BOUNDS FOR THE GRADIENT OF THE TRANSITION KERNEL FOR ELLIPTIC OPERATORS WITH UNBOUNDED DIFFUSION, DRIFT AND POTENTIAL TERMS

Global Sobolev regularity and pointwise upper bounds for the gradient of transition densities associated with second order elliptic operators in R-d was obtained in the case of bounded diffusion coefficients in [13, Section 5]. In this paper we generalize these results to the case of unbounded diffusions. Our technique is based on an approximation procedure and a De Giorgi type regularity result. We like to point out, that such an approximation procedure cannot be applied to the result in [13], since the constants in the estimates obtained in [13] depend on the infinity norm of the diffusion coefficients.