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Regularity of Solution of the Schr\"odinger Equation on Symmetric Space
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abstract
In this article, we investigate the behavior of solutions \( u(x,t) \) to the fractional Schr\"odinger equation on rank symmetric spaces of non-compact type. We proved that as time \( t \) approaches $0$, then $u(x,t)$ converges pointwise almost everywhere to the initial radial data \( f \), provided that \( f \in H^s(\mathbb{X}) \) with \( s > \frac{1}{2} \). This result extends Sj\"olin's results in this setting.
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Regularity and pointwise convergence for dispersive equations with asymptotically concave phase on Damek-Ricci spaces
On Damek-Ricci spaces, radial initial data in H^β with β>a/4 give almost everywhere pointwise convergence for dispersive equations with asymptotically concave phase of degree a; the threshold is sharp up to the endpoint.
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