Uniform Sobolev inequalities for second order non-elliptic differential operators
Uniform Sobolev inequalities for second order non-elliptic differential operators
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二阶非椭圆微分算子的一致 Sobolev 不等式
DOI：
10.1016/j.aim.2016.07.016
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Sanghyuk Lee
中科院分区：
文献类型：
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作者：
Eunhee Jeong;Yehyun Kwon;Sanghyuk Lee
研究非椭圆型二阶微分算子P (D)的一致Sobolev不等式。d≥3我们证明水列夫类型估计为u为L q (R d)≤C为P (d) u为L和C (R d)持有独立的常数项的一阶和P (d)当且仅当1 q = 2 / d / P−1 d和2 d (d−1)2 + 2 d−4 < P < 2 d (d−1)。我们也获得限制弱类型端点估计的关键(P, q) = (2 d−1 d, 2 d (d−1)(d−2)2),(2 d (d−1)d 2 + 2 d−4、2 d (d−1)−2)。结果扩展了不等式| P (D) u|≤| V u|的唯一延拓成立的函数类。
We study uniform Sobolev inequalities for the second order differential operators P (D) of non-elliptic type. For d≥ 3 we prove that the Sobolev type estimate‖ u‖ L q (R d)≤ C‖ P (D) u‖ L p (R d) holds with C independent of the first order and the constant terms of P (D) if and only if 1/p− 1/q= 2/d and 2 d (d− 1) d 2+ 2 d− 4< p< 2 (d− 1) d. We also obtain restricted weak type endpoint estimates for the critical (p, q)=(2 (d− 1) d, 2 d (d− 1)(d− 2) 2),(2 d (d− 1) d 2+ 2 d− 4, 2 (d− 1) d− 2). As a consequence, the result extends the class of functions for which the unique continuation for the inequality| P (D) u|≤| V u| holds.