified
the optimal steps of complex reasoning by balancing specificity and accuracy.

##  2\. Related Work

###  2.1. LLM Reasoning

Prompt strategies attempt to decompose a complicated problem into a sequence
of smaller sub-problems so that the problem becomes more manageable (Zhou et
al., 2023). One popular line of study is the Chain-of-Thought (CoT) (Wei et
al., 2023) series, structuring prompts to encourage the LLM to step through
its reasoning process, such as Least-to-Most prompting (Zhou et al., 2023),
and Self-Consistency with CoT (CoT-SC) (Wang et al., 2023). However, these
prompting strategies only utilize linear reasoning paths and struggle in tasks
that require exploration and strategic lookahead. Alternatively, Tree of
Thoughts (ToT) (Yao et al., 2023) and Graph of Thoughts (GoT) (Besta et al.,
2023) excel in these sorts of tasks. LLM-based prompting frameworks’
effectiveness is hindered by inherent limitations such as self-bias and
hallucination. To address this, through in-context learning, our work
introduces the semantics of edges within our Thought Graph, offering
structural information.

###  2.2. Knowledge Graph for LLM Reasoning

LLMs exhibit limitations in integrating new knowledge and occasionally
generate hallucinations. A survey (Agrawal et al., 2023) on knowledge-graph-
based knowledge augmentation in LLMs reveals using knowledge graphs (KGs) as a
source of external information has promising results in reducing
hallucinations. For example, MindMap (Wen et al., 2023) has developed a prompt
pipeline enabling LLMs to comprehend and integrate KG input with their
implicit knowledge. In our approach, we give LLM examples from the gene
ontology knowledge graph to enable the edge semantics.

###  2.3. LLM Reasoning in Biomedical Domain

With the rise of LLMs, recent studies explore LLMs’ application in various
biomedical tasks. The gene set biological process was formulated by (Hu et
al., 2023) as inputting a gene set to an LLM and outputting a biological
process name that is predominant in the system and correctly describing the
function of the gene set. It’s challenging because it requires the LLM to
accurately understand and interpret complex biological concepts, including the
nuanced roles of genes in various cellular contexts and their interactions
within intricate biological networks. Although their results (Hu et al., 2023)
have shown that GPT-4 provides better biological process names than the
conventional Gene Set Enrichment Analysis (GSEA) (Subramanian et al., 2005),
the performance is still far from perfect.

##  3\. Methodology

###  3.1. Problem Formulation

Given a gene set $X=\\{x_{1},x_{2},...,x_{n}\\}$, where each $x_{i}$ is a
gene, the objective $G=F(X)$ is to design a framework $F$ to generate a tree
structure graph $G=(N,E)$ that represents the terms (e.g., biological
processes or pathways) associated with the genes in $X$. In this graph, $N$ is
the set of nodes, and $E$ is the set of edges between these nodes.

###  3.2. Infrastructure of Thought Graph

Our framework Thought Graph adapts ToT (Yao et al., 2023) as a graph generator
to generate a curated tree graph $G$, named Thought Graph. Thought Graph
contains terms as the nodes $N$ and their dependencies as edges $E$. ToT uses
self-reflection to prune and only explore relevant paths. The result, after
exploration, is a graph Thought Graph that illustrates the reasoning path and
a final answer chosen selected from the last layer of the graph as the term
that best describes the gene set $X=\\{x_{1},x_{2},...,x_{m}\\}$.

####  3.2.1. Thoughts expansion

Thought Graph process with $n$ steps proceeds in a breadth-first fashion to
generate a tree of depth $n$. At each step, the process expands the tree by
generating a set of candidate nodes. The first step generates a set of general
“high-level” terms that describe the gene set, and subsequent steps iterate on
the candidate terms by proposing more specific but related terms.

Step 1 (Initial Expansion). The first step is unique from all subsequent steps
because its task is to generate the initial set of $k$ candidate terms
$T_{i}={t^{1}_{1},\dots,t^{1}_{k}}$, where $t^{i}_{j}$ denotes the term $j$
from layer $i$. This set of candidate terms is generated with an “initial
prompt” that takes the gene set as input: $T_{i}\sim p_{\theta}(t^{1}_{1\dots
k}\mid x_{1}\dots,x_{n})$.

Subsequent Steps (Recursive Expansion). In step $i$, we use a Voter ($V$) to
examine and vote across the candidate terms $T_{i}$:
$V(p_{\theta},T_{i})(t^{i}_{j})=1[t^{i}_{j}=t^{i}_{j}*]$, where a good term
$t^{i}_{j}*\sim p^{vote}_{\theta}(t^{i}_{j}*|B)$ is based on comparing the
candidate terms $T_{i}$ in the vote prompt, and select two best terms. For
each selected term from the previous step, $t^{i-1}_{j}$ and gene set $X$ are
added to the “subsequent prompt” for the LLM generates $k$ new terms:
$\\{t^{i}_{1},\dots,t^{i}_{k}\\}\sim p_{\theta}(t^{i}_{1\dots k}\mid x_{1\dots
n},t^{i-1}_{j})$. This process will be conducted recursively for $n-1$ times
(minus the initial expansion). For the final layer, $t^{n}_{1}\sim t^{n}_{k}$
are presented to the LLM to choose the final answer.

###  3.3. Thought Graph

The Thought Graph output provides a representation of the step-wise reasoning
process and integrates edge and node semantics for domain-specific context.
Each node $n_{i}\in N$ is a unique biological process, arranged hierarchically
to reflect varying levels of specificity. The edges $E$ represent the
relationships between these processes. Specifically, we use four pre-defined
relations from the Gene Ontology (GO): is a, part of, has part, and regulates.
These relations establish a hierarchy where, for instance, if A is a subtype
of B, A is deemed more specific than B. This approach helps to elucidate the
nuanced relationships between different biological processes, as detailed in
the GO database.111https://geneontology.org/docs/ontology-relations/

##  4\. Experiment & Evaluation

###  4.1. Data Collection

The GO database (Ashburner et al., 2000) forms the basis of our study. We
specifically use a dataset compiled by Hu et al. (Hu et al., 2023) from the
Biological Process branch of Gene Ontology consisting of 12,214 human gene
sets, each annotated with a biological process name and description. Due to
constraints in financial and computational resources, we randomly select 100
samples from this dataset for evaluation.

###  4.2. Baselines and Model Description

Our evaluation framework includes one domain-specific tool and five LLM
baselines. GSEA (gene set enrichment analysis) (Subramanian et al., 2005) is a
statistical method for associating the expression of groups of genes with
biological processes. Our LLM baselines involve different approaches. Input-
Output (IO) Prompting with zero-shot and zero-shot-9 prompts generate one and
nine unique terms for a single gene set, respectively, with no examples, while
few-shot includes five question-answer examples. Chain-of-Thought (CoT)
employs the two top pathways from Thought Graph for detailed step-by-step
prompting. The approach by Hu et al. (Hu et al., 2023) integrates expert-
curated prompts with specific guidelines that solicit post-hoc critical
analysis. For all LLM instances, we use GPT-4 (gpt-4-1106-preview) in Chat
Completion mode with temperature 0.7. In Thought Graph, we set the number of
steps to five and vote on two samples at each step to proceed.

###  4.3. Evaluation Methods

We use two evaluation metrics: cosine similarity and similarity percentile.
Cosine similarity measures the semantic similarity of the predicted term to
ground-truth term from 0 (no similarity) to 1 (identical). We calculate
similarity using embeddings from SapBERT (Liu et al., 2021), a masked language
model trained to model medical entity relations. After calculating the
similarity between the predicted and ground-truth terms, we also calculate the
similarity between the predicted term and all 12,214 terms in our dataset to
form a null distribution. The percentile score is the percentile of the
similarity between the predicted term and the ground-truth term in our null
distribution. We also include the proportion of similarity percentiles greater
than 99% as a proxy for accuracy.

Among the nine nodes that receive positive votes (indicated as green nodes in
Fig. 1), the one with the highest similarity score is selected as the best
score (b), while the score of the node predicted by Thought Graph is recorded
as the predicted score (p). To establish a fair baseline comparison, we
implemented IO zero-shot-9 to generate nine answers, and select the best of
these for evaluation.

Figure 2. The distribution of the mean similarity score at each layer using Thought Graph (p). The blue line denotes the median of layer 3. Method | Similarity | Percentile | Percentile ¿ 99%
---|---|---|---
GSEA | 24.78% | 52.00% | 17%
IO zero-shot | 45.75% | 77.00% | 27%
IO zero-shot-9 (b) | 59.68% | 91.42% | 61%
IO few-shot | 48.73% | 81.85% | 32%
CoT | 28.83% | 43.71% | 0%
Hu et al. (Hu et al., 2023) | 52.31% | 84.44% | 43%
Thought Graph (p) | 48.53% | 80.90% | 42%
Thought Graph (b) | 65.06% |  95.05% | 65%
Table 1. Mean cosine similarity, mean cosine similarity percentile, and
proportion percentile above 99% of a domain-specific tool and seven LLM
methods on 100 GO data samples.

###  4.4. Performance Evaluation

Overall Performance: Table 1 indicates that Thought Graph (b) achieves the top
performance in both cosine similarity (65.06%) and similarity percentile
(95.05%). In particular, we want to posit that IO zero-shot learning
emphasizes coverage across a wide range of biological process names
(diversity), while the CoT focuses on an in-depth exploration of these names
(specificity), whereas our framework is designed to balance both. Thought
Graph (b) outperforms IO zero-shot-9 (b) and CoT, indicating that depth
without breadth, or vice versa, is insufficient. Thought Graph and other LLM
baselines outperform GSEA, and we also noticed GSEA cannot provide any terms
for 26% of the time, highlighting the advantage of the LLMs. In addition,
Thought Graph (p) scores lower than few-shot and Hu et al. baselines. This may
be the result of our decision to constrain the final answer to the last layer.
However, that Thought Graph (b) outperforms all baselines, including zero-
shot-9, assures us that our approach to generating candidate sets of terms is
promising, and that it is adept at generating a correct answer, but further
optimization is needed.

Thought Graph Analysis: Layer-by-layer analysis in Fig.2 demonstrates
increasing performance from layers 1 to 3, followed by a decrease in layers 4
and 5. This trend suggests a trade-off between specificity and accuracy, with
layer 3 the optimal level by a small margin. While the performance at layer 1
is lower, this is largely because our initial prompt specifically requests
“high-level” terms, and only generates three of them. As expected, the
variance in mean similarity scores increases with the number of layers, as
deeper layers explore deeper and more distant parts of the ontology, but
stabilize after layer 3. In the latter layers, more specific terms are often
voted out in favor of more accurate, general terms, demonstrating the ability
of the voting mechanism to dynamically moderate specificity. Though our
results reflect a modest sample size, layer 3 emerges as an early candidate
for the optimal depth.

##  5\. Conclusion

Thought Graph represents an advancement in the field of gene ontology and
bioinformatics. Integrating gene set analysis with semantic graphs allows for
a more nuanced and comprehensive understanding of biological processes. The
effectiveness of the Thought Graph in mapping complex gene interactions and
functions has been demonstrated, showing its potential to outperform existing
methods. This novel method not only enhances the accuracy of gene set analysis
but also opens avenues for research in understanding genetic influences on
various BPs. Future work can expand on this foundation, exploring broader
applications and measuring uncertainty in complex reasoning.

##  6\. Acknowledgement

We thank the support from NIH (OTA-21-008, R01LM014306-01) and NSF (NSF
2303038, NSF 2333703).

## References

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# Regularity of the Radon-Nikodym Derivative of a Convolution of Orbital
Measures on Noncompact Symmetric Spaces

Boudjemâa Anchouche  Department of Mathematics, College of Science, Kuwait
University, P. O. Box 5969, 13060 Safat, Kuwait<EMAIL_ADDRESS>
###### Abstract.

Let $G/K$ be a Riemannian symmetric space of noncompact type, and let
$\nu_{a_{j}}$, $j=1,...,r$ be some orbital measures on $G$ (see the definition
below). The aim of this paper is to study the $L^{2}$-regularity (resp.
$C^{k}$-smoothness) of the Radon-Nikodym derivative of the convolution
$\nu_{a_{1}}\ast...\ast\nu_{a_{r}}$ with respect to a fixed left Haar measure
$\mu_{G}$ on $G$. As a consequence of a result of Ragozin, [11], we prove that
if $r\geq\,\max_{1\leq i\leq s}\dim{G_{i}}/K_{i}$, then
$\nu_{a_{1}}\ast...\ast\nu_{a_{r}}$ is absolutely continuous with respect to
$\mu_{G}$, i.e., $d\big{(}\nu_{a_{1}}\ast...\ast\nu_{a_{r}}\big{)}/d\mu_{G}$
is in $L^{1}(G)$, where $G_{i}/K_{i}$, $i=1,...,s$, are the irreducible
components in the de Rham decomposition of $G/K$. The aim of this paper is to
prove that $d\big{(}\nu_{a_{1}}\ast...\ast\nu_{a_{r}}\big{)}/d\mu_{G}$ is in
$L^{2}(G)$ (resp. $C^{k}\left(G\right)$) for $r\geq\max_{1\leq i\leq
s}\dim\left({G_{i}}/{K_{i}}\right)+1$ (resp. $r\geq\max_{1\leq i\leq
s}\dim\left({G_{i}}/{K_{i}}\right)+k+1$). The case of a compact symmetric
space of rank one was considered in [2] and [3], and the case of a complex
Grassmannian was considered in [1].

###### Key words and phrases:

Convolution of Orbital Measures, Radon-Nikodym Derivative, Symmetric Spaces of
Noncompact Type

###### 2010 Mathematics Subject Classification:

43A85, 28C10, 43A77, 43A90, 53C35

###### Contents

  1. 1 Introduction
  2. 2 Some Preliminary Results
  3. 3 Spherical Transform of the Density Function
  4. 4 $L^{2}$-regularity of the Radon-Nikodym derivative
  5. 5 $C^{k}$-regularity of the Radon-Nikodym derivative
  6. 6 Case of an Arbitrary Symmetric Space of Noncompact Type

111This paper was uploaded on Researchagate on August 2018, DOI:
10.13140/RG.2.2.34657.97122

##  1\. Introduction

Let $G$ be a real, connected, noncompact semisimple Lie group with finite
center, and $K$ a maximal compact subgroup of $G$, hence $G/K$ is a symmetric
space of noncompact type. Unless otherwise stated, we assume in all what
follows that $G/K$ is irreducible. Let $a_{1}$, $\cdots$, $a_{r}$ be points in
$G-N_{G}\left(K\right)$, where $N_{G}\left(K\right)$ is the normalizer of $K$
in $G$. For each integer $j$, $1\leq j\leq r$, let

${\mathscr{I}}_{a_{j}}(f)=\int_{K}\int_{K}f(k_{1}a_{j}k_{2})d\mu_{K}(k_{1})d\mu_{K}(k_{2})$

where $f$ is a continuous function with compact support in $G$ and $\mu_{K}$ a
normalized Haar measure on $K$.

To the linear functional ${\mathscr{I}}_{a_{j}}$ corresponds, by Riesz
representation theorem, see [5], a measure, which will be denoted by
$\nu_{a_{j}}$, i.e., there exists a Borel measure on $G$ such that

${\mathscr{I}}_{a_{j}}(f)=\int_{G}f(g)d\nu_{a_{j}}(g).$

Since

${\mathscr{I}}_{a_{j}}={\mathscr{I}}_{k_{1}a_{j}k_{2}},$

for any $k_{1},k_{2}$ in $K$, the measure $\nu_{a_{j}}$ is $K$-bi-invariant.

The aim of this paper is to study the regularity of the Radon-Nikodym
derivative of the convolution $\nu_{a_{1}}\ast...\ast\nu_{a_{r}}$ with respect
to a fixed Haar measure $\mu_{\mathsf{G}}$ of $G$. More precisely, the aim is
to prove the following

######  Theorem (Main Theorem).

Let $G/K$ be an irreducible symmetric space of noncompact type, $a_{1}$, …,
$a_{r}$ points in $G-N_{G}(K)$, $\nu_{a_{1}}$,…,$\nu_{a_{r}}$ be the
associated orbital measures.

  1. (1)

If

$r\geq\dim\,G/K+1,$

then

$\frac{d\left(\nu_{a_{1}}\ast...\ast\nu_{a_{r}}\right)}{d\mu_{G}}\in
L^{2}\left(G\right).$

  2. (2)

If

$r\geq\dim\,G/K+k+1,$

then

$\frac{d\left(\nu_{a_{1}}\ast...\ast\nu_{a_{r}}\right)}{d\mu_{G}}\in
C^{k}\left(G\right).$

The paper is organized as follows: Section 2 consists of some preliminary
results. In section 3, we state some results on spherical Transform of the
density function. In section 4, we investigate the $L^{2}$-regularity of the
Radon-Nikodym derivative. In section 5 we study the smoothness of the Radon-
Nikodym derivative. In section 6 we consider the case of a reducible symmetric
space of noncompact type.

##  2\. Some Preliminary Results

Let $G$ be a real, connected, noncompact semisimple Lie group with finite
center, and $K$ a maximal compact subgroup of $G$. Fix a Cartan involution
$\theta$ and let ${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{p}}$ be the
corresponding Cartan decomposition of the Lie algebra ${\mathfrak{g}}$ of $G$,
where ${\mathfrak{k}}$ is the Lie algebra of $K$ and $\mathfrak{p}$ is the
orthogonal complement of $\mathfrak{k}$ with respect to the Killing form of
$\mathfrak{g}$. It is well known that $G/K$ has a structure of a symmetric
space of noncompact type, where the metric is induced from the restriction of
the Killing form of ${\mathfrak{g}}$ to ${\mathfrak{p}}$. Let $\mathfrak{a}$
be a maximal abelian subspace of $\mathfrak{p}$, ${\mathfrak{a}}^{*}$ its
dual, and ${\mathfrak{a}}_{\mathbb{C}}^{*}$ the complexification of
${\mathfrak{a}}^{*}$, i.e., ${\mathfrak{a}}_{\mathbb{C}}^{*}$ is the set of
linear $\mathbb{R}$ forms on ${\mathfrak{a}}$ with values in $\mathbb{C}$. The
dimension of $\mathfrak{a}$ is independent of the choice of a Cartan
decomposition and $\dim\mathfrak{a}$ is called the rank of the symmetric space
$G/K$ and denoted by $l=\operatorname{rank(G\,/\,K)}$.

The Killing form $B$ of $\mathfrak{g}$ is non-degenerate on $\mathfrak{a}$, so
it induces an isomorphism between $\mathfrak{a}$ and ${\mathfrak{a}}^{*}$. The
extension of the inner product on $\mathfrak{a}$ induced by the Killing form
to ${\mathfrak{a}}_{\mathbb{C}}^{*}$ will be denoted also by
$\left\langle.,.\right\rangle$. For an element
$\lambda\in{\mathfrak{a}}_{\mathbb{C}}^{*}$, we denote by $H_{\lambda}$ the
corresponding element in ${\mathfrak{a}}_{\mathbb{C}}$., i.e., by Riesz
Theorem, there exist $H_{\lambda}$ in $\mathfrak{a}$ such that
$\lambda(H)=\left\langle H,H_{\lambda}\right\rangle$, for all
$H\in{\mathfrak{a}}_{\mathbb{C}}$. Under this correspondence,
${\mathfrak{a}}^{*}$ corresponds to $\mathfrak{a}$. We transfer the inner
product defined on ${\mathfrak{a}}_{\mathbb{C}}$ to an inner product on
${\mathfrak{a}}_{\mathbb{C}}^{*}$, via
$\left\langle\lambda,\mu\right\rangle:=\left\langle
H_{\lambda},H_{\mu}\right\rangle$.

For $\alpha$ in ${\mathfrak{a}}^{*}$, we put

${\mathfrak{g}}_{\alpha}=\Big{\\{}X\in{\mathfrak{g}}\mid\operatorname{ad}(H)X=\alpha(H)X,\text{
for all }H\text{ in }{\mathfrak{a}}\Big{\\}}.$

A nonzero element $\alpha$ in ${\mathfrak{a}}^{*}$ is called a restricted root
if ${\mathfrak{g}}_{\alpha}\neq 0$. So we have

${\mathfrak{g}}={\mathfrak{g}}_{0}\oplus\sum_{\alpha\in\Sigma}{\mathfrak{g}}_{\alpha},$

where

${\mathfrak{g}}_{0}=\Big{\\{}X\in{\mathfrak{g}}\mid\operatorname{ad}(H)X=0,\text{
for all }H\text{ in }{\mathfrak{a}}\Big{\\}}.$

Denote by $\Sigma$ the set of restricted roots on $\mathfrak{a}$, and let

${\mathfrak{a}}^{{}^{\prime}}=\Big{\\{}X\in{\mathfrak{a}}\mid\alpha\left(X\right)\neq
0,\text{ for all }\alpha\text{ in }\Sigma\Big{\\}}.$

The connected components of ${\mathfrak{a}}^{{}^{\prime}}$, which are open
convex sets, are called Weyl chambers. Let $M^{\prime}$ (resp. M) be the
normalizer (resp. centralizer) of $A$ in $K$. Then the group
$\mathcal{W}=M^{\prime}/M$, called the Weyl group, is a finite group acting
transitively on the set of Weyl chambers. The induced action of $\mathcal{W}$
on $\mathfrak{a}^{*}$ is given by $w\lambda(H)=\lambda(w^{-1}H)$.

Fix a connected component $\mathfrak{a}^{+}$ of $\mathfrak{a}^{\prime}$, and
call it a positive Weyl chamber, and let

$\Sigma^{+}=\Big{\\{}\alpha\in\Sigma\mid\alpha\left(X\right)>0,\text{ for all
}X\text{ in }{\mathfrak{a}^{+}}\Big{\\}},\text{ and
}\,\,\Sigma^{-}=\Big{\\{}-\alpha\mid\alpha\in\Sigma^{+}\Big{\\}}.$

The set $\Sigma^{+}$ (resp. $\Sigma^{-}$) is called the set of positive (resp.
negative) restricted roots with respect to the Weyl chamber
${\mathfrak{a}}^{+}$. For $\alpha$ in $\Sigma$, we put
$m_{\alpha}=\dim{\mathfrak{g}}_{\alpha}$, and let

$\varrho=\frac{1}{2}\operatorname{Tr}{\operatorname{ad}_{\mathfrak{n}}}_{\mid\mathfrak{a}}=\frac{1}{2}\sum_{\alpha\in\Sigma^{+}}m_{\alpha}\alpha,\hskip
14.22636pt{}\mathfrak{n}=\sum_{\alpha\in\Sigma^{+}}{\mathfrak{g}}_{\alpha}.$

Let $A,K$, and $N$ be Lie subgroups of $G$ with Lie algebras $\mathfrak{a}$,
$\mathfrak{k}$, and $\mathfrak{n}$. Then we have the Iwasawa decomposition

$G=KAN=K\exp\left(\mathfrak{a}\right)N.$

The Iwasawa projection

$H:G\longrightarrow\mathfrak{a}$

is the map which to each $g$ in $G$ associates the unique element $H(g)$ in
$\mathfrak{a}$ such that $g\in K\exp(H(g))N$. Each element $g$ in $G$ can be
written,

$g=k\left(g\right)\exp\big{(}H\left(g\right)\big{)}n\left(g\right),$

where $k\left(g\right)\in K$, and $n\left(g\right)\in N$. We have also the
Cartan decompostion

$G=KAK,$

or more precisely, the decomposition

$G=K\overline{\exp(\mathfrak{a}^{+})}K,$

where $\overline{\exp(\mathfrak{a}^{+})}$ is th closure of
$\exp(\mathfrak{a}^{+})$. So every $K$-bi-invariant function can be considered
as a function on $A$ or a function on $\overline{\exp(\mathfrak{a}^{+})}$. The
following result of Harish-Chandra gives a characterization of spherical
functions on $G$.

######  Theorem 2.1.

[8, Harish-Chandra] The spherical functions on $G$ are parametrized by
$\mathfrak{a}^{*}_{\mathbb{C}}$. More precisely, for each $\lambda$ in
${\mathfrak{a}}_{\mathbb{C}}^{*}$ corresponds a spherical function
$\varphi_{\lambda}$ on $G$ given by

$\varphi_{\lambda}\left(g\right)=\int_{K}\operatorname{e}^{(\sqrt{-1}\lambda-\varrho)\left(H(gk)\right)}d\mu_{K}\left(k\right).$

Moreover, $\varphi_{\lambda}=\varphi_{\nu}$ if and only if $\lambda$ and $\nu$
are in the same orbit under the action of the Weyl group, i.e., there exists
$s$ in the Weyl group $\mathcal{W}$ such that $\lambda=s\nu$.

Let $L^{1}(K\diagdown G\diagup K)$ be the space of $K$-bi-invariant $L^{1}$
functions on $G$. The Harish-Chandra transform of a function $f$ in
$L^{1}(K\diagdown G\diagup K)$, denoted by ${\mathcal{H}}(f)$, is given by

${\mathcal{H}}(f)\left(\lambda\right)=\int_{G}f\left(g\right)\varphi_{\lambda}\left(g^{-1}\right)d\mu_{G}(g).$

It is known that if $\lambda$ is in $\mathfrak{a}^{*}$, then the $(G,K)$
spherical function $\varphi_{\lambda}$ is positive definite, hence bounded,
[12], Corollary 11.5.11 & Proposition 8.4.2 (i). Therefore
${\mathcal{H}}(f)(\lambda)$ is well defined for $f$ in $L^{1}(G)$ and
$\lambda$ in $\mathfrak{a}^{*}$. The Harish-Chandra transform gives a map from
$L^{1}\left(K\diagdown G\diagup K\right)$ to the $\mathcal{W}$-invariant
function on ${\mathfrak{a}}^{*}$.

Let $B(K\diagdown G\diagup K)$ be the space of all linear combinations of
continuous positive definite $K$-bi-invariant functions
$f:G\longrightarrow\mathbb{C}$. Then we have the following

######  Theorem 2.2.

[12, Theorem 11.5.26, (Plancherel Theorem)] For $f\in B(K\diagdown G\diagup
K)\,\cap\,L^{1}(K\diagdown G\diagup K)$,

$\int_{G}\left|f\left(g\right)\right|^{2}d\mu_{G}\left(g\right)={\frac{1}{\left|\mathcal{W}\right|}}\int_{\mathfrak{a}^{*}}\left|{\mathcal{H}}(f)\left(\lambda\right)\right|^{2}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda$

where $\left|\mathcal{W}\right|$ is the number of elements of the Weyl group
$\mathcal{W}$, $\operatorname{c}$ is the Harish-Chandra function.

The following inversion for the spherical transform is also needed in this
paper

######  Theorem 2.3.

[12, Theorem $11.5.26$, (Inversion Formula)] For $f\in B(K\diagdown G\diagup
K)\,\cap\,L^{1}(K\diagdown G\diagup K)$,

$f\left(g\right)={\frac{1}{\left|\mathcal{W}\right|}}\int_{\mathfrak{a}^{*}}{\mathcal{H}}(f)\left(\lambda\right)\varphi_{\lambda}\left(g\right)\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda$

where $\left|\mathcal{W}\right|$ is the number of elements of the Weyl group
$\mathcal{W}$, $\operatorname{c}$ is the Harish-Chandra function.

##  3\. Spherical Transform of the Density Function

The notations are as in section 2. Let $\varphi_{\lambda}$ be the spherical
function for the Gelfand pair $\left(G,K\right)$ corresponding to $\lambda$ in
${\mathfrak{a}}^{*}$. As was mentioned above, the spherical, or Harish-
Chandra, transform of a function $f$ in $L^{1}\left(G\right)$ is defined by

${\mathcal{H}}(f)\left(\lambda\right)=\int_{G}f\left(g\right)\varphi_{\lambda}\left(g^{-1}\right)d\mu_{G}(g).$

We define the spherical transform of a compactly supported measure $\mu$ by

${\mathcal{H}}(\mu)\left(\lambda\right)=\int_{G}\varphi_{\lambda}\left(g^{-1}\right)d\mu(g).$

It is clear that if $\mu$ is absolutely continuous with respect to a fixed
left Haar measure $\mu_{G}$ of $G$, i.e., $d\,\mu=fd\,\mu_{G}$, then

(1) ${\mathcal{H}}(\mu)={\mathcal{H}}(f).$

To simplify the notation, we denote $\nu_{a_{j}}$ simply by $\nu_{j}$ and
hence, denote the convolution $\nu_{a_{1}}\ast...\ast\nu_{a_{r}}$ by
$\nu_{1}\ast...\ast\nu_{r}$.

######  Proposition 3.1.

${\mathcal{H}\big{(}\nu_{1}\ast...\ast\nu_{r}\big{)}(\lambda)}=\prod_{i=1}^{r}\varphi_{\lambda}(a_{i}^{-1}).$

###### Proof.

Let $r=1$. Then

$\displaystyle{\mathcal{H}}\big{(}\nu_{1}\big{)}(\lambda)$
$\displaystyle=\int_{G}\varphi_{\lambda}(g^{-1})d\nu_{1}(g)$
$\displaystyle=\int_{K}\int_{K}\varphi_{\lambda}\big{(}(k_{1}a_{1}k_{2})^{-1}\big{)}d\mu_{K}(k_{1})d\mu_{K}(k_{2}).$
$\displaystyle=\varphi_{\lambda}(a_{1}^{-1})\,\,\,(\text{ since
$\varphi_{\lambda}$ is $K$-bi-invariant and $\mu_{K}(K)=1$}).$

Consider the case $r=2$, i.e, the spherical transform of $\nu_{1}\ast\nu_{2}$.

$\displaystyle{\mathcal{H}}\big{(}\nu_{1}\ast\nu_{2}\big{)}(\lambda)$
$\displaystyle=\int_{G}\varphi_{\lambda}(g^{-1})d\left(\nu_{1}\ast\nu_{2}\right)(g)$
$\displaystyle=\int_{G}\int_{G}\varphi_{\lambda}(g_{2}^{-1}g_{1}^{-1})d\nu_{1}(g_{1})d\nu_{2}(g_{2})$
$\displaystyle=\int_{G}\left(\int_{K}\int_{K}\varphi_{\lambda}(g_{2}^{-1}k_{2}^{-1}a_{1}^{-1}k_{1}^{-1})d\mu_{K}(k_{1})d\mu_{K}(k_{2})\right)d\nu_{2}(g_{2})$
(2)
$\displaystyle=\int_{G}\left(\int_{K}\int_{K}\varphi_{\lambda}(g_{2}^{-1}k_{2}^{-1}a_{1}^{-1})d\mu_{K}(k_{1})d\mu_{K}(k_{2})\right)d\nu_{2}(g_{2})$
(3)
$\displaystyle=\varphi_{\lambda}(a_{1}^{-1})\int_{G}\varphi_{\lambda}(g_{2}^{-1})d\nu_{2}(g_{2})$
(4)
$\displaystyle=\varphi_{\lambda}(a_{1}^{-1})\int_{K}\int_{K}\varphi_{\lambda}(k_{2}^{-1}a_{2}^{-1}k_{1}^{-1})d\mu_{K}(k_{1})d\mu_{K}(k_{2})$
(5)
$\displaystyle=\varphi_{\lambda}(a_{1}^{-1})\varphi_{\lambda}(a_{2}^{-1}).$

To get $(\ref{123b})$ from $(\ref{123a})$, we used the fact that
$\varphi_{\lambda}$ satisfies

$\int_{K}\varphi_{\lambda}\left(xky\right)d\mu_{K}(k)=\varphi_{\lambda}(x)\varphi_{\lambda}(y),$

and to get $(\ref{123d})$ from $(\ref{123c})$, we used the fact
$\varphi_{\lambda}$ is $K$-bi-invariant.
The argument goes by induction for arbitrary $r$. ∎

It is easy to see that the measures $\nu_{{1}},...,\nu_{{r}}$ are supported on
$Ka_{1}K,...,$ $Ka_{r}K$ and, from [1], we know that the measure
$\nu_{{1}}\ast...\ast\nu_{{r}}$ is absolutely continuous with respect to the
Haar measure of the group $G$ if and only if the set $Ka_{1}K\dots Ka_{r}K$ is
of non-empty interior.

Suppose that $G/K$ is an irreducible symmetric space, hence the linear
isotropy representation of $K$ on the tangent space
$T_{eK}\left(G/K\right)\simeq{\mathfrak{g}}/{\mathfrak{k}}$ is irreducible and
non trivial. Then by Theorem 2.5 in [11], if $r\geq\dim G/K$, then
$\nu_{1}\ast...\ast\nu_{r}$ is absolutely continuous with respect to
$\mu_{G}$. If we denote by $\varrho_{a_{1},\cdots,a_{r}}$ the Radon-Nikodym
derivative of $\nu_{1}\ast...\ast\nu_{r}$ with respect to the Haar measure
$\mu_{G}$ of $G$, then

$\varrho_{a_{1},\cdots,a_{r}}=\frac{d\left(\nu_{1}\ast...\ast\nu_{r}\right)}{d\mu_{G}}\in
L^{1}(G).$

From what was said above, the function $\varrho_{a_{1},\cdots,a_{r}}$ is
$K$-bi-invariant, i.e.,

$\varrho_{a_{1},\cdots,a_{r}}\left(k_{1}gk_{2}\right)=\varrho_{a_{1},\cdots,a_{r}}\left(g\right),\,\,\text{
for all $k_{1}$ and $k_{2}$ in $K$.}$

Moreover, from

$\operatorname{supp}\big{(}\varrho_{a_{1},\cdots,a_{r}}\big{)}=\operatorname{supp}\bigg{(}\nu_{1}\ast...\ast\nu_{r}\bigg{)}=Ka_{1}K...Ka_{r}K,$

we see that $\varrho_{a_{1},\cdots,a_{r}}$ is compactly supported. In what
follows, we will denote by $L^{p}\left({K\diagdown G\diagup K}\right)$ the
space of $K$-bi-invariant functions which are in $L^{p}(G)$. Hence we have the
following

######  Proposition 3.2.

If $r\geq\dim G\diagup K$, then $\nu_{1}\ast...\ast\nu_{r}$ is absolutely
continuous with respect to $\mu_{G}$ and its density function
$\varrho_{a_{1},\cdots,a_{r}}$ is in $L^{1}\left({K\diagdown G\diagup
K}\right)$.

##  4\. $L^{2}$-regularity of the Radon-Nikodym derivative

######  Theorem 4.1.

Let $a_{i}=\exp(H_{i})$, where $H_{i}$ is in $\mathfrak{a}^{+}$ for
$i=1,...,r$. If $r\geq\dim\,G/K+1$, then $\varrho_{a_{1},\cdots,a_{r}}$ is in
$L^{2}\left(G\right)$.

To prove the theorem we need some preparatory results.

######  Lemma 4.1.

There exists a positive constant $c$ such that for all $y$ in
$\mathfrak{a}^{*}$, $y\neq 0$, there exist $i$, $1\leq i\leq r$ such that

(6) $\left|\left<\frac{y}{\left\|y\right\|},\alpha_{i}\right>\right|\geq c.$

###### Proof.

Suppose that the inequality (6) is not true, then there exists a sequence
$(y_{p})_{p\geq 1}$ in $\mathfrak{a}^{*}$, $y_{p}\neq 0,\forall p\geq 1$, such
that for all $i$, $1\leq i\leq r$

$\left|\left<\frac{y_{p}}{\left\|y_{p}\right\|},\alpha_{i}\right>\right|<\frac{1}{p},\,\forall\,p\geq
1.$

Since the sequence $\frac{y_{p}}{\left\|y_{p}\right\|}$ is in the unit sphere
$\mathbb{S}^{l-1}$ in $\mathfrak{a}^{*}$, after extracting a subsequence, if
necessary, we can assume without loss of generality that the sequence
$\frac{y_{p}}{\left\|y_{p}\right\|}$ converges to $u\in\mathbb{S}^{l-1}$.
Hence

$\left<u,\alpha_{i}\right>=0.$

Since $\\{\alpha_{i}\\}_{i=1}^{r}$ is a basis for $\mathfrak{a}^{*}$, there
exist real numbers $c_{i}$, $i=1,...,r$ such that

$u=\sum_{i=1}^{r}c_{i}\alpha_{i}.$

Hence

$\left\|u\right\|^{2}=\sum_{i=1}^{r}c_{i}\left<u,\alpha_{i}\right>=0.$

A contradiction, since $u\in\mathbb{S}^{l-1}$. Therefore, the inequality (6)
is established. ∎

Let $\left(a_{j}^{-1}\right)_{j}$ be a sequence of points contained in a
compact subset $\mathcal{C}$ of $A$, such that the $a_{j}\not\in N_{G}(K)$ for
all $j$. For an element $w$ of $\mathcal{W}$, we put

$\Sigma^{+}_{w}(\mathcal{C})=\bigg{\\{}\alpha\in\Sigma^{+}\mid
w\alpha\big{(}\log a\big{)}\neq 0\,\,\text{ for all
}a\in\mathcal{C}\bigg{\\}}.$

The following result is implicit in [6].

######  Proposition 4.1.

For each $j$, there exists a positive constant $C(a_{j})$ such that for all
$\lambda\in\mathfrak{a}^{*}$,

$\left|\varphi_{\lambda}\left(a_{j}^{-1}\right)\right|\leq
C(a_{j})\sum_{w\in\mathcal{W}}\prod_{\alpha\in\Sigma^{+}_{w}(\mathcal{C})}\bigg{(}1+\left|\left<\lambda,\alpha\right>\right|\bigg{)}^{-\frac{1}{2}m_{\alpha}}.$

###### Proof.

For $\lambda$ in $\mathfrak{a}^{*}$, take
$F_{a,H_{\lambda}}(k)=\operatorname{e}^{\sqrt{-1}\left<H(ak),H_{\lambda}\right>}$,
where $H_{\lambda}$ is defined by $\lambda(H)=\left<H_{\lambda},H\right>$ for
all $H$ in $\mathfrak{a}$, and let
$g(k)=\operatorname{e}^{-\rho\left(H(ak)\right)}$. Then, by Theorem 2.1, we
have

$\varphi_{\lambda}(a)=\int_{K}\operatorname{e}^{\left(\sqrt{-1}\lambda-\rho\right)H(ak)}dk=\int_{K}\operatorname{e}^{\sqrt{-1}F_{a,H_{\lambda}}(k)}g(k)dk.$

The proposition follows from [6, Theorem 11.1]. ∎

As a consequence of Proposition 4.1 we have the following:

######  Corollary 4.1.

Let $a_{i}$, $i=1,...,r$ be elements of $A$ such that
$\Sigma^{+}_{w}(\mathcal{C})=\Sigma^{+}$ for all $w$ in $\mathcal{W}$. Then
there exist positive constants
$\widetilde{C(a_{j})}=\left|\Sigma^{+}\right|C(a_{j})$, with $C(a_{j})$ as in
Proposition 4.1 such that

$\left|\varphi_{\lambda}\left(a_{j}^{-1}\right)\right|\leq\widetilde{C(a_{j})}\prod_{\alpha\in\Sigma^{+}}\bigg{(}1+\left|\left<\lambda,\alpha\right>\right|\bigg{)}^{-\frac{1}{2}m_{\alpha}}.$

######  Lemma 4.2.

[12, Lemma 9.3.3] $B(K\diagdown G\diagup K)\cap L^{1}(K\diagdown G\diagup K)$
is dense in $L^{1}(K\diagdown\,G\,\diagup K)$.

Since $\varrho_{a_{1},\cdots,a_{r}}$ is in $L^{1}(K\diagdown\,G\,\diagup K)$,
by Lemma 4.2, there exist a sequence
$\left(\varrho_{a_{1},\cdots,a_{r}}^{j}\right)_{j}$ in $B(K\diagdown G\diagup
K)\cap L^{1}(K\diagdown G\diagup K)$ converging in $L^{1}$ to
$\varrho_{a_{1},\cdots,a_{r}}$. Since $\varrho_{a_{1},\cdots,a_{r}}$ is of
compact support, we can assume that $\varrho_{a_{1},\cdots,a_{r}}^{j}$ is of
compact support. Then, we have the following

######  Proposition 4.2.

$\lim_{j\rightarrow\infty}{\mathcal{H}\big{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}={\mathcal{H}\big{(}\nu_{1}\ast...\ast\nu_{r}\big{)}(\lambda)}$.

###### Proof.

Since

${\mathcal{H}\big{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}=\int_{G}\varrho^{j}_{a_{1},\cdots,a_{r}}\left(g\right)\varphi_{\lambda}\left(g^{-1}\right)d\mu_{G}(g),$

and since $\varphi_{\lambda}$ is bounded for $\lambda$ in $\mathfrak{a}^{*}$,
we get

$\displaystyle\left|{\mathcal{H}\big{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}-{\mathcal{H}\big{(}\nu_{1}\ast...\ast\nu_{r}\big{)}(\lambda)}\right|$
$\displaystyle=\left|\int_{G}\bigg{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\left(g\right)-\varrho_{a_{1},\cdots,a_{r}}\left(g\right)\bigg{)}\varphi_{\lambda}\left(g^{-1}\right)d\mu_{G}(g)\right|$
$\displaystyle\leq
c\int_{G}\left|\bigg{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\left(g\right)-\varrho_{a_{1},\cdots,a_{r}}\left(g\right)\bigg{)}\right|d\mu_{G}(g).$

The Lemma follows from
$\bigints_{G}\left|\bigg{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\left(g\right)-\varrho_{a_{1},\cdots,a_{r}}\left(g\right)\bigg{)}\right|d\mu_{G}(g)\longrightarrow
0$. ∎

###### Proof of Theorem 4.1.

Plancherel Theorem applied to $\varrho_{a_{1},\cdots,a_{r}}^{j}$ says that

(7)
$\int_{G}\left|\varrho_{a_{1},\cdots,a_{r}}^{j}\left(g\right)\right|^{2}d\mu_{G}\left(g\right)={\frac{1}{\left|\mathcal{W}\right|}}\int_{\mathfrak{a}^{*}}\left|{\mathcal{H}}(\varrho_{a_{1},\cdots,a_{r}}^{j})\left(\lambda\right)\right|^{2}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda.$

Combining Proposition 3.1 and Proposition 4.2, we get

$\left|{\mathcal{H}}(\varrho_{a_{1},\cdots,a_{r}}^{j})\left(\lambda\right)\right|^{2}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}\longrightarrow\left|\prod_{i=1}^{r}\varphi_{\lambda}(a_{i}^{-1})\right|^{2}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}.$

By Proposition 7.2 in [9], there exists a positive constant $C$ such that

(8) $\left|\operatorname{c}\left(\lambda\right)\right|^{-2}\leq
C\big{(}1+\left\|\lambda\right\|\big{)}^{n-l},$

for all $\lambda\in\mathfrak{a}^{*}$, where $n=\dim\,G/K$ and
$l=\operatorname{rank(G\,/\,K)}$. Combining (8), Lemma 4.1, and Corollary 4.1,
we get

$\left|\prod_{i=1}^{r}\varphi_{\lambda}(a_{i}^{-1})\right|^{2}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}\leq
C\frac{\big{(}1+\left\|\lambda\right\|\big{)}^{n-l}}{\big{(}1+c\left\|\lambda\right\|\big{)}^{r}},$

where $C$ is a positive constant, and $c$ is the constant which appears in
$\left(\ref{ineq-import}\right)$. Hence

$\left|\prod_{i=1}^{r}\varphi_{\lambda}(a_{i}^{-1})\right|^{2}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}\text{
is in }L^{1}(\mathfrak{a}^{*})\text{ for }r>n.$

Moreover, since $\varrho_{a_{1},\cdots,a_{r}}^{j}$ is of compact support, by
the Paley-Wiener Theorem for spherical functions on semisimple Lie groups
([7], Theorem 3.5), for each positive integer $N$, there exist a constant
$C_{N,j}$ such that

$\left|{\mathcal{H}\big{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}\right|\leq
C_{N,j}\left(1+\left\|\lambda\right\|\right)^{-N},\text{ for arbitrary
$\lambda$ in $\mathfrak{a}^{*}$}.$

Then, we can chose $N$ such that

$\left(1+\left\|\lambda\right\|\right)^{-2N}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}\text{
is in }L^{1}(\mathfrak{a}^{*}).$

As a consequence of Proposition 4.2, without loss of generality, we can take
the sequence $\left(C_{N,j}\right)_{j}$ to be uniformly bounded, i.e., there
exist a positive constant $C$ such that $C_{N,j}\leq C$, for all $j$. Then by
the Lebesgue dominated convergence Theorem, for $r>n$, we have

(9)
$\lim_{j\longrightarrow\infty}\int_{\mathfrak{a}^{*}}\left|{\mathcal{H}}(\varrho_{a_{1},\cdots,a_{r}}^{j})\left(\lambda\right)\right|^{2}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda=\int_{\mathfrak{a}^{*}}\left|\prod_{i=1}^{r}\varphi_{\lambda}(a_{i}^{-1})\right|^{2}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda.$

Using (9) and Fatou’s Lemma, we get

(10)
$\displaystyle\int_{G}\left|\varrho_{a_{1},\cdots,a_{r}}(g)\right|^{2}d\mu_{G}(g)$
$\displaystyle\leq\liminf_{j\rightarrow\infty}\int_{G}\left|\varrho^{j}_{a_{1},\cdots,a_{r}}(g)\right|^{2}d\mu_{G}(g)$
(11)
$\displaystyle\leq\frac{1}{\left|\mathcal{W}\right|}\bigintssss_{{\mathfrak{a}}^{*}}\left(\left|\operatorname{c}(\lambda)\right|^{-1}\prod_{i=1}^{r}\left|\varphi_{\lambda}(a_{i}^{-1})\right|\right)^{2}d\lambda.$

Combining Corollary 4.1 and the estimate (8), we get

$\displaystyle\bigintssss_{G}\left|\varrho_{a_{1},\cdots,a_{r}}(g)\right|^{2}d\mu_{G}(g)$
$\displaystyle\leq\frac{1}{\left|\mathcal{W}\right|}\bigg{(}\prod_{i=1}^{r}\left(\widetilde{C(a_{i})}\right)\bigg{)}^{2}{\bigintss_{{\mathfrak{a}}^{*}}}\big{(}1+\left\|\lambda\right\|\big{)}^{n-l}\Bigg{(}\prod_{\alpha\in\Sigma^{+}}\bigg{(}1+\left|\left<\lambda,\alpha\right>\right|\bigg{)}^{-m(\alpha)}\Bigg{)}^{r}d\lambda$
$\displaystyle\leq
C(a)\bigintssss_{{\mathfrak{a}}^{*}}\big{(}1+\left\|\lambda\right\|\big{)}^{n-l}\prod_{\alpha\in\Sigma^{+}}\bigg{(}1+\left|\left<\lambda,\alpha\right>\right|\bigg{)}^{-rm(\alpha)}\,d\lambda,$

where

$C(a)=C\left(a_{1},...,a_{r}\right)=\frac{1}{\left|\mathcal{W}\right|}\bigg{(}\prod_{i=1}^{r}\left(\widetilde{C(a_{i})}\right)\bigg{)}^{2}$

is a constant which depends only on the points $a_{1},...,a_{r}$.

From the inequality (6), we deduce that

$\bigintssss_{{\mathfrak{a}}^{*}}\big{(}1+\left\|\lambda\right\|\big{)}^{n-l}\prod_{\alpha\in\Sigma^{+}}\bigg{(}1+\left|\left<\lambda,\alpha\right>\right|\bigg{)}^{-r\,m_{\alpha}}\,d\lambda$
$=\bigintss_{0}^{\infty}t^{l-1}\Bigg{(}\bigintss_{\mathbb{S}^{l-1}}\frac{\big{(}1+\left\|t\xi\right\|\big{)}^{n-l}}{\prod_{\alpha\in\Sigma^{+}}\bigg{(}1+\left|\left<t\xi,\alpha\right>\right|\bigg{)}^{r\,m_{\alpha}}}d\sigma(\xi)\Bigg{)}d\,t$
$=\bigintss_{0}^{\infty}t^{l-1}(1+t)^{n-l}\Bigg{(}\bigintss_{\mathbb{S}^{l-1}}\frac{d\sigma(\xi)}{\prod_{\alpha\in\Sigma^{+}}\bigg{(}1+t\left|\left<\xi,\alpha\right>\right|\bigg{)}^{r\,m_{\alpha}}}\Bigg{)}d\,t$
$\leq
C\bigintss_{0}^{\infty}\frac{t^{l-1}(1+t)^{n-l}}{(1+ct)^{r\min\,{m_{\alpha_{i}}}}}d\,t\\\
\leq C\bigintss_{0}^{\infty}\frac{t^{l-1}\,(1+t)^{n-l}}{(1+ct)^{r}}d\,t,$

where $d\,\sigma$ is the induced Lebesgue measure on the unit sphere
$\mathbb{S}^{l-1}$. The Theorem follows from the fact that the integral

$\bigintss_{0}^{\infty}\frac{t^{l-1}(1+t)^{n-l}}{(1+ct)^{r}}d\,t$

is convergent for

$r>n.$

∎

##  5\. $C^{k}$-regularity of the Radon-Nikodym derivative

The aim of this section is to prove part $(2)$ of the Main Theorem. As a
consequence of Lemma 4.2, we have the following

######  Proposition 5.1.

Let $X$ be an element of $\mathfrak{g}$. If $r>n+1$, then

$X\varrho^{j}_{a_{1},\cdots,a_{r}}(g)\longrightarrow
X\varrho_{a_{1},\cdots,a_{r}}(g).$

###### Proof.

Applying the inversion formula for the spherical transform, Theorem 2.3, to
the function $\varrho_{a_{1},\cdots,a_{r}}^{j}$, we get

$\displaystyle\varrho^{j}_{a_{1},\cdots,a_{r}}(g)$
$\displaystyle=c\,\bigintssss_{{\mathfrak{a}}^{*}}{\mathcal{H}\big{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}\varphi_{\lambda}\left(g\right)\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda,$

where $c$ is a constant independent of the function
$\varrho^{j}_{a_{1},\cdots,a_{r}}$.

Hence, for $X$ in $\mathfrak{g}$, we have,

$\displaystyle X\varrho^{j}_{a_{1},\cdots,a_{r}}(g)$
$\displaystyle=\frac{d}{dt}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{(}g\exp(tX)\big{)}_{\mid
t=0}$
$\displaystyle=c\,\frac{d}{dt}\bigg{(}\bigintssss_{{\mathfrak{a}}^{*}}{\mathcal{H}\big{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}\varphi_{\lambda}\left(g\exp(tX)\right)\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda,\bigg{)}_{\mid\,t=0}$
$\displaystyle=c\,\bigintssss_{{\mathfrak{a}}^{*}}{\mathcal{H}\big{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}\frac{d}{dt}\bigg{(}\varphi_{\lambda}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda.$

The justification of the derivation under the integral sign follows the same
argument as in section 4 and uses Gangolli-Varadarajan estimate (see [4],
Proposition 3, (vi)).

From Lemma 4.2, we know that

$\tiny{{\mathcal{H}\big{(}\varrho^{j}_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}\frac{d}{dt}\bigg{(}\varphi_{\lambda}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}\rightarrow{\mathcal{H}\big{(}\varrho_{a_{1},\cdots,a_{r}}\big{)}(\lambda)}\frac{d}{dt}\bigg{(}\varphi_{\lambda}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}}.$

Moreover, as a consequence of Proposition 3.1 we get

$\displaystyle\mathcal{H}\big{(}\varrho_{a_{1},\cdots,a_{r}}\big{)}(\lambda)\frac{d}{dt}\bigg{(}\varphi_{\lambda}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}$
$\displaystyle=\prod_{i=1}^{r}\varphi_{\lambda}(a_{i}^{-1})\frac{d}{dt}\bigg{(}\varphi_{\lambda}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}.$

With the help of Gangolli-Varadarajan estimate (see [4], Proposition 3, (vi)),
we get

$\left|\prod_{i=1}^{r}\varphi_{s\xi}(a_{i}^{-1})\frac{d}{dt}\bigg{(}\varphi_{s\xi}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(s\xi\right)\right|^{-2}\right|\leq
C\,\frac{s^{l-1}(1+s)(1+s)^{n-l}}{(1+cs)^{r}},$

where $C>0$ is independent of $g$, and since for $r>n+1$, the function

$\prod_{i=1}^{r}\varphi_{s\xi}(a_{i}^{-1})\frac{d}{dt}\bigg{(}\varphi_{s\xi}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\operatorname{c}\left(s\xi\right)^{-2}\,\in
L^{1}(\mathfrak{a}^{*}).$

The Proposition follows from the Paley-Wiener Theorem for spherical functions
on semisimple Lie groups ([7], Theorem 3.5) and the Lebesgue dominated
convergence theorem. ∎

Let $r>n+1$. Using Proposition 5.1, by passing to the limit, we get

$\displaystyle X\varrho_{a_{1},\cdots,a_{r}}(g)$
$\displaystyle=c\bigintssss_{{\mathfrak{a}}^{*}}{\mathcal{H}\big{(}\nu_{1}\ast...\ast\nu_{r}\big{)}(\lambda)}\frac{d}{dt}\bigg{(}\varphi_{\lambda}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda,$
$\displaystyle=c\,\bigintssss_{{\mathfrak{a}}^{*}}\prod_{i=1}^{r}\varphi_{\lambda}(a_{i}^{-1})\frac{d}{dt}\bigg{(}\varphi_{\lambda}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(\lambda\right)\right|^{-2}d\,\lambda\big{(}\text{
by Proposition \ref{widehatofprod}}\big{)},$
$\displaystyle=\int_{0}^{\infty}\int_{\mathbb{S}^{l-1}}\prod_{i=1}^{r}\varphi_{s\xi}(a_{i}^{-1})\frac{d}{dt}\bigg{(}\varphi_{s\xi}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(s\xi\right)\right|^{-2}\,d\sigma(\xi)\,ds.$

Since the function

$g\longmapsto\prod_{i=1}^{r}\varphi_{s\xi}(a_{i}^{-1})\frac{d}{dt}\bigg{(}\varphi_{s\xi}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(s\xi\right)\right|^{-2}$

is infinitely differentiable (see [4, Proposition 3]) and

$\left|\prod_{i=1}^{r}\varphi_{s\xi}(a_{i}^{-1})\frac{d}{dt}\bigg{(}\varphi_{s\xi}\left(g\exp(tX)\right)\bigg{)}_{\mid\,t=0}\left|\operatorname{c}\left(s\xi\right)\right|^{-2}\right|\leq
C\,\frac{s^{l-1}(1+s)(1+s)^{n-l}}{(1+cs)^{r}},$

where $C>0$ is independent of $g$, the differentiation under integral sign
theorem implies that if $r>n+1$, then

$\varrho_{a_{1},\cdots,a_{r}}\in C^{1}(G).$

We conclude by iteration that if $r>n+k$, then

$\varrho_{a_{1},\cdots,a_{r}}\in C^{k}(G).$

##  6\. Case of an Arbitrary Symmetric Space of Noncompact Type

Suppose that $G/K$ is an arbitrary symmetric space of noncompact type, not
necessarily irreducible. Then by [10], each irreducible factor in the de Rham
decomposition of $G/K$ is again a Riemannian symmetric space of the noncompact
type. Hence we can write

(12) $G/K=G_{1}/K_{1}\times\cdots\times G_{s}/K_{s},$

where $G_{i}/K_{i}$, $i=1,...,s$ are irreducible symmetric spaces of
noncompact type. Fix left Haar measures $\mu_{G_{i}}$ on $G_{i}$, $i=1,...,s$.
Then

$\mu_{G}=\mu_{G_{1}}\times\cdots\times\mu_{G_{s}}$

is a left Haar measure on $G$.
Let

$a_{i}=\left(a_{i}^{1},\cdots,a_{i}^{s}\right)$

be an element of $G$ and assume that $a_{i}^{j}\not\in N_{G_{j}}(K_{j})$,
where $N_{G_{j}}(K_{j})$ is the normalizer of $K_{j}$ in $G_{j}$,
$j=1,\cdots,s$. Then it can be seen that the measures $\nu_{a_{i}}$, defined
in section 1, can be written as

$\nu_{a_{i}}=\nu_{a_{i}^{1}}\times\cdots\times\nu_{a_{i}^{s}},$

where $\nu_{a_{i}^{j}}$ is the measure corresponding to the linear functional

${\mathscr{I}}_{a_{i}^{j}}(f)=\int_{K_{j}}\int_{K_{j}}f(k_{1}a_{i}^{j}k_{2})d\mu_{K_{j}}(k_{1})d\mu_{K_{j}}(k_{2}),$

where $\mu_{K_{j}}$ a fixed Haar measure on $K_{j}$, and $f$ is a continuous
function with compact suppost on $G_{j}$.

Applying Ragozin’s result to each component of the de Rham decomposition of
$G/K$, we deduce that if

$r\geq\max_{1\leq i\leq s}{\dim G_{i}/K_{i}},$

then $\nu_{a_{1}^{j}}\ast...\ast\nu_{a_{r}^{j}}$ is absolutely continuous with
respect to $\mu_{G_{j}}$ for $j=1,\cdots s$. If we denote by
$\varrho_{a_{1}^{j},\cdots,a_{r}^{j}}$ the Radon-Nikodym derivative of
$\nu_{a_{1}^{j}}\ast...\ast\nu_{a_{r}^{j}}$ with respect to the Haar measure
$\mu_{G_{j}}$, then

$\varrho_{a_{1}^{j},\cdots,a_{r}^{j}}=\frac{d\left(\nu_{a_{1}^{j}}\ast...\ast\nu_{a_{r}^{j}}\right)}{d\mu_{G_{j}}}\in
L^{1}(G_{j}).$

Since

$\nu_{a_{1}}\ast...\ast\nu_{a_{r}}=\left(\nu_{a_{1}^{1}}\ast...\ast\nu_{a_{r}^{1}}\right)\times\cdots\times\left(\nu_{a_{1}^{s}}\ast...\ast\nu_{a_{r}^{s}}\right),$

we deduce that if $r\geq\max_{1\leq i\leq s}{\dim G_{i}/K_{i}}$ then the
measure $\nu_{a_{1}}\ast...\ast\nu_{a_{r}}$ is absolutely continuous with
respect to $\mu_{G}=\mu_{G_{1}}\times\cdots\times\mu_{G_{s}}$. If we denote by
$\varrho_{a_{1},\cdots,a_{r}}$ the Radon-Nikodym derivative of
$\nu_{a_{1}}\ast...\ast\nu_{a_{r}}$ with respect to $\mu_{G}$, then

$\displaystyle\varrho_{a_{1},\cdots,a_{r}}\left(x_{1},\cdots,x_{s}\right)$
$\displaystyle=\frac{d\left(\nu_{a_{1}}\ast...\ast\nu_{a_{r}}\right)}{d\mu_{G}}$
$\displaystyle=\frac{d\bigg{(}\left(\nu_{a_{1}^{1}}\ast...\ast\nu_{a_{r}^{1}}\right)\times\cdots\times\left(\nu_{a_{1}^{s}}\ast...\ast\nu_{a_{r}^{s}}\right)\bigg{)}}{d\left(\mu_{G_{1}}\times\cdots\times\mu_{G_{s}}\right)}$
$\displaystyle=\frac{d\left(\nu_{a_{1}^{1}}\ast...\ast\nu_{a_{r}^{1}}\right)}{d\mu_{G_{1}}}\left(x_{1}\right)\cdots\frac{d\left(\nu_{a_{1}^{s}}\ast...\ast\nu_{a_{r}^{s}}\right)}{d\mu_{G_{s}}}\left(x_{s}\right)$
$\displaystyle=\varrho_{a_{1}^{1},\cdots,a_{r}^{1}}(x_{1})\cdots\varrho_{a_{1}^{s},\cdots,a_{r}^{s}}(x_{s})\in
L^{1}(G).$

Hence

(13)
$\displaystyle\int_{G}\left|\varrho_{a_{1},\cdots,a_{r}}\right|^{2}d\,\mu_{G}$
$\displaystyle=\prod_{i=1}^{s}\int_{G_{i}}\left|\varrho_{a_{1}^{i},\cdots,a_{r}^{i}}\right|^{2}d\,\mu_{G_{i}}.$

As a consequence of $(\ref{produit})$ and part (i) of the Main Theorem, we
deduce that if

$r>\max_{1\leq i\leq s}{\dim G_{i}/K_{i}},$

then

$\varrho_{a_{1},\cdots,a_{r}}\in L^{2}(G).$

Similarly, applying part (ii) of the Main Theorem, we deduce that if

$r>\max_{1\leq i\leq s}{\dim G_{i}/K_{i}}+k,$

then

$\varrho_{a_{1},\cdots,a_{r}}\in C^{k}(G).\\\ $

######  Acknowledgment.

It is a great pleasure for me to thank Kaïs Ammari for helpful suggestions and
his careful reading of the paper. It’s also a great pleasure to thank Mahmoud
Al-Hashami for pointing out some misprints in an earlier version of the paper.

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