Weighted $L_p$ 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 $\Bbb B^d$ 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 $w_\mu(x)=(1-|x|^2)^{\mu-1/2}, \ \mu\ge0$, the exponent $2$ is sharp. We also study the average case Markov factor for $L_{2,w}$ on random polynomials with independent $N(0, \sigma^2)$ 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.