We continue our study of the Widom factors for $L_p\left(\mathrm{\mu }\right)$ extremal polynomials initiated in (Alpan and Zinchenko, 2020). In this work we characterize sets for which the lower bounds obtained in (Alpan and Zinchenko, 2020) are saturated, establish continuity of the Widom factors with respect to the measure $\mathrm{\mu }$, and show that despite the lower bound ${\left[W_{2,n}\left({\mathrm{\mu }}_K\right)\right]}^2\ge 2S\left({\mathrm{\mu }}_K\right)$ for the equilibrium measure ${\mathrm{\mu }}_K$ on a compact set $K\subset \mathbb{R}$ the general lower bound ${\left[W_{p,n}\left(\mathrm{\mu }\right)\right]}^p\ge S\left(\mathrm{\mu }\right)$ is optimal even for measures $d\mathrm{\mu }=wd{\mathrm{\mu }}_K$ with polynomial weights $w$ on $K\subset \mathbb{R}$. We also study pull-back measures under polynomial pre-images introduced in (Geronimo and Van Assche,1988), (Peherstorfer and Steinbauer, 2001) and obtain invariance of the Widom factors for such measures. Lastly, we study in detail the Widom factors for orthogonal polynomials with respect to the equilibrium measure on a circular arc and, in particular, find their limit, infimum, and supremum and show that they are strictly monotone increasing with the degree and strictly monotone decreasing with the length of the arc.

我们继续研究Widom因素， ${\mathrm{大号}}_p（\mathrm{\mu }）$极值多项式始于（Alpan和Zinchenko，2020）。在这项工作中，我们描述了在（Alpan和Zinchenko，2020）中获得的下界饱和的集合的特征，建立了Widom因子关于度量的连续性$\mathrm{\mu }$，并显示尽管下界 ${\left[{\mathrm{w\; ^}}_{2，ñ}（{\mathrm{\mu }}_{ķ}）\right]}^2\ge 2\mathrm{小号}（{\mathrm{\mu }}_{ķ}）$ 用于均衡测度 ${\mathrm{\mu }}_{ķ}$ 紧凑的套装 $ķ\subset \mathbb{[R}$ 一般下限 ${\left[{\mathrm{w\; ^}}_{p，ñ}（\mathrm{\mu }）\right]}^p\ge \mathrm{小号}（\mathrm{\mu }）$ 即使是措施也是最佳的 $d\mathrm{\mu }=wd{\mathrm{\mu }}_{ķ}$ 多项式权重 $w$ 上 $ķ\subset \mathbb{[R}$。我们还研究了在（Geronimo和Van Assche，1988），（Peherstorfer和Steinbauer，2001）中引入的多项式原像下的回拉测度，并获得了Widom因子的不变性。最后，针对圆弧上的平衡测度，我们详细研究了正交多项式的Widom因子，特别是找到了它们的极限，最小和极值，并表明它们是严格单调的，随度数的增加而严格的是单调的。与弧的长度。