论文标题
圆锥形域和三角形的伯恩斯坦不平等
Bernstein inequality on conic domains and triangle
论文作者
论文摘要
我们在$ l^p $ space中建立了加权的伯恩斯坦不等式，用于圆锥表面$ \ mathbb {v} _0^{d+1} = \ {（x，x，t）：\ | x \ | | = t，x \ in \ mathbb {r}^d，t \ in [0,1] \} $以及在圆锥表面和超平面$ t = 1 $界的实心锥上，当$ d = 1 $时，它变成了平面上的三角形。尽管$ t $变量中衍生物的不平等现象是按预期的，但$ x $变量中的衍生物的不平等现象比人们预期的要强。例如，在三角形上$ \ {（x_1，x_2）：x_1 \ ge 0，\，\，x_2 \ ge 0，\，\，x_1+x_1+x_2 \ le 1 \} $，通常的伯恩斯坦不平等，用于衍生$ \ partial_1 $ \ partial_1 $ \ \ f \ | c n \ | f \ | _ {p，w} $带有$ ϕ_1（x_1，x_2）：= x_1（1-x_1-x_2）$，而我们的新结果给出$ \ | （1-x_2）^{ - 1/2} ϕ_1 \ partial_1 f \ | _ { 新的不平等更强，并指出了多边形域的一种现象未观察到的现象。
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.
