On extremizers for Strichartz estimates for higher order Schrödinger equations

Oliveira e Silva, Diogo; Quilodrán, René

doi:10.1090/tran/7223

On extremizers for Strichartz estimates for higher order Schrödinger equations

Authors: Diogo Oliveira e Silva and René Quilodrán
Journal: Trans. Amer. Math. Soc. 370 (2018), 6871-6907
MSC (2010): Primary 42B10
DOI: https://doi.org/10.1090/tran/7223
Published electronically: February 26, 2018
MathSciNet review: 3841835
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Abstract:

For an appropriate class of convex functions $\phi$, we study the Fourier extension operator on the surface {(𝑦,|𝑦|²+𝜙(𝑦)):𝑦∈ℝ

²}

equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations and compute the corresponding operator norms exactly where only lower bounds were previously known.

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