Small cap decoupling for the parabola with logarithmic constant

Abstract

We note that the subpolynomial factor for the ${\ell }^q{L}^p$ small cap decoupling constants for the truncated parabola ${\mathbb{P}}^1=\{(t,t^2):|t|\le 1\}$ may be controlled by a suitable power of $\log R$. This is achieved by considering a suitable amplitude-dependent wave envelope estimate, as was introduced in a recent paper of Guth and Maldague to demonstrate a small cap decoupling for the $(2+1)$ cone; we demonstrate that the version for ${\mathbb{P}}^1$ may be taken with a loss controlled by a power of $\log R$ as well.

34 pages. This version includes a broad/narrow step missing in a previous version. Wave packets have been tweaked. Improved exposition in overview. Various small numerical changes. Central claims are unchanged, except for updated numerology. The author thanks Zane Li and Po-Lam Yung for pointing out the major error to be corrected in previous version. Comments on current version are welcome