论文标题
加权$ l_p $马尔可夫因子，球上的重量增加了一倍
Weighted $L_p$ Markov factors with doubling weights on the ball
论文作者
论文摘要
令$ l_ {p，w}，\ 1 \ 1 \ le p <\ infty，$表示加权$ l_p $ l_p $单位球上的函数空间$ \ bbb b^d $，重量$ w $ in $ \ bbb bbb b^d $。多项式$ p $上$ l_ {p，w} $的马尔可夫因子由$ \ frac {\ | \，| \ | \ nabla p | \，\ | _ {p，w}}} {\ | p \ | p \ | ________ {p，w}} $，$ \ nabla p $ p $我们研究了$ l_ {p，w} \（1 \ le p <\ infty）$的最坏情况马尔可夫因素，并获得这些因素的程度最多是$ 2 $。特别是，对于jacobi重量$w_μ（x）=（1- | x |^2）^{μ-1/2}，\μ\ ge0 $，指数$ 2 $很尖。我们还研究了具有独立$ n（0，σ^2）$系数的随机多项式上的$ l_ {2，w} $的平均案例马尔可夫因子，并获得平均（预期）马尔可夫因子的上限是$ 3/2 $的订单订单度，与程度的上层最高率最差的情况下。
Let $L_{p,w},\ 1 \le p<\infty,$ denote the weighted $L_p$ space of functions on the unit ball $\Bbb B^d$ with a doubling weight $w$ on $\Bbb B^d$. The Markov factor for $L_{p,w}$ on a polynomial $P$ is defined by $\frac{\|\, |\nabla P|\,\|_{p,w}}{\|P\|_{p,w}}$, where $\nabla P$ is the gradient of $P$. We investigate the worst case Markov factors for $L_{p,w}\ (1\le p<\infty)$ and obtain that the degree of these factors are at most $2$. In particular, for the Jacobi weight $w_μ(x)=(1-|x|^2)^{μ-1/2}, \ μ\ge0$, the exponent $2$ is sharp. We also study the average case Markov factor for $L_{2,w}$ on random polynomials with independent $N(0, σ^2)$ coefficients and obtain that the upper bound of the average (expected) Markov factor is order degree to the $3/2$, as compared to the degree squared worst case upper bound.
