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      A sharp \(k\)-plane Strichartz inequality for the Schr\"odinger equation

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          Abstract

          We prove that \[ \|X(|u|^2)\|_{L^3_{t,\ell}}\leq C\|f\|_{L^2(\mathbb{R}^2)}^2, \] where \(u(x,t)\) is the solution to the linear time-dependent Schr\"odinger equation on \(\mathbb{R}^2\) with initial datum \(f\), and \(X\) is the (spatial) X-ray transform on \(\mathbb{R}^2\). In particular, we identify the best constant \(C\) and show that a datum \(f\) is an extremiser if and only if it is a gaussian. We also establish bounds of this type in higher dimensions \(d\), where the X-ray transform is replaced by the \(k\)-plane transform for any \(1\leq k\leq d-1\). In the process we obtain sharp \(L^2(\mu)\) bounds on Fourier extension operators associated with certain high-dimensional spheres, involving measures \(\mu\) supported on natural "co-\(k\)-planarity" sets.

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          Schrödinger equations: pointwise convergence to the initial data

          Luis Vega (1988)
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            Bilinear virial identities and applications

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              Distance Measures for Well-Distributed Sets

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                Author and article information

                Journal
                2016-11-11
                Article
                1611.03692
                5ac2285d-ded6-421c-a32a-24ef57a78742

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                19 pages
                math.CA

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