Journal article
Riesz Transforms Outside a Convex Obstacle
International mathematics research notices, Vol.2016(19), pp.5875-5921
01/01/2016
DOI: 10.1093/imrn/rnv338
Abstract
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].
Details
- Title: Subtitle
- Riesz Transforms Outside a Convex Obstacle
- Creators
- Rowan Killip - University of California, Los AngelesMonica Visan - University of California, Los AngelesXiaoyi Zhang - University of Iowa
- Resource Type
- Journal article
- Publication Details
- International mathematics research notices, Vol.2016(19), pp.5875-5921
- Publisher
- OXFORD UNIV PRESS
- DOI
- 10.1093/imrn/rnv338
- ISSN
- 1073-7928
- eISSN
- 1687-0247
- Number of pages
- 47
- Grant note
- Simons Collaboration Grant Sloan Foundation DMS-1001531; DMS-0901166; DMS-1161396 / NSF
- Language
- English
- Date published
- 01/01/2016
- Academic Unit
- Mathematics
- Record Identifier
- 9984241045902771
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