Uniform Sobolev inequalities for second order non-elliptic differential operators

详细信息    查看全文

• 作者：Eunhee Jeong ^{moonshine10@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Yehyun Kwon ^{kwonyh27@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Sanghyuk Lee ; ^{shklee@snu.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B45 ; 42B15
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：22 October 2016
• 年：2016
• 卷：302
• 期：Complete
• 页码：323-350
• 全文大小：545 K

文摘

We study uniform Sobolev inequalities for the second order differential operators P(D)$P(D)$ of non-elliptic type. For d≥3$d\ge 3$ we prove that the Sobolev type estimate ‖u‖_L^q(R^d)≤C‖P(D)u‖_L^p(R^d)${\Vert u\Vert }_{L^q({\mathbb{R}}^d)}\le C{\Vert P(D)u\Vert }_{L^p({\mathbb{R}}^d)}$ holds with C   independent of the first order and the constant terms of P(D)$P(D)$ if and only if 1/p−1/q=2/d$1/p-1/q=2/d$ and $\frac{2d(d-1)}{d^2+2d-4}<p<\frac{2(d-1)}{d}$. We also obtain restricted weak type endpoint estimates for the critical $(p,q)=(\frac{2(d-1)}{d},\frac{2d(d-1)}{{(d-2)}^2})$, $(\frac{2d(d-1)}{d^2+2d-4},\frac{2(d-1)}{d-2})$. As a consequence, the result extends the class of functions for which the unique continuation for the inequality |P(D)u|≤|Vu|$|P(D)u|\le |Vu|$ holds.