In this paper we study elliptic and parabolic boundary value problems with inhomogeneous boundary conditions in weighted function spaces of Sobolev, Bessel potential, Besov and Triebel-Lizorkin type. As one of the main results, we solve the problem of weighted L_q-maximal regularity in weighted Besov and Triebel-Lizorkin spaces for the parabolic case, where the spatial weight is a power weight in the Muckenhoupt \(A_{\infty }\)-class. In the Besov space case we have the restriction that the microscopic parameter equals to q. Going beyond the A_p-range, where p is the integrability parameter of the Besov or Triebel-Lizorkin space under consideration, yields extra flexibility in the sharp regularity of the boundary inhomogeneities. This extra flexibility allows us to treat rougher boundary data and provides a quantitative smoothing effect on the interior of the domain. The main ingredient is an analysis of anisotropic Poisson operators.

在本文中，我们研究了 Sobolev、Bessel 势、Besov 和 Triebel-Lizorkin 型加权函数空间中具有非齐次边界条件的椭圆和抛物线边值问题。作为主要结果之一，我们解决了抛物线情况下加权 Besov 和 Triebel-Lizorkin 空间中的加权L _{q -}最大正则性问题，其中空间权重是 Muckenhoupt \(A_{\infty }\ ) -类。在 Besov 空间情况下，我们有微观参数等于q的限制。超出A _{p 范围}，其中p是所考虑的 Besov 或 Triebel-Lizorkin 空间的可积性参数，在边界不均匀性的尖锐规律性方面产生额外的灵活性。这种额外的灵活性使我们能够处理更粗糙的边界数据，并在域内部提供定量平滑效果。主要成分是对各向异性泊松算子的分析。