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

In earlier work, Gupta and Spronk established an L1–L2 dichotomy for Type A singular orbital measures on the flat symmetric spaces SU(2,q)0/S(U(2)×U(q)). Their analysis reduced the unresolved endpoint case q=2, k=2 to the study of an integral over a region W1 of the Weyl chamber. In this paper we revisit the contribution from the region W1. We decompose W1 into near-diagonal and off-diagonal subregions according to the size of λ1−λ2. 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 W1 is finite for every q≥2 and every integer k≥2. Combined with their earlier analysis of the complementary region W2, this yields the endpoint case q=2, k=2, thereby resolving the open problem stated in Remark 4.4 of Gupta and Spronk.