Vol. 15, No. 5, 2022

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

Simon Bortz, Steve Hofmann, José Luis Luna García, Svitlana Mayboroda and Bruno Poggi

Vol. 15 (2022), No. 5, 1215–1286
Abstract

This is the first part of a series of two papers where we study perturbations of divergence form second-order elliptic operators $-\mathrm{div}A\nabla$ 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 ${L}^{2}$ control of the square function via a vector-valued $Tb$ 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 $-\left(\nabla -ia\right)A\left(\nabla -ia\right)+V$ when the magnetic potential $a$ and the electric potential $V$ 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
Mathematical Subject Classification
Primary: 35B20, 35B25, 35J15, 35J25, 35J75
Secondary: 31B10, 35B33, 42B37, 47B90