  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.
