(untitled) (Communications in Mathematical Research, 2020)
Let L be the infinitesimal generator of an analytic semigroup on L^2(R^n) 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 Schrödinger operators and second-order elliptic operators of divergence form.