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 div A 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 L2 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 ( 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.

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
Milestones
Received: 6 April 2020
Revised: 23 November 2020
Accepted: 31 December 2020
Published: 29 September 2022
Authors
Simon Bortz
Department of Mathematics
University of Alabama
Tuscaloosa, AL
United States
Steve Hofmann
Department of Mathematics
University of Missouri
Columbia, MO
United States
José Luis Luna García
Department of Mathematics
University of Missouri
Columbia, MO
United States
Svitlana Mayboroda
School of Mathematics
University of Minnesota
Minneapolis, MN
United States
Bruno Poggi
School of Mathematics
University of Minnesota
Minneapolis, MN
United States