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Almost everywhere convergence of the inverse spherical transform on \(\text{SL}(2,\mathbb{R})\). (English) Zbl 0821.43002

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 \(\varphi(t,\lambda) = | \lambda \tanh (\pi \lambda)|^{1/2} (\sinh t)^{1/2} \varphi_ \lambda (t).\)
Reviewer: K.Saka (Akita)

MSC:

43A50 Convergence of Fourier series and of inverse transforms
43A90 Harmonic analysis and spherical functions
43A80 Analysis on other specific Lie groups
40A30 Convergence and divergence of series and sequences of functions
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