This is the first part of a series of two papers where we study
perturbations of divergence form second-order elliptic operators
by complex-valued first- and zeroth-order terms, whose coefficients lie in
critical spaces, via the method of layer potentials. In the present paper, we establish
control of the square
function via a vector-valued
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
when the magnetic
potential and the
electric potential
are accordingly small in the norm of a scale-invariant Lebesgue space.
Keywords
second-order elliptic equation, elliptic equation with
lower-order terms, boundary value problems, layer
potentials, $Tb$ theorem, equation with drift terms