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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 |
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Vol. 15 (2022), No. 5, 1215–1286
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Abstract |
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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
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Mathematical Subject Classification
Primary: 35B20, 35B25, 35J15, 35J25, 35J75
Secondary: 31B10, 35B33, 42B37, 47B90
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Milestones
Received: 6 April 2020
Revised: 23 November 2020
Accepted: 31 December 2020
Published: 29 September 2022
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Authors |
| Simon Bortz | |
| Department of Mathematics University of Alabama Tuscaloosa, AL United States |
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| Steve Hofmann | |
| Department of Mathematics University of Missouri Columbia, MO United States |
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| José Luis Luna García | |
| Department of Mathematics University of Missouri Columbia, MO United States |
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| Svitlana Mayboroda | |
| School of Mathematics University of Minnesota Minneapolis, MN United States |
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| Bruno Poggi | |
| School of Mathematics University of Minnesota Minneapolis, MN United States |
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