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The wave-Sobolev spaces $H^{s,b}$ are $L^2$-based Sobolev spaces on theMinkowski space-time $\R^{1+n}$, with Fourier weights are adapted to the symbolof the d'Alembertian. They are a standard tool in the study of regularityproperties of nonlinear wave equations, and in such applications the needarises for product estimates in these spaces. Unfortunately, it seems that withevery new application some estimates come up which have not yet appeared in theliterature, and then one has to resort to a set of well-established proceduresfor proving the missing estimates. To relieve the tedium of having toconstantly fill in such gaps "by hand", we make here a systematic effort todetermine the complete set of estimates in the bilinear case. We determine aset of necessary conditions for a product estimate $H^{s_1,b_1} \cdotH^{s_2,b_2} \hookrightarrow H^{-s_0,-b_0}$ to hold. These conditions define apolyhedron $\Omega$ 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 allpoints in the interior of $\Omega$, and all points on the faces minus theedges, give product estimates. We can also allow some but not all points on theedges, but here we do not claim to have the sharp result. The correspondingresult for $n=2$ and $n=1$ will be published elsewhere.

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