Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations
| Autor: | René Quilodrán, Diogo Oliveira e Silva, Gianmarco Brocchi |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Class (set theory)
Pure mathematics fractional Schrödinger equation Entire function 35B38 01 natural sciences extremizers Schrödinger equation symbols.namesake Mathematics - Analysis of PDEs Dimension (vector space) convolution of singular measures 35Q41 0103 physical sciences Order (group theory) 0101 mathematics 42B37 Mathematics Numerical Analysis Applied Mathematics 010102 general mathematics Strichartz inequalities Fourier transform Compact space Mathematics - Classical Analysis and ODEs sharp Fourier restriction theory symbols 010307 mathematical physics Complex plane Analysis |
| Zdroj: | Anal. PDE 13, no. 2 (2020), 477-526 |
| ISSN: | 1948-206X 2157-5045 |
| 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: | OpenAIRE |
| Externí odkaz: |
|