Sobolev-Morrey Spaces Research Articles

Local Sobolev–Morrey estimates for nondivergence operators with drift on homogeneous groups

Let \(G\) be a homogeneous group, \(X_0,X_1,X_2,\ldots ,X_{p_0} \) be left invariant real vector fields on \(G\) and satisfy Hormander’s rank condition. Assume that \(X_1,X_2,\ldots ,X_{p_0}\) are homogeneous of degree one and \(X_0\) is homogeneous of degree two. In this paper, we study the following operator with drift: $$\begin{aligned} L=\sum _{i,j=1}^{p_0} a_{ij}(x)X_i X_j+a_0(x)X_0, \end{aligned}$$ where \(a_{ij}(x)\) are real valued, bounded measurable functions defined in \(G\), satisfying the uniform ellipticity condition and \(a_0(x)\) is bounded away from zero in \(G\). Moreover, we assume that the coefficients \(a_{ij},\ a_0\) belong to the space \({ VMO}(G)\) (vanishing mean oscillation) with respect to the subelliptic metric induced by the vector fields \(X_0,X_1,X_2\ldots ,X_{p_0}\). For this class of operators, we obtain the local Sobolev–Morrey estimates by establishing the boundedness on Morrey spaces for singular integrals under homogeneous type spaces and proving interpolation results on Sobolev–Morrey spaces.

Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems

We prove local existence, uniqueness, Holder regularity in space and time, and smooth dependence in Holder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data. As a result the gap between low smoothness of the data, which is typical for many applications, and high smoothness of the solutions, which is necessary for the applicability of differential calculus to abstract formulations of the initial boundary value problems, has been closed. The theory works for any space dimension, and the nonlinearities in the equations as well as in the boundary conditions are allowed to be nonlocal and to have any growth. The main tools are new maximal regularity results (Griepentrog in Adv Differ Equ 12:781–840, 1031–1078, 2007) in Sobolev–Morrey spaces for linear parabolic initial boundary value problems with nonsmooth data, linearization techniques and the Implicit Function Theorem.

Embedding theorems in the Sobolev-Morrey type spaces S p,a,ℵτ W W(G) with dominant mixed derivatives

We construct Sobolev-Morrey type spaces with dominant mixed derivatives and, using the obtained integral representation, prove some embedding theorems in these spaces.