A T1 theorem for weakly singular integral operators

Abstract: We establish conditions in the spirit of the T1 theorem of David and Journ\'e which guarantee the boundedness of \nabla T on L^p(\R^n), where T is an integral transformation and 1<p<\infty. These are natural size and regularity conditions for the kernel of the integral transformation, along with the sharp condition T1,T^t1\in\mathcal{I}^1(\mathrm{BMO}). A simple example satisfying these conditions is the Riesz potential denoted by \mathcal{I}^1.