Almost everywhere convergence of the inverse spherical transform on \(\text{SL}(2,\mathbb{R})\) (Q1329664)

The almost everywhere convergence of the inverse spherical transform on \(\text{SL}(2, \mathbb{R})\) is proved. \(G\) will denote \(\text{SL}(2, \mathbb{R})\). Inside \(G\) there are the compact subgroup \(K = \text{SO}(2)\), consisting of all orthogonal matrices in \(G\) and the subgroup \(A\) of diagonal elements of \(G\). Then we have a Cartan decomposition: \(G = KAK\). Let \(\mu\) denote the Haar measure on \(G\), normalized according to the following integral formula: \[ \int_ G f(g) d\mu(g) = {1\over 2\pi} \int^ \infty_ 0 \int^{2\pi} _ 0 \int^{2\pi}_ 0 f(k(\theta_ 1) a(s) k(\theta_ 2)) \sinh s d\theta_ 1 d\theta_ 2 ds. \] The spherical transform of a bi-\(K\)- invariant function in \(G\) is given by \[ {\mathcal F} f(\lambda) = \int_ G f(g) \varphi_ \lambda (g) d\mu(g) = 2\pi \int^ \infty_ 0 f(a(s)) \varphi_ \lambda (a(s)) \sinh s ds \] where a continuous function \(\varphi_ \lambda(g)\) is an elementary spherical function on \(G\). The inverse spherical transform is \[ f(a(s)) = {1\over \pi} \int^ \infty_ 0 {\mathcal F} f(\lambda) \varphi_ \lambda (a(s)) \lambda \tanh (\pi \lambda) d\lambda. \] We set \[ S_ R f(t) = \int^ R_ 0 {\mathcal F} f(\lambda) \varphi_ \lambda (a(t)) \lambda \tanh (\pi \lambda) d\lambda / \pi\quad \text{ and } \quad S * f(t) = \sup_{R > 1} | S_ Rf(t)|. \] The following results are obtained. Theorem. \(S * f\) is a bounded operator on the space of all bi-\(K\)-invariant \(L^ p\) functions on \(G\) for \({4\over 3} < p \leq 2\). Corollary. If \(f\) is a bi-\(K\)- invariant \(L^ p\) function on \(G\) for \({4\over 3} < p \leq 2\), then \(S_ R f(t) \to f(t)\) a.e. as \(R \to \infty\). These proofs are based on Schindler's asymptotic estimate for \(\varphi(t,\lambda) = | \lambda \tanh (\pi \lambda)|^{1/2} (\sinh t)^{1/2} \varphi_ \lambda (t).\)