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Dynamics of Partial Differential Equations
Volume 19 (2022)
Number 2
A remark on the Strichartz inequality in one dimension
Pages: 163 – 175
DOI: https://dx.doi.org/10.4310/DPDE.2022.v19.n2.a4
Authors
Abstract
In this paper, we study the extremal problem for the Strichartz inequality for the Schrödinger equation on $\mathbb{R}^2$. We show that the solutions to the associated Euler–Lagrange equation are exponentially decaying in the Fourier space and thus can be extended to be complex analytic. Consequently we provide a new proof to the characterization of the extremal functions: the only extremals are Gaussian functions, which was investigated previously by Foschi [7] and Hundertmark–Zharnitsky [11].
Keywords
Schrödinger equations, Strichartz’s inequality, extremizers
2010 Mathematics Subject Classification
Primary 35-xx. Secondary 42-xx.
Received 15 January 2021
Published 19 May 2022