Critical perturbations for second-order elliptic operators, I: Square function bounds for layer potentials

Abstract

This is the first part of a series of two papers where we study perturbations of divergence form second order elliptic operatorsdiv A∇ by complexvalued first and zeroth order terms, whose coefficients lie in critical spaces, via the method of layer potentials.In the present paper, we establish L 2 control of the square function via a vector-valued T b theorem and abstract layer potentials, and use these square function bounds to obtain uniform slice bounds for solutions.For instance, an operator for which our results are new is the generalized magnetic Schrödinger operator -(∇ia)A(∇ia) + V when the magnetic potential a and the electric potential V are accordingly small in the norm of a scale-invariant Lebesgue space.

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