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Transactions of the American Mathematical Society

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

Authors: Piero D’Ancona, Damiano Foschi and Sigmund Selberg
Journal: Trans. Amer. Math. Soc. 364 (2012), 31-63
MSC (2010): Primary 35L05, 46E35
Published electronically: August 9, 2011
MathSciNet review: 2833576
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Abstract: The wave-Sobolev spaces $H^{s,b}$ are $L^2$-based Sobolev spaces on the Minkowski space-time $\mathbb {R}^{1+n}$, with Fourier weights 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 $\mathbb {R}^6$ 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 results for $n=2$ and $n=1$ will be published elsewhere.

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Additional Information

Piero D’Ancona
Affiliation: Department of Mathematics, University of Rome “La Sapienza”, Piazzale Aldo Moro 2, I-00185 Rome, Italy

Damiano Foschi
Affiliation: Department of Mathematics, University of Ferrara, Via Macchiavelli 35, I-44100 Ferrara, Italy

Sigmund Selberg
Affiliation: Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway

Received by editor(s): March 26, 2009
Received by editor(s) in revised form: October 7, 2009
Published electronically: August 9, 2011
Additional Notes: This paper was written as part of the international research program on Nonlinear Partial Differential Equations at the Centre for Advanced Study at the Norwegian Academy of Science and Letters in Oslo during the academic year 2008–2009.
Article copyright: © Copyright 2011 P. D’Ancona, D. Foschi, S. Selberg