锥域与三角形上的Bernstein不等式
Bernstein inequality on conic domains and triangle
摘要：
在$L^p$空间中建立了二次权在圆锥面$\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = t, x \in \mathbb{R}^d, t\in [0,1]\}$上的加权Bernstein不等式，以及在由圆锥面和超平面$t =1$所围成的实锥上的加权Bernstein不等式，当$d=1$在平面上变成三角形时.虽然$t$变量中导数的不等式与预期一致，但$x$变量中导数的不等式比预期的更强。作为一个例子，在三角形$\{(x_1,x_2): x_1 \ge 0, \, x_2 \ge 0,\, x_1+x_2 \le 1\}$上，通常的导数$\partial_1$的Bernstein不等式指出$\|ϕ_1 \partial_1 f\|_{p,w} \le c n \|f\|_{p,w}$与$ϕ_1(x_1,x_2):= x_1(1-x_1-x_2)$，而我们的新结果给出了$$\| (1-x_2)^{-1/2} ϕ_1 \partial_1 f\|_{p,w} \le c n \|f\|_{p,w}.$$。新的不等式更强，并指出了多边形域上迄今未观察到的现象。
英文摘要：
We establish weighted Bernstein inequalities in $L^p$ space for the doubling weight on the conic surface $\mathbb{V}_0^{d+1} = \{(x,t): \|x\| = t, x \in \mathbb{R}^d, t\in [0,1]\}$ as well as on the solid cone bounded by the conic surface and the hyperplane $t =1$, which becomes a triangle on the plane when $d=1$. While the inequalities for the derivatives in the $t$ variable behave as expected, there are inequalities for the derivatives in the $x$ variables that are stronger than what one may have expected. As an example, on the triangle $\{(x_1,x_2): x_1 \ge 0, \, x_2 \ge 0,\, x_1+x_2 \le 1\}$, the usual Bernstein inequality for the derivative $\partial_1$ states that $\|ϕ_1 \partial_1 f\|_{p,w} \le c n \|f\|_{p,w}$ with $ϕ_1(x_1,x_2):= x_1(1-x_1-x_2)$, whereas our new result gives $$\| (1-x_2)^{-1/2} ϕ_1 \partial_1 f\|_{p,w} \le c n \|f\|_{p,w}.$$ The new inequality is stronger and points out a phenomenon unobserved hitherto for polygonal domains.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
