On almost-everywhere convergence of inverse spherical transforms (Q1919866)

Suppose that \(G\) is a noncompact, connected, semisimple Lie group with finite center and real rank one, and with a maximal compact subgroup \(K\). We assume that an Iwasawa decomposition \(G=ANK\) is fixed. Let \({\mathcal A}\) denote the Lie algebra of \(A\), so that \({\mathcal A}\) is isomorphic to the real line. We fix an element \(H_0\) of \({\mathcal A}\) so that \({\mathcal A} = \mathbb{R} H_0\). Then every element of \(G\) can be written as \(g=k_1 a(t)k_2\) for some \(k_1\) and \(k_2\) in \(K\) and \(t\geq 0\), where \(a(t)=\exp (tH_0)\). For all bi-\(K\)-invariant integrable functions \(f\) on \(G\) there is a spherical transform \[ {\mathcal F} f(\lambda) = \int^\infty_0 f\bigl(a(t)\bigr) \varphi_\lambda \bigl(a(t)\bigr) D(t)d t, \] where \(\varphi_\lambda\) is the spherical function on \(G\) and \(D(t)\) is the density on \([0,\infty)\) which corresponds to the Haar measure on \(G\). If \(f\) is a bi-\(K\)-invariant square integrable function on \(G\), then we have the Plancherel formula \[ \int^\infty_0 \bigl|{\mathcal F} f(x)\bigr|^2 \bigl|c(\lambda) \bigr|^{-2} d\lambda = \int_G \bigl|f(x)\bigr|^2 dx \] and we define the partial sum of the inverse spherical transform of \(f\) by \[ S_R f\bigl(a(t)\bigr) = \int^R_0{\mathcal F} f(\lambda) \varphi_\lambda \bigl(a(t)\bigr) \bigl|c(\lambda) \bigr|^{-2} d\lambda \] for \(R>0\). The main result is the following Theorem. For every bi-\(K\)-invariant square integrable function \(f\) on \(G\), the partial sums of the inverse spherical transform \(S_Rf(x)\) converge almost everywhere on \(G\). -- To prove this result, it is sufficient to prove that \[ t\mapsto \bigl( D(t)\bigr)^{1/2} \sup_{R>1} \left|\int^R_1{\mathcal F} f(\lambda) \varphi_\lambda \bigl(a(t)\bigr) \bigl|c(\lambda)\bigr|^{-2} d\lambda \right| \] is square integrable on \([0,\infty)\). To prove this result the authors transform the problem to one about Hankel and Fourier transforms.