Improved Integrability Estimates on the Region W₁ for Type A Singular Orbital Measures on SU(2, q)₀/S(U(2) × U(q))

Abstract

In earlier work, Gupta and Spronk established an $L^1$–$L^2$ dichotomy for Type A singular orbital measures on the flat symmetric spaces $SU{(2,q)}_0/S(U\left(2\right)\times U\left(q\right))$. Their analysis reduced the unresolved endpoint case $q=2$, $k=2$ to the study of an integral over a region $W_1$ of the Weyl chamber. In this paper we revisit the contribution from the region $W_1$. We decompose $W_1$ into near-diagonal and off-diagonal subregions according to the size of ${\lambda }_1-{\lambda }_2$. Applying the Mean Value Theorem in the near-diagonal region and standard decay estimates in the off-diagonal region, we prove that the contribution from $W_1$ is finite for every $q\ge 2$ and every integer $k\ge 2$. Combined with their earlier analysis of the complementary region $W_2$, this yields the endpoint case $q=2$, $k=2$, thereby resolving the open problem stated in Remark 4.4 of Gupta and Spronk.

1. Introduction

The study of convolution powers of orbital measures occupies an important place in harmonic analysis on Lie groups, symmetric spaces and Cartan motion groups. Questions concerning absolute continuity, smoothness and $L^p$-integrability of convolution products have attracted considerable attention since the pioneering work of Ragozin [1] and have subsequently been investigated by many authors; see, for example, [2,3,4,5,6].

The general problem has a natural common formulation across compact, noncompact and flat symmetric spaces. If $(G,K)$ is a Riemannian symmetric pair with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}$ (see [7,8] for general background), then the associated noncompact symmetric space $G/K$, its compact dual, and the flat symmetric space attached to the Cartan motion group $K⋉\mathfrak{p}$ form three closely related settings in which orbital measures arise as invariant probability measures supported on lower-dimensional orbits. Such measures are singular with respect to the ambient invariant measure, and a basic smoothing question is to determine how many convolution powers are required before absolute continuity, or the stronger property of $L^2$-integrability, occurs.

The roots of this circle of problems go back to Dunkl’s work on convolution structures associated with spheres [9]. Ragozin established fundamental absolute-continuity results for convolution powers of central and orbital measures on compact Lie groups and symmetric spaces [1,10]. These results initiated a broad study of sharp convolution exponents. In the noncompact symmetric-space setting, substantial improvements and sharp criteria were obtained by Graczyk and Sawyer [4,11,12,13]; see also their survey [5]. Related $L^2$-regularity and dichotomy questions have been studied for compact Lie groups and Lie algebras, compact symmetric spaces, and noncompact symmetric spaces; see, for example, [14,15,16,17,18]. These developments show that the passage from $L^1$-regularity to $L^2$-regularity is often sensitive to the geometry and singularity type of the underlying orbit.

For flat symmetric spaces associated with Cartan motion groups, orbital measures arise naturally as invariant probability measures supported on K-orbits in the tangent space. Their convolution powers encode important geometric and analytic information, and the determination of their $L^2$-integrability has become a central problem in the subject.

The harmonic analysis of flat symmetric spaces was developed in a structural form by Ben Saïd and Ørsted [19]. Within this framework, Gupta and Spronk [20] initiated a systematic study of the $L^1$–$L^2$ dichotomy for orbital measures on flat symmetric spaces. Their results include the rank-one setting and detailed low-rank analysis for flat symmetric spaces of type AIII, where explicit spherical functions reduce the $L^2$ problem to convergence questions for integrals involving Bessel functions and the Plancherel density.

In Ref. [20], Gupta and Spronk established an $L^1$–$L^2$ dichotomy for Type A singular orbital measures associated with the flat symmetric spaces

$$SU{(2,q)}_0/S(U\left(2\right)\times U\left(q\right)).$$

For $q\ge 3$, they obtained a complete description of the $L^2$-integrability of convolution powers of Type A singular orbital measures. More precisely, they proved that

$${\mu }_H^{*k}\in L^2$$

if and only if

$$k\ge {\displaystyle \frac{3}{4}}+{\displaystyle \frac{q}{2}}.$$

Their proof reduces the problem to the convergence of an explicit integral over a Weyl chamber and then decomposes the chamber into two regions, denoted by $W_1$ and $W_2$.

The only unresolved case appears in Remark 4.4 of [20]. After a sequence of reductions, the authors showed that the endpoint case

$$q=2,k=2,$$

depends on the convergence of a particular integral over the region $W_1$. They further observed that the principal difficulty occurs when ${\lambda }_1-{\lambda }_2$ is close to zero and left the problem open.

The purpose of the present paper is to remove precisely this remaining obstruction. The novelty lies in a sharper local analysis of the $W_1$-integral. The estimate used in [20] on $W_1$ gives finiteness only for $k\ge 3$ and therefore does not reach the endpoint $k=2$. We separate the near-diagonal and off-diagonal regimes. Near the diagonal, the difference $f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right)$ is estimated by the Mean Value Theorem before taking the $2k$-th power. Writing $u={\lambda }_1-{\lambda }_2$ and $v={\lambda }_1+{\lambda }_2$, this produces a factor $u^{2k}$ in the numerator, while the denominator contributes only $u^{2k-2}$; the residual factor $u^2$ removes the apparent singularity at $u=0$. Away from the diagonal, no such cancellation is needed and the standard Bessel decay gives separate integrability in u and v. This is the technical improvement that lowers the sufficient exponent on $W_1$ from $k\ge 3$ to $k\ge 2$ and is decisive for the unresolved endpoint.

Following Gupta and Spronk, the positive Weyl chamber is decomposed into two regions $W_1$ and $W_2$, where

$$W_1=\{({\lambda }_1,{\lambda }_2):{\lambda }_2\ge {\lambda }_1/2\},$$

and

$$W_2=\{({\lambda }_1,{\lambda }_2):{\lambda }_2<{\lambda }_1/2\}.$$

To obtain a more refined analysis, we further decompose

$$W_1=W_{11}\cup W_{12},$$

where

$$W_{11}=\left\{({\lambda }_1,{\lambda }_2)\in W_1:0<{\lambda }_1-{\lambda }_2<1\right\},$$

and

$$W_{12}=\left\{({\lambda }_1,{\lambda }_2)\in W_1:{\lambda }_1-{\lambda }_2\ge 1\right\}.$$

The near-diagonal region $W_{11}$ is treated using the Mean Value Theorem together with asymptotic estimates for the functions $f_r$, while the off-diagonal region $W_{12}$ is handled using standard decay estimates.

The principal result of the paper is that the contribution from $W_1$ is finite for every $q\ge 2$ and every integer $k\ge 2$. This improves the analysis of Gupta and Spronk on the region $W_1$, where the corresponding argument yields finiteness only under the condition $k\ge 3$.

The endpoint consequence is stated explicitly as follows.

Theorem 1.

Let $q=2$, let H be a Type A singular point represented by $H\cong (x,0)$ with $x>0$, and let ${\mu }_H$ be the corresponding orbital measure on the flat symmetric space

$$SU{(2,2)}_0/S(U\left(2\right)\times U\left(2\right)).$$

Then

$${\mu }_H^{*2}\in L^2\left(\mathfrak{p}\right).$$

Thus the endpoint case $q=2$, $k=2$ left open in Remark 4.4 of [20] is resolved.

Theorem 1 follows by combining the new $W_1$ estimate proved here with the already established analysis of the complementary region $W_2$ in [20]. Thus the present paper does not re-prove the $W_2$ theory; its contribution is the missing $W_1$ estimate and the consequent resolution of the endpoint.

The paper is organized as follows. In Section 2 we recall the notation and estimates needed from [20]. Section 3 summarizes the analysis of the region $W_2$. Section 4 and Section 5 contain the analysis of the regions $W_{11}$ and $W_{12}$, respectively. The main theorem and its endpoint consequence are proved in Section 6. Concluding remarks are given in Section 7.

2. Preliminaries

In this section we recall the notation and estimates required for the analysis of the region $W_1$. Throughout the paper we follow the notation of Gupta and Spronk [20]. For Type A singular orbital measures on the flat symmetric spaces associated with $SU{(2,q)}_0/S(U\left(2\right)\times U\left(q\right))$, Gupta and Spronk showed that the $L^2$-integrability problem for convolution powers of orbital measures may be reduced to the convergence of an explicit integral over the positive Weyl chamber. Since our purpose is to refine one step of their argument, we shall use their notation and formulation throughout and refer the reader to [20] for the underlying geometric construction of the orbital measures.

Let

$$r=q-2,$$

so that $r\ge 0$ whenever $q\ge 2$.

For $r\ge 0$, define

$$f_r\left(t\right)={\displaystyle \frac{J_r\left(t\right)}{t^r}},$$

where $J_r$ denotes the Bessel function of the first kind of order r.

Throughout the sequel, H is a Type A singular point represented by $H\cong (x,0)$, where $x>0$ is fixed. Thus x is the nonzero coordinate of H in the notation of [20]. Constants in estimates involving $f_r\left(x\lambda \right)$ may depend on this fixed parameter x (and on r and k), but they are independent of the spectral variables ${\lambda }_1, {\lambda }_2$.

The proof of Theorem 4.3 in [20] reduces the $L^2$-integrability problem for Type A singular orbital measures to the study of an integral involving the functions $f_r$.

As in [20], compact subsets of the Weyl chamber contribute a finite amount to the integral under consideration. Consequently, it suffices to analyze the asymptotic region where the spectral parameters are sufficiently large. Throughout the paper, all estimates are understood in this asymptotic sense.

We shall repeatedly use the following estimates established in [20].

Proposition 1.

There exists a constant $C>0$ such that for sufficiently large t,

$$\left|f_r\left(t\right)\right|\le C{t}^{-(r+{\displaystyle \frac{1}{2}})}$$

and

$$\left|f_r^{\prime }\left(t\right)\right|\le C{t}^{-(r+{\displaystyle \frac{1}{2}})}.$$

Let

$${\mathfrak{a}}^{+}=\left\{({\lambda }_1,{\lambda }_2):{\lambda }_1>{\lambda }_2>0\right\}$$

denote the positive Weyl chamber.

Following [20], we decompose

$${\mathfrak{a}}^{+}=W_1\cup W_2,$$

where

$$W_1=\left\{({\lambda }_1,{\lambda }_2)\in {\mathfrak{a}}^{+}:{\lambda }_2\ge {\displaystyle \frac{{\lambda }_1}{2}}\right\},$$

and

$$W_2=\left\{({\lambda }_1,{\lambda }_2)\in {\mathfrak{a}}^{+}:{\lambda }_2<{\displaystyle \frac{{\lambda }_1}{2}}\right\}.$$

The quantity to be estimated is

$$\varphi ({\lambda }_1,{\lambda }_2)={\displaystyle \frac{|f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right){|}^{2k}{\left({\lambda }_1{\lambda }_2\right)}^{1+2r}}{{({\lambda }_1^2-{\lambda }_2^2)}^{2k-2}}}.$$

The convergence of

$${\int \int }_{{\mathfrak{a}}^{+}}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2$$

determines the $L^2$-integrability of the k-fold convolution power of the associated orbital measure.

More precisely, this is the Plancherel-integrability criterion obtained in [20]. We use it here only in the form needed to complete the previously reduced Type A problem. The bounded part of the Weyl chamber is harmless; the issue is the asymptotic integrability of the explicit Plancherel integrand, and the only estimate that must be improved for the endpoint is the one on $W_1$.

The contribution from $W_2$ was analyzed sharply in [20] (Equations (4.15) and (4.16) in the proof of Theorem 4.3), where the exact integrability threshold was obtained.

Accordingly, the principal task of the present paper is to obtain a refined analysis of the contribution from the region $W_1$.

3. The Region ${\mathit{W}}_{\mathbf{2}}$

In [20], Gupta and Spronk decomposed the region $W_2$ into the subregions

$$W_2=W_{21}\cup W_{22},$$

where

$$W_{21}=\left\{({\lambda }_1,{\lambda }_2)\in W_2:{\lambda }_2>c\right\},$$

and

$$W_{22}=\left\{({\lambda }_1,{\lambda }_2)\in W_2:0<{\lambda }_2\le c\right\},$$

for a suitable positive constant c.

The contribution from these regions was analyzed completely in [20]. More precisely, it was shown that for $q\ge 3$,

$${\int \int }_{W_2}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2<\infty$$

if and only if

$$k\ge {\displaystyle \frac{3}{4}}+{\displaystyle \frac{q}{2}}.$$

For $q=2$, Equations (4.15) and (4.16) in the proof of Theorem 4.3 of [20] prove convergence when $k=2$.

Since the sharp analysis of $W_2$, including the necessity and sufficiency of the above condition for $q\ge 3$, is already available in [20], we shall not repeat it here. No new estimate for $W_2$ is claimed in the present paper; this section is included only to make clear how the new $W_1$ analysis combines with the earlier result to settle the endpoint $q=2$, $k=2$. Accordingly, the remaining task is to investigate the contribution from the region $W_1$.

To this end, we further decompose $W_1$ into near-diagonal and off-diagonal regions. This decomposition is the key new ingredient in our argument.

4. The Region ${\mathit{W}}_{\mathbf{11}}$

In this section we consider the near-diagonal region

$$W_{11}=\left\{({\lambda }_1,{\lambda }_2)\in W_1:0<{\lambda }_1-{\lambda }_2<1\right\}.$$

Our goal is to prove the convergence of

$${\int \int }_{W_{11}}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2$$

for all integers $k\ge 2$. As recalled in Section 2, the contribution from the bounded part of the Weyl chamber is finite. It therefore suffices in this section to estimate the portion of $W_{11}$ for which $v={\lambda }_1+{\lambda }_2>R$, with R sufficiently large. The bounded portion is accounted for in Proposition 2. Since $({\lambda }_1,{\lambda }_2)\in W_1$, we have

$${\lambda }_2\ge {\displaystyle \frac{{\lambda }_1}{2}}.$$

Consequently,

$${\lambda }_1\asymp {\lambda }_2.$$

Introduce the variables

$$u={\lambda }_1-{\lambda }_2,v={\lambda }_1+{\lambda }_2.$$

The inverse transformation has Jacobian determinant $1/2$; hence $d{\lambda }_{1}d{\lambda }_2={\displaystyle \frac{1}{2}}dudv$, and this constant factor is absorbed into the constant C appearing below.

Then

$$0<u<1.$$

The remaining condition defining $W_1$ becomes

$${\lambda }_2\ge {\displaystyle \frac{{\lambda }_1}{2}}⟺v\ge 3u.$$

Thus, in $(u,v)$-coordinates, the asymptotic part of $W_{11}$ is described by $0<u<1$, $v\ge 3u$, and $v>R$.

Moreover,

$${\lambda }_1^2-{\lambda }_2^2=({\lambda }_1-{\lambda }_2)({\lambda }_1+{\lambda }_2)=uv.$$

Hence

$${({\lambda }_1^2-{\lambda }_2^2)}^{2k-2}=u^{2k-2}{v}^{2k-2}.$$

Since ${\lambda }_1\asymp {\lambda }_2$, it follows that

$${\lambda }_1{\lambda }_2\asymp v^2.$$

Therefore

$${\left({\lambda }_1{\lambda }_2\right)}^{1+2r}\asymp v^{2+4r}.$$

Substituting these estimates into the definition of $\varphi$, we obtain

$$\varphi ({\lambda }_1,{\lambda }_2)\le C{|f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right)|}^{2k}{u}^{-(2k-2)}{v}^{4r+4-2k}.$$

Next, applying the Mean Value Theorem, there exists $\xi \in ({\lambda }_2,{\lambda }_1)$ such that

$$f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right)=x({\lambda }_1-{\lambda }_2)f_r^{\prime }\left(x\xi \right).$$

Since ${\lambda }_1\asymp {\lambda }_2$, we also have

$$\xi \asymp v.$$

Using Proposition 1, we obtain

$$\left|f_r^{\prime }\left(x\xi \right)\right|\le C{v}^{-(r+{\displaystyle \frac{1}{2}})}.$$

Hence

$$|f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right)|\le Cu{v}^{-(r+{\displaystyle \frac{1}{2}})}.$$

Raising both sides to the power $2k$ gives

$$|f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right){|}^{2k}\le C{u}^{2k}{v}^{-2kr-k}.$$

Substituting into the preceding estimate for $\varphi$, we obtain

$$\varphi ({\lambda }_1,{\lambda }_2)\le C{u}^2{v}^{4r+4-3k-2kr}.$$

This is the crucial near-diagonal gain. The factor $u^{2k}$ produced by the Mean Value Theorem cancels the potentially singular factor $u^{2k-2}$ from ${({\lambda }_1^2-{\lambda }_2^2)}^{2k-2}$ and leaves the integrable factor $u^2$, independently of k. It is exactly this cancellation that is missing from the coarser $W_1$ estimate and that allows the endpoint $k=2$. More precisely, the asymptotic part of $W_{11}$ is contained in

$$\left\{\right(u,v):0<u<1,v\ge 3u,v>R\}.$$

For the purpose of separating the variables, we enlarge this domain to the product region

$$\left\{\right(u,v):0<u<1,v>R\}.$$

This enlargement is legitimate because the majorant is nonnegative. Since the integrand is nonnegative, Tonelli’s theorem implies that, for some sufficiently large R,

$${\int \int }_{W_{11}\cap \{v>R\}}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2\le C{\int }_0^1{u}^{2}du{\int }_R^{\infty }{v}^{4r+4-3k-2kr}dv.$$

Therefore it suffices to examine the convergence of the product integral on the right-hand side.

The latter converges provided

$$4r+4-3k-2kr<-1.$$

Equivalently,

$$4r+5<k(2r+3).$$

For $k=2$ this becomes

$$4r+5<4r+6,$$

which is always satisfied. Since $2r+3>0$, the condition therefore holds for every integer $k\ge 2$.

Hence

$${\int \int }_{W_{11}}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2<\infty$$

for every $r\ge 0$ and every integer $k\ge 2$.

We summarize the above discussion in the following proposition.

Proposition 2.

For every $r\ge 0$ and every integer $k\ge 2$, the integral over $W_{11}\cap \{v>R\}$ is finite for all sufficiently large R. The complementary bounded portion contributes finitely by the compact-region argument recalled in Section 2. Consequently, the full contribution from $W_{11}$ is finite.

5. The Region ${\mathit{W}}_{\mathbf{12}}$

In this section we consider the off-diagonal region

$$W_{12}=\{({\lambda }_1,{\lambda }_2)\in W_1:{\lambda }_1-{\lambda }_2\ge 1\}.$$

Our goal is to prove the convergence of

$${\displaystyle {\int \int }_{W_{12}}}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2$$

for all integers $k\ge 2$. As in Section 4, the bounded part has finite contribution. We need only estimate $W_{12}\cap \{v>R\}$ for sufficiently large R; the bounded portion is included again in Proposition 3. Since $({\lambda }_1,{\lambda }_2)\in W_1$, we again have

$${\lambda }_2\ge {\displaystyle \frac{{\lambda }_1}{2}},$$

and hence

$${\lambda }_1\asymp {\lambda }_2.$$

Using the same change of variables as in Section 4,

$$u={\lambda }_1-{\lambda }_2,v={\lambda }_1+{\lambda }_2,$$

we now have

$$u\ge 1.$$

As above, the condition ${\lambda }_2\ge {\lambda }_1/2$ is equivalent to $v\ge 3u$. Hence the asymptotic part of $W_{12}$ is described by $u\ge 1$, $v\ge 3u$, and $v>R$.

As in Section 4,

$${\left({\lambda }_1{\lambda }_2\right)}^{1+2r}\asymp v^{2+4r},$$

and

$${({\lambda }_1^2-{\lambda }_2^2)}^{2k-2}=u^{2k-2}{v}^{2k-2}.$$

Consequently,

$$\varphi ({\lambda }_1,{\lambda }_2)\le C{|f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right)|}^{2k}{u}^{-(2k-2)}{v}^{4r+4-2k}.$$

Unlike the near-diagonal region, we do not apply the Mean Value Theorem. Here $u={\lambda }_1-{\lambda }_2$ is bounded away from zero, so the denominator creates no near-diagonal singularity. It is therefore more efficient to retain the full factor $u^{-(2k-2)}$ and use only the standard decay of $f_r$. Indeed, Proposition 1 gives

$$\left|f_r\left(t\right)\right|\le C{t}^{-(r+{\displaystyle \frac{1}{2}})}$$

for sufficiently large t. Therefore,

$$|f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right)|\le |f_r\left(x{\lambda }_1\right)|+|f_r\left(x{\lambda }_2\right)|\le C{v}^{-(r+{\displaystyle \frac{1}{2}})}.$$

Raising both sides to the power $2k$ yields

$$|f_r\left(x{\lambda }_1\right)-f_r\left(x{\lambda }_2\right){|}^{2k}\le C{v}^{-2kr-k}.$$

Substituting into the preceding estimate for $\varphi$, we obtain

$$\varphi ({\lambda }_1,{\lambda }_2)\le C{u}^{-(2k-2)}{v}^{4r+4-3k-2kr}.$$

More precisely, the asymptotic part of $W_{12}$ is contained in

$$\left\{\right(u,v):u\ge 1,v\ge 3u,v>R\}.$$

We enlarge this set to the product region

$$\left\{\right(u,v):u\ge 1,v>R\},$$

which is legitimate because the majorant is nonnegative. Tonelli’s theorem implies

$${\int \int }_{W_{12}\cap \{v>R\}}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2\le C{\int }_1^{\infty }{u}^{-(2k-2)}du{\int }_R^{\infty }{v}^{4r+4-3k-2kr}dv.$$

The integral in the variable u converges provided

$$2k-2>1,$$

that is,

$$k>{\displaystyle \frac{3}{2}}.$$

Hence it converges for every integer $k\ge 2$.

The integral in the variable v converges provided

$$4r+4-3k-2kr<-1,$$

or equivalently,

$$4r+5<k(2r+3).$$

For $k=2$, this becomes

$$4r+5<4r+6,$$

which is always satisfied. Since $2r+3>0$, the condition therefore holds for every integer $k\ge 2$.

Therefore,

$${\int \int }_{W_{12}}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2<\infty$$

for every $r\ge 0$ and every integer $k\ge 2$.

We summarize the above discussion in the following proposition.

Proposition 3.

For every $r\ge 0$ and every integer $k\ge 2$, the integral over $W_{12}\cap \{v>R\}$ is finite for all sufficiently large R. The complementary bounded portion contributes finitely by the compact-region argument recalled in Section 2. Consequently, the full contribution from $W_{12}$ is finite.

6. Main Results

We first record the improved estimate on $W_1$, which is the new technical result of the paper.

Theorem 2.

The contribution from the region $W_1$ is finite for every $q\ge 2$ and every integer $k\ge 2$. Equivalently,

$${\int \int }_{W_1}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2<\infty .$$

Proof. 

Since $W_1=W_{11}\cup W_{12}$ and the two subregions are disjoint,

$${\int \int }_{W_1}\varphi ={\int \int }_{W_{11}}\varphi +{\int \int }_{W_{12}}\varphi .$$

Proposition 2 proves finiteness of the first integral for every $r\ge 0$ and every integer $k\ge 2$, while Proposition 3 proves the same for the second integral. Since $r=q-2$, the assertion follows for every $q\ge 2$ and $k\ge 2$. □

We now prove the main theorem stated in the Introduction, so that the endpoint conclusion is presented directly as the $L^2$-regularity statement for the convolution square.

Proof of Theorem 1.

When $q=2$, we have $r=0$. By Theorem 2,

$${\int \int }_{W_1}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2<\infty$$

for $k=2$.

Since the contribution from $W_2$ is finite for $q=2$ and $k=2$ by Equations (4.15) and (4.16) in the proof of Theorem 4.3 of [20], the corresponding integral over the entire Weyl chamber converges. By the Plancherel reduction established in [20], this convergence is precisely the remaining condition needed for the convolution square of the Type A singular orbital measure. Consequently,

$${\mu }_H^{*2}\in L^2\left(\mathfrak{p}\right),$$

which proves the theorem. □

Remark 1.

For Type A singular orbital measures, the analysis of the region $W_1$ in [20] yields finiteness only under the condition $k\ge 3$. The present argument shows that the contribution from $W_1$ is finite for every $q\ge 2$ and every integer $k\ge 2$. The improvement comes from treating the two regimes of $W_1$ separately. On $W_{11}$ the Mean Value Theorem exposes the vanishing of the Bessel-type difference at the diagonal and leaves the factor ${({\lambda }_1-{\lambda }_2)}^2$ after cancellation with the denominator; on $W_{12}$ ordinary decay estimates suffice because ${\lambda }_1-{\lambda }_2\ge 1$. Thus the argument is a refinement of the integrability analysis, but it is a refinement that changes the admissible endpoint from $k\ge 3$ to $k\ge 2$ on $W_1$ and thereby settles the case that was previously unresolved.

7. Concluding Remarks

In this paper we revisited the contribution from the region $W_1$ arising in the analysis of Type A singular orbital measures on $SU{(2,q)}_0/S(U\left(2\right)\times U\left(q\right))$.

The essential point is that the original $W_1$ estimate can be sharpened once the near-diagonal and off-diagonal behaviours are separated. On $W_{11}$, the Mean Value Theorem makes the diagonal cancellation explicit: after taking the $2k$-th power, the resulting factor ${({\lambda }_1-{\lambda }_2)}^{2k}$ dominates the factor ${({\lambda }_1-{\lambda }_2)}^{2k-2}$ arising from the denominator and leaves a uniformly integrable quadratic factor. On $W_{12}$, where the spectral parameters are separated, the standard decay estimate for $f_r$ gives the required integrability directly. Together these two estimates prove that

$${\int \int }_{W_1}\varphi ({\lambda }_1,{\lambda }_2)d{\lambda }_{1}d{\lambda }_2<\infty$$

for every $q\ge 2$ and every integer $k\ge 2$. This improves the $W_1$ range obtained in [20], where the corresponding estimate required $k\ge 3$. Combining the improved $W_1$ estimate with the already established $W_2$ analysis in Equations (4.15) and (4.16) of the proof of Theorem 4.3 in [20] gives ${\mu }_H^{*2}\in L^2\left(\mathfrak{p}\right)$ for the Type A singular orbit when $q=2$, thereby resolving the endpoint case left open in Remark 4.4 of [20].

The present result also illustrates a general feature of endpoint problems for orbital measures: a global decay estimate may lose the local vanishing that occurs when spectral parameters coalesce. Isolating that near-diagonal geometry can therefore change the critical convolution exponent. It would be interesting to determine whether analogous local refinements resolve other borderline $L^2$ problems for singular orbital measures on higher-rank flat symmetric spaces.