Siegel–Radon transforms of transverse dynamical systems

The paper’s Theorem 3.12 proves the adjointness ⟨S_{ψ_ξ} f, ϕ⟩_X = ⟨f, S^*_{ψ_ξ} ϕ⟩_{H\G} by applying Corollary 2.5 to the H-invariant kernel F_{f,ϕ}(g,y)=f(g)ϕ(g^{-1}.y)ψ_ξ(y), which turns the H\G×Y integral into the X-fiber sum directly; this uses the global unimodularity assumptions on G and H and the standard quotient integration formula fixed in §1.1 and Remark 2.4 (eq. (2.5)) . The candidate solution obtains the same identity via Proposition A’s transverse measure identity (1.2), inserting a compactly supported cutoff κ and a normalizing factor J(Hg)=∫_H κ((hg)^{-1})dm_H(h) to match the disintegration over H\G; the ξ-covariances cancel after unfolding by (2.5), yielding exactly the desired pairing. Under the paper’s standing unimodularity and integrability assumptions, the model’s steps (Tonelli/Fubini, boundedness of ψ_ξ, finiteness of σ, quotient integration) are justified by Proposition A and Remark 2.4 (and by the very definitions of S_{ψ_ξ} and S^*_{ψ_ξ}) . Minor details the model leaves implicit (positivity/continuity of J on the compact support and explicit invocation of unimodularity) are standard and align with the paper’s hypotheses, so both arguments are correct and independent.