THE SOLUTION OF THE KATO PROBLEM FOR DEGENERATE ELLIPTIC OPERATORS WITH GAUSSIAN BOUNDS

We prove the Kato conjecture for degenerate elliptic operators on R-n. More precisely, we consider the divergence form operator L-w = -w(-1) divA del, where w is a Muckenhoupt A(2) weight and A is a complex-valued n x n matrix such that w(-1)A is bounded and uniformly elliptic. We show that if the heat kernel of the associated semigroup e(-tLw) satisfies Gaussian bounds, then the weighted Kato square root estimate, parallel to L(w)(1/2)f parallel to(L2(w)) approximate to parallel to del f parallel to(L2(w)), holds.