Abstract
This is the first part of a series of two papers where we study perturbations of divergence form secondorder 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 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 −(∇−i a)A(∇−i a)+V when the magnetic potential a and the electric potential V are accordingly small in the norm of a scale-invariant Lebesgue space.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1215-1286 |
| Number of pages | 72 |
| Journal | Analysis and PDE |
| Volume | 15 |
| Issue number | 5 |
| DOIs | |
| State | Published - 2022 |
Bibliographical note
Funding Information:This material is based upon work supported by National Science Foundation under grant DMS-1440140 while the authors were in residence at the MSRI in Berkeley, California, during the Spring 2017 semester. S. Bortz and S. Mayboroda were partly supported by NSF INSPIRE Award DMS-1344235. S. Mayboroda was also supported in part by the NSF RAISE-TAQS grant DMS-1839077 and the Simons Foundation grant 563916, SM. S. Hofmann was supported by NSF grant DMS-1664047. S. Bortz would like to thank Moritz Egert for some helpful conversations.
Publisher Copyright:
© 2022 Mathematical Sciences Publishers
Keywords
- Boundary value problems
- Elliptic equation with lower-order terms
- Equation with drift terms.
- Layer potentials
- Second-order elliptic equation
- T b theorem
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