论文标题
关于与强烈退化的抛物线型托架物类型的抛物线措施的良好特性
On the fine properties of parabolic measures associated to strongly degenerate parabolic operators of Kolmogorov type
论文作者
论文摘要
我们认为表格的强烈退化抛物线运营商\ [ \ Mathcal {l}：= \ nabla_x \ cdot（a（x，x，y，t）\ nabla_x）+x \ cdot \ nabla_y- \ partial_t \] ω= \ {（x，x，y，t）=（x，x，x_ {m}，y，y，y_ {m}，t）\ in \ mathbb r^{m-1} \ times \ times \ times \ times \ times \ times \ times \ times \ times \ mathbb r^{m-1} \ times \ mathb r \ times \ mathb r \ times \ mathb r \ times \ times \ mathb r \ m \] We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix in $\mathbb R^{m}$) and concerning $ψ$ and $Ω$ we assume that $Ω$ is what we call an (unbounded) Lipschitz domain: $ψ$ satisfies a uniform Lipschitz condition adapted to the dilation structure and the （非euclidean）操作员$ \ mathcal {l} $的基础。 We prove, assuming in addition that $ψ$ is independent of the variable $y_m$, that $ψ$ satisfies an additional regularity condition formulated in terms of a Carleson measure, and additional conditions on $A$, that the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon-Nikodym derivative defines an $A_\infty$-weight with respect to the surface measure.
We consider strongly degenerate parabolic operators of the form \[ \mathcal{L}:=\nabla_X\cdot(A(X,Y,t)\nabla_X)+X\cdot\nabla_Y-\partial_t \] in unbounded domains \[ Ω=\{(X,Y,t)=(x,x_{m},y,y_{m},t)\in\mathbb R^{m-1}\times\mathbb R\times\mathbb R^{m-1}\times\mathbb R\times\mathbb R\mid x_m>ψ(x,y,t)\}. \] We assume that $A=A(X,Y,t)$ is bounded, measurable and uniformly elliptic (as a matrix in $\mathbb R^{m}$) and concerning $ψ$ and $Ω$ we assume that $Ω$ is what we call an (unbounded) Lipschitz domain: $ψ$ satisfies a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator $\mathcal{L}$. We prove, assuming in addition that $ψ$ is independent of the variable $y_m$, that $ψ$ satisfies an additional regularity condition formulated in terms of a Carleson measure, and additional conditions on $A$, that the associated parabolic measure is absolutely continuous with respect to a surface measure and that the associated Radon-Nikodym derivative defines an $A_\infty$-weight with respect to the surface measure.
