Xiaojing Feng

Regularity in Orlicz spaces for non-divergence degenerate elliptic equations on homogeneous groups

Let G be a homogeneous group, and let X1, X2, · · ·, Xp0 be left-invariant real vector fields on G that are homogeneous of degree one with respect to the dilation group of G and satisfy Hörmander’s condition. We establish a regularity result in the Orlicz spaces for the following equation:

GLOBAL WEAK MORREY ESTIMATES FOR SOME ULTRAPARABOLIC OPERATORS OF KOLMOGOROV-FOKKER-PLANCK TYPE

Lu\left( x \right) = \sum\limits_{j = 1}^{{p_0}} {{a_{ij}}\left( x \right){X_i}{X_J}u\left( x \right)} = f\left( x \right)

Global Orlicz regularity for sub-Laplace equations on homogeneous groups☆

, where aij(x) are real valued, bounded measurable functions defined on G, satisfying the uniform ellipticity condition, and belonging to the space VMO(G) with respect to the subelliptic metric induced by the vector fields X1, X2 · · ·, Xp0.

Existence of solutions for 2mth-order periodic boundary value problems ☆

In this paper, we consider a class of hypoelliptic Ornstein-Uhlenbeck operators in ℝN given by

Orlicz Regularity for Non-Divergence Parabolic Systems with Partially Vmo Coefficients

\mathcal{A} = \sum\limits_{i,j = 1}^{p_0 } {a_{ij} \partial _{x_i x_j }^2 + } \sum\limits_{i,j = 1}^N {b_{ij} x_i \partial _x } ,

Regularity lifting of weak solutions for nonlinear sub-Laplace equations on homogeneous groups

where (aij), (bij) are N × N constant matrices, and (aij) is symmetric and positive semidefinite. We deduce global Morrey estimates forA from similar estimates of its evolution operator L on a strip domain S = ℝN × [−1, 1].

Multiple solutions of some boundary value problems with parameters

We consider a class of hypoelliptic operators of the following type L = Sigma(p0)(i,j=1) a(ij)partial derivative(2)(xixj) + Sigma(N)(i,j=1) b(ij)x(i)partial derivative(xj) - partial derivative(t), where (a(ij)), (b(ij)) are constant matrices and (a(ij)) is symmetric positive definite on R-p0 (p(0) <= N). By establishing global Morrey estimates of singular integral on the homogenous space and the relation between Morrey space and weak Morrey space, we obtain the global weak Morrey estimates of the operator L on the whole space RN+1.