${\displaystyle B^q}$ for parabolic measures

If Ω is a Lip(1,1//2) domain, μ a doubling measure on ${\displaystyle {\partial }_p\Omega ,\partial /\partial t-L_i}$, i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures ${\displaystyle {\omega }_0}$, ${\displaystyle {\omega }_1}$ have the property that ${\displaystyle {\omega }_0\in B^q(\mu )}$ implies ${\displaystyle {\omega }_1}$ is absolutely continuous with respect to ${\displaystyle {\omega }_0}$ whenever a certain Carleson-type condition holds on the difference function of the coefficients of ${\displaystyle L_1}$ and ${\displaystyle L_0}$. Also ${\displaystyle {\omega }_0\in B^q(\mu )}$ implies ${\displaystyle {\omega }_1\in B^q(\mu )}$ whenever both measures are center-doubling measures. This is B. Dahlberg's result for elliptic measures extended to parabolic-type measures on time-varying domains. The method of proof is that of Fefferman, Kenig and Pipher.