Small cap decoupling for the parabola with logarithmic constant

Abstract: We note that the subpolynomial factor for the $\ell<sup>qL<sup>p$ small cap decoupling constants for the truncated parabola $$\mathbb{P}^{1={(t,t^2):|t|\leq} 1}$$ 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 paper of Guth and Maldague to demonstrate a small cap decoupling for the $(2+1)$ cone; we demonstrate that the version for $\mathbb{P}<sup>1$ may be taken with a loss controlled by a power of $\log R$ as well.