Sharp Strichartz inequalities for fractional and higher order Schr\'odinger equations

Autor: Brocchi, Gianmarco, Silva, Diogo Oliveira e, Quilodrán, René
Rok vydání: 2018
Předmět:
Zdroj: Analysis & PDE 13 (2020) 477-526
Druh dokumentu: Working Paper
DOI: 10.2140/apde.2020.13.477
Popis: We investigate a class of sharp Fourier extension inequalities on the planar curves $s=|y|^p$, $p>1$. We identify the mechanism responsible for the possible loss of compactness of nonnegative extremizing sequences, and prove that extremizers exist if $14$. In particular, this resolves the dichotomy of Jiang, Pausader & Shao concerning the existence of extremizers for the Strichartz inequality for the fourth order Schr\"odinger equation in one spatial dimension. One of our tools is a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$, developed in a companion paper. We further show that any extremizer exhibits fast $L^2$-decay in physical space, and so its Fourier transform can be extended to an entire function on the whole complex plane. Finally, we investigate the extent to which our methods apply to the case of the planar curves $s=y|y|^{p-1}$, $p>1$.
Comment: 50 pages, 2 figures; v3: extended remark after Proposition 6.7, typos corrected
Databáze: arXiv
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