Weighted Lp Markov factors with doubling weights on the ball

Let $L_{p,w},1\le p<\infty ,$ denote the weighted $L_p$ space of functions on the unit ball $B^d$ with a doubling weight $w$ on $B^d$. The Markov factor for $L_{p,w}$ of a polynomial $P$ is defined by $\frac{{\Vert |\nabla P|\Vert }_{p,w}}{{\Vert P\Vert }_{p,w}},$ where $\nabla P$ is the gradient of $P$. We investigate the worst case Markov factors for $L_{p,w}$ and prove that the degree of these factors is at most 2. In particular, for the Gegenbauer weight $w_{\mu }\left(x\right)={(1-{\left|x\right|}^2)}^{\mu -1/2},\mu \ge 0,$ the exponent 2 is sharp. We also study the average case Markov factor for $L_{2,w}$ on random polynomials with independent $N(0,{\sigma }^2)$ 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.