Weighted $L_p$ Markov factors with doubling weights on the ball

Let $L_{p,w},1\le p<\mathrm{\infty },$ denote the weighted $L_p$ space of functions on the unit ball ${\mathbb{B}}^d$ with a doubling weight $w$ on ${\mathbb{B}}^d$. The Markov factor for $L_{p,w}$ on a polynomial $P$ is defined by $\frac{|, |\nabla P|,|{p,w}}{|P|{p,w}}$,where$\nabla P$isthegradientof$P$. We investigate the worst case Markov factors for $L_{p,w}(1\le p<\mathrm{\infty })$ and obtain that the degree of these factors are at most $2$. In particular, for the Jacobi weight $w_{\mu }(x)=(1-\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{\mu -1\mathrm{/}2},\mu \ge 0$, 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 $3\mathrm{/}2$, as compared to the degree squared worst case upper bound.