Sobolev improving for averages over curves in $\mathbb{R}^4$

David  Beltran; Shaoming Guo; Jonathan Hickman; Andreas Seeger

Sobolev improving for averages over curves in $\mathbb{R}^4$

David Beltran, Shaoming Guo, Jonathan Hickman, Andreas Seeger

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study Lp-Sobolev improving for averaging operators Aγ given by convolution with a compactly supported smooth density μγ on a non-degenerate curve. In particular, in 4 dimensions we show that Aγ maps Lp(R4) the Sobolev space Lp1/p(R4) for all 6<p<∞. This implies the complete optimal range of Lp-Sobolev estimates, except possibly for certain endpoint cases. The proof relies on decoupling inequalities for a family of cones which decompose the wave front set of μγ. In higher dimensions, a new non-trivial necessary condition for Lp(Rn)→Lp1/p(Rn) boundedness is obtained, which motivates a conjectural range of estimates.

Original language	English
Number of pages	60
Journal	Advances in Mathematics
Publication status	Accepted/In press - 5 Oct 2021

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2102.08806Accepted author manuscript, 738 KB

https://arxiv.org/abs/2102.08806

Cite this

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title = "Sobolev improving for averages over curves in $\mathbb{R}^4$",

abstract = "We study Lp-Sobolev improving for averaging operators Aγ given by convolution with a compactly supported smooth density μγ on a non-degenerate curve. In particular, in 4 dimensions we show that Aγ maps Lp(R4) the Sobolev space Lp1/p(R4) for all 6<p<∞. This implies the complete optimal range of Lp-Sobolev estimates, except possibly for certain endpoint cases. The proof relies on decoupling inequalities for a family of cones which decompose the wave front set of μγ. In higher dimensions, a new non-trivial necessary condition for Lp(Rn)→Lp1/p(Rn) boundedness is obtained, which motivates a conjectural range of estimates.",

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AU - Beltran, David

AU - Guo, Shaoming

AU - Hickman, Jonathan

AU - Seeger, Andreas

PY - 2021/10/5

Y1 - 2021/10/5

N2 - We study Lp-Sobolev improving for averaging operators Aγ given by convolution with a compactly supported smooth density μγ on a non-degenerate curve. In particular, in 4 dimensions we show that Aγ maps Lp(R4) the Sobolev space Lp1/p(R4) for all 6<p<∞. This implies the complete optimal range of Lp-Sobolev estimates, except possibly for certain endpoint cases. The proof relies on decoupling inequalities for a family of cones which decompose the wave front set of μγ. In higher dimensions, a new non-trivial necessary condition for Lp(Rn)→Lp1/p(Rn) boundedness is obtained, which motivates a conjectural range of estimates.

AB - We study Lp-Sobolev improving for averaging operators Aγ given by convolution with a compactly supported smooth density μγ on a non-degenerate curve. In particular, in 4 dimensions we show that Aγ maps Lp(R4) the Sobolev space Lp1/p(R4) for all 6<p<∞. This implies the complete optimal range of Lp-Sobolev estimates, except possibly for certain endpoint cases. The proof relies on decoupling inequalities for a family of cones which decompose the wave front set of μγ. In higher dimensions, a new non-trivial necessary condition for Lp(Rn)→Lp1/p(Rn) boundedness is obtained, which motivates a conjectural range of estimates.

M3 - Article

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -

Sobolev improving for averages over curves in $\mathbb{R}^4$

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