Discrete restriction for $(x,x^3)$ and related topics
Abstract
Defining the truncated extension operator $E$ for a sequence $a(n)$ with $n \in {\mathbb Z}$ by putting \[ E{a}(\alpha,\beta):=\sum_{|n|\leq N}a(n) e(\alpha n^3 + \beta n), \] we obtain the conjectured tenth moment estimate \[ \| E{a} \|_{L^{10}({\mathbb T}^2)}\lesssim_\epsilon N^{\frac{1}{10}+\epsilon} \|a\|_{\ell^2({\mathbb Z})}. \] We obtain related conclusions when the curve $(x,x^3)$ is replaced by $(\phi_1(x), \phi_2(x))$ for suitably independent polynomials $\phi_1(x),\phi_2(x)$ having integer coefficients.
- Publication:
-
arXiv e-prints
- Pub Date:
- November 2019
- DOI:
- 10.48550/arXiv.1911.12262
- arXiv:
- arXiv:1911.12262
- Bibcode:
- 2019arXiv191112262H
- Keywords:
-
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Analysis of PDEs;
- Mathematics - Number Theory;
- 42B05;
- 11L07;
- 42B37;
- 35Q53
- E-Print:
- 13 pages. Comments welcome :)