Weighted Sobolev–Morrey Estimates for Nondivergence Degenerate Operators with Drift on Homogeneous Groups

Let X0,X1,…,Xq(q<N) be real vector fields, which are left invariant on homogeneous group G, provided that X0 is homogeneous of degree two and X1,…,Xq are homogeneous of degree one. We consider the following nondivergence degenerate operator with drift L=∑i,j=1qaij(x)XiXj+a0(x)X0, where the coefficients aij(x), a0(x) belonging to vanishing mean oscillation space are bounded measurable functions. Furthermore, aij(x) satisfies the uniform ellipticity condition on Rq and a0(x)≠0. We obtain the local weighted Sobolev–Morrey estimates by applying the boundedness of commutators and interpolation inequalities on weighted Morrey spaces.

Weighted Lebesgue and central Morrey estimates for -adic multilinear Hausdorff operators and its commutators

UDC 517.9 We establish the sharp boundedness of -adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of -adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Morrey estimates for a class of elliptic equations with drift term

We consider the following boundary value problem $$\begin{array}{} \displaystyle \begin{cases} - {\rm div}{[M(x)\nabla u - E(x) u]} =f(x) & \text{in}~~ {\it\Omega} \\ u =0 & \text{on}~~ \partial{\it\Omega}, \end{cases} \end{array}$$ where Ω is a bounded open subset of ℝN, with N > 2, M : Ω → ℝN2 is a symmetric matrix, E(x) and f(x) are respectively a vector field and function both belonging to suitable Morrey spaces and we study the corresponding regularity of u and D u.

Estimates for parabolic equations with measure data in generalized Morrey spaces

In this paper, we prove the optimal generalized Morrey estimates for the spatial gradient of the solutions obtained by limits of approximations (SOLA) for a class of parabolic problems with right-hand side measure in a very general irregular domain. The nonlinearity is assumed to be merely measurable only in the time variable [Formula: see text] and belongs to the small bounded mean oscillation (BMO) class as functions of the spatial variable [Formula: see text].