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We establish conditions in the spirit of the T1 theorem of David and Journ\'ewhich guarantee the boundedness of \nabla T on L^p(\R^n), where T is anintegral transformation and 1<p<\infty. These are natural size and regularityconditions for the kernel of the integral transformation, along with the sharpcondition T1,T^t1\in\mathcal{I}^1(\mathrm{BMO}). A simple example satisfyingthese conditions is the Riesz potential denoted by \mathcal{I}^1.

A T1 theorem for weakly singular integral operators