Morrey Estimates Research Articles

Sobolev and Morrey Estimates for Non-Smooth Vector Fields of Step Two

We prove Sobolev-type and Morrey-type inequalities for Sobolev spaces related to a family of non-smooth vector fields which formally satisfy the Hörmander condition of step 2. The coefficients of the vector fields are not regular enough to define the Carnot-Carathéodory distance. Thus the result is proved by developing a real analysis technique which is based on an approximation procedure of Lipschitz continuous vector fields with a family of left-invariant first order operators on a nilpotent Lie group.

Some interior regularity results for solutions of Hessian equations

We prove monotonicity formulae related to degenerate k-Hessian equations which yield Morrey type estimates for certain integrands involving the second derivatives of the solution. In the special case k=2 we deduce that weak solutions in $W^{2,p}(\Omega)$ , $p>n-1$ , have locally Holder continuous gradients. In the nondegenerate case we also show that weak solutions in $W^{2,p}(\Omega)$ , $p>kn/2$ , have locally bounded second derivatives.