An endpoint version of uniform Sobolev inequalities
An endpoint version of uniform Sobolev inequalities
复制标题
统一索博列夫不等式的端点版本
DOI：
10.1515/forum-2018-0042
复制
发表时间：
2016
影响因子：
0.8
通讯作者：
Cheng Zhang
中科院分区：
文献类型：
--
作者：
Tianyi Ren;Yakun Xi;Cheng Zhang
本文证明了[C.E.Kenig，A.Ruiz和C.D.Sogge]中的一致Sobolev不等式的一个端点形式，一致Sobolev不等式和二阶常系数微分算子的唯一连续性，
杜克·马思。J.55 1987,329-347]。
已知强型不等式在端点处不再成立；然而，我们证明了约束弱型不等式在端点处成立，这暗示了用实数内插得到的早期经典结果。在我们的证明中的关键成分是由Bourain[J.Bourain，
估计确定了函数的最大值，
C.R.Acad.SCI。巴黎310 1985,499-502]。
我们还证明了在五角形边界的某些部分上的限制弱型Stein-Tomas限制不等式，它完全刻画了该限制弱型限制不等式所满足的指数范围。
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.