Atlas of products for wave-Sobolev spaces on $\mathbf R^{1+3}$

TL;DR

This paper systematically characterizes the full set of bilinear product estimates in wave-Sobolev spaces on Minkowski space-time, providing a comprehensive framework for their application in nonlinear wave equations.

Contribution

It determines necessary conditions and proves the sufficiency of these conditions for product estimates in wave-Sobolev spaces in three spatial dimensions.

Findings

Complete set of bilinear product estimates in $H^{s,b}$ spaces for $n=3$

Identification of a polyhedral region where estimates hold

Partial results for lower dimensions $n=2,1$

Abstract

The wave-Sobolev spaces $H^{s,b}$ are $L^2$-based Sobolev spaces on the Minkowski space-time $\R^{1+n}$, with Fourier weights are adapted to the symbol of the d'Alembertian. They are a standard tool in the study of regularity properties of nonlinear wave equations, and in such applications the need arises for product estimates in these spaces. Unfortunately, it seems that with every new application some estimates come up which have not yet appeared in the literature, and then one has to resort to a set of well-established procedures for proving the missing estimates. To relieve the tedium of having to constantly fill in such gaps "by hand", we make here a systematic effort to determine the complete set of estimates in the bilinear case. We determine a set of necessary conditions for a product estimate $H^{s_1,b_1} \cdot H^{s_2,b_2} \hookrightarrow H^{-s_0,-b_0}$ to hold. These conditions define a polyhedron $\Omega$ in the space $\R^6$ of exponents $(s_0,s_1,s_2,b_0,b_1,b_2)$. We then show, in space dimension $n=3$, that all points in the interior of $\Omega$, and all points on the faces minus the edges, give product estimates. We can also allow some but not all points on the edges, but here we do not claim to have the sharp result. The corresponding result for $n=2$ and $n=1$ will be published elsewhere.