An Off-DiagonalT1 Theorem and Applications

We give a general boundedness criterion, analogous to theT1 Theorem, for singular integrals mappingLptoLq,q<p. As an easy corollary of this result, we deduce the Kato–Ponce “Leibniz Rule” for fractional order derivatives. We also study commutators of parabolic singular integrals and deduce pointwise a.e. convergence of truncations of a class of parabolic singular integrals which includes the caloric double-layer potential on the boundary of a non-cylindrical domain.