Endp oint Estimates for Commutators of Fractional Integrals Asso ciated to Op erators with Heat Kernel Bounds

Let L be the infinitesimal generator of an analytic semigroup on L2(Rn) with pointwise upper bounds on heat kernel, and denote by L?α/2 the fractional integrals of L. For a BMO function b(x), we show a weak type LlogL estimate of the commutators b, L?α/2](f)(x) = b(x)L?α/2(f)(x)?L?α/2(bf)(x). We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.