TY - JOUR
T1 - Critical Perturbations for Second Order Elliptic Operators—Part II
T2 - Non-tangential Maximal Function Estimates
AU - Bortz, Simon
AU - Hofmann, Steve
AU - Luna Garcia, José Luis
AU - Mayboroda, Svitlana
AU - Poggi, Bruno
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH, DE, part of Springer Nature 2024.
PY - 2024/6
Y1 - 2024/6
N2 - This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators -divA∇ by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the L2 well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi–Nash–Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-Lp “N<S” estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full L2 bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class. As a corollary, we claim the first results in an unbounded domain concerning the Lp-solvability of boundary value problems for the magnetic Schrödinger operator -(∇-ia)2+V when the magnetic potential a and the electric potential V are accordingly small in the norm of a scale-invariant Lebesgue space.
AB - This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators -divA∇ by first and zero order terms, whose complex coefficients lie in critical spaces, via the method of layer potentials. In particular, we show that the L2 well-posedness (with natural non-tangential maximal function estimates) of the Dirichlet, Neumann and regularity problems for complex Hermitian, block form, or constant-coefficient divergence form elliptic operators in the upper half-space are all stable under such perturbations. Due to the lack of the classical De Giorgi–Nash–Moser theory in our setting, our method to prove the non-tangential maximal function estimates relies on a completely new argument: We obtain a certain weak-Lp “N<S” estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrapping argument to recover the full L2 bound. Finally, we show the existence and uniqueness of solutions in a relatively broad class. As a corollary, we claim the first results in an unbounded domain concerning the Lp-solvability of boundary value problems for the magnetic Schrödinger operator -(∇-ia)2+V when the magnetic potential a and the electric potential V are accordingly small in the norm of a scale-invariant Lebesgue space.
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U2 - 10.1007/s00205-024-01977-x
DO - 10.1007/s00205-024-01977-x
M3 - Article
AN - SCOPUS:85189453593
SN - 0003-9527
VL - 248
JO - Archive For Rational Mechanics And Analysis
JF - Archive For Rational Mechanics And Analysis
IS - 3
M1 - 31
ER -