Journal
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
Volume 2016, Issue 19, Pages 5875-5921Publisher
OXFORD UNIV PRESS
DOI: 10.1093/imrn/rnv338
Keywords
-
Categories
Funding
- NSF [DMS-1001531, DMS-0901166, DMS-1161396]
- Sloan Foundation
- Simons Collaboration Grant
Ask authors/readers for more resources
The goal of this paper was to develop some basic harmonic analysis tools for the Dirichlet-Laplacian in the exterior domain associated to a smooth convex obstacle in dimensions. Specifically, we will discuss analogues of the Mikhlin Multiplier Theorem, Littlewood-Paley Theory, and Hardy inequalities, culminating in a proof that homogeneous Sobolev norms defined with respect to the Dirichlet and whole-space Laplacians are equivalent for the sharp ranges of integrability exponent and regularity. Counterexamples are included to show that these results are indeed sharp. In particular, we precisely settle the question of boundedness of Riesz transforms on , including the endpoint. The utility of such results in the study of nonlinear partial differential equation (PDE) is that they allow us to deduce important results, such as the fractional product and chain rules for the Dirichlet-Laplacian, directly from the classical Euclidean setting. As an application, we discuss the local well-posedness and stability problems for energy-critical NLS. All the results of this paper play an essential role in the authors' proof of large-data global well-posedness and scattering for the energy-critical NLS in three-dimensional exterior domains; see [29].
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available