An endpoint version of uniform Sobolev inequalities
-
Tianyi Ren
,
Yakun Xi
Abstract
We prove an endpoint version of the uniform Sobolev inequalities in [C. E. Kenig, A. Ruiz and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 1987, 329–347]. It was known that strong type inequalities no longer hold at the endpoints; however, we show that restricted weak type inequalities hold there, which imply the earlier classical result by real interpolation. The key ingredient in our proof is a type of interpolation first introduced by Bourgain [J. Bourgain, Esitmations de certaines functions maximales, C. R. Acad. Sci. Paris 310 1985, 499–502]. We also prove restricted weak type Stein–Tomas restriction inequalities on some parts of the boundary of a pentagon, which completely characterizes the range of exponents for which the inequalities hold.
Acknowledgements
The authors would like to express their gratitude to their advisor, Professor Christopher D. Sogge, for bringing this research topic and Bourgain’s interpolation to their attention, and also for the invaluable guidance and suggestions he provided.
References
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© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
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- Petersson norm of cusp forms associated to real quadratic fields
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Articles in the same Issue
- Frontmatter
- Upper bounds for geodesic periods over rank one locally symmetric spaces
- Additive bases with coefficients of newforms
- A note on split extensions of bialgebras
- Petersson norm of cusp forms associated to real quadratic fields
- A realization theorem for sets of lengths in numerical monoids
- On the non-existence of the Mackey topology for locally quasi-convex groups
- The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
- The cup product of Brooks quasimorphisms
- Heat kernel estimates for time fractional equations
- Extreme non-Arens regularity of the group algebra
- Rational homology and homotopy of high-dimensional string links
- Global integrability for solutions to some anisotropic problem with nonstandard growth
- Hardy operators on Musielak–Orlicz spaces
- Contractibility of the stability manifold for silting-discrete algebras
- Order of the canonical vector bundle over configuration spaces of spheres
- An endpoint version of uniform Sobolev inequalities
- Space-time L2 estimates, regularity and almost global existence for elastic waves
- k-spaces and duals of non-archimedean metrizable locally convex spaces
- Path homology theory of multigraphs and quivers
