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Critical Perturbations for Second Order Elliptic Operators—Part II: Non-tangential Maximal Function Estimates

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Abstract

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators \(-\textrm{div}A \nabla \) 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 \(L^2\) 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-\(L^p\) “\(N<S\)” estimate, which we eventually couple with square function bounds, weighted extrapolation theory, and a bootstrap** argument to recover the full \(L^2\) 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 \(L^p\)-solvability of boundary value problems for the magnetic Schrödinger operator \(-(\nabla -i\textbf{a})^2+V\) when the magnetic potential \(\textbf{a}\) and the electric potential V are accordingly small in the norm of a scale-invariant Lebesgue space.

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Notes

  1. This means \(L^p\) bounds on certain square functions or non-tangential maximal functions.

  2. More generally, \(L_0\) could be an operator of the same form as L, whose lower order terms are small enough to ensure that the non-tangential maximal function estimates hold (see Theorem 1.11).

  3. Note that in Lemma 5.2, we may use that \(\Vert \mathbb {V}(\Theta _{t,1} f)\Vert _{L^p({\mathbb {R}^n})} \lesssim _{m} \Vert \mathbb {V}(\Theta _{t,m+1} f)\Vert _{L^p({\mathbb {R}^n})}\), by the aforementioned integration by parts argument. The subscript m refers to the number of transversal derivatives.

  4. Since the solutions u do not satisfy pointwise bounds, non-tangential convergence is also understood in an averaged sense; see Definition 2.5.

  5. The boundary data is achieved in the distributional sense, see Sect. 2.

  6. See Remark 1.14.

  7. Solvability throughout this paper means that we have accompanying \(L^p\) bounds for the non-tangential maximal function.

  8. Theorem 1.11 also has an appropriate analogue for p sufficiently near 2, but then the boundary operators have different domains and ranges; see Sect. 9.

  9. See Sect. 7 for the definitions of the operators \(\mathcal {S}^{\mathcal {L}}_0\), K, and \({\widetilde{K}}\).

  10. Again, see Remark 1.14.

  11. The Dirichlet problem is a consequence of semigroup theory, since the double layer potential in this setting is a constant multiple of the Poisson semigroup associated to \(\mathcal {L}_0\).

  12. We remind the reader that the notation CQ means the concentric dilate of Q by a factor \(C>0\).

  13. Strictly speaking, the statement of [59, Theorem 2.11] explicitly excludes the case under consideration (indeed the proof given does not apply in this case); however as is mentioned immediately after the statement of said Theorem, we may run an argument similar to the standard proof of the Riesz–Thorin Theorem, employing instead the three line lemma for sub-harmonic functions as in [14].

  14. In fact we will only need the first and second estimates in Definition 2.39 for \(T_t\), in the range \(|t|\approx \ell (Q)\).

  15. This is one of the few places where we may require additional smallness in addition to that imposed in [12], prior to discussing existence and uniqueness of the solutions to the boundary value problems.

  16. More precisely, the dependence on the ellipticity of \(\mathcal {L}_{\Vert }\) is through the constants appearing in Theorem 2.62.

  17. Notice that, since we already have good unweighted \(L^2\) square function estimates for \(\Theta _{t,m}\), the John-Nirenberg lemma gives us that this object is under control; as opposed to the unweighted case, where we were forced to hide this term.

  18. See Definition 2.5 for the truncated maximal function \(\widetilde{\mathcal {N}}_2^{(\varepsilon )}\).

  19. We note that, having obtained the map** property \({\mathcal {D}} \rightarrow S^2_+\), the equality of (7.11) holds on every t-slice in the space \(Y^{1,2}({\mathbb {R}^n})\) therefore the weak \(L^2({\mathbb {R}^n})\) limits, in t, of the co-normal derivatives \(\partial _{\nu _t}\) are the same.

  20. We may modify this Lemma, using now the function space \(W^{1,2}(\Sigma _a^b)\) instead of \(Y^{1,2}(\Sigma _a^b)\) to obtain the desired continuity in \(L^2({\mathbb {R}^n})\) instead of \(L^{2^*}({\mathbb {R}^n})\).

  21. For a definition of the Besov space, see Definition 14.1 of [46]

  22. Technically, Leoni considers the non-homogeneous case; however his proof easily gives the result stated here.

  23. An application of Sneiberg’s Lemma, and the known \(L^2\) results (see the introduction), reduces the invertibility of the boundary operators in Hypothesis B to the uniform boundedness of said operators in \(L^p\) for p in a neighborhood of 2. In turn this last is achieved by the methods of this paper.

  24. Of course, the definitions of good \({\mathcal {D}}\)/\({\mathcal {N}}\)/\({\mathcal {R}}\) solutions have to be appropriately modified to work with the exponent p and the slice spaces \(D^p_+\) and \(S^p_+\).

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Acknowledgements

S. Bortz would like to thank Moritz Egert and Olli Saari for some helpful conversations. B. Poggi would like to thank Max Engelstein for some helpful discussions.

Funding

This material is based upon work supported by National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the MSRI in Berkeley, California, during the Spring 2017 semester. S. Bortz was supported by the Simons foundation Grant “Travel support for Mathematicians” (Grant Number 959861). S. Bortz and S. Mayboroda were partly supported by NSF INSPIRE Award DMS-1344235. S. Hofmann was supported by NSF Grant DMS-2000048. S. Mayboroda and B. Poggi were supported in part by the NSF RAISE-TAQS Grant DMS-1839077 and the Simons foundation Grant 563916, SM. B. Poggi was also supported by the University of Minnesota Doctoral Dissertation Fellowship, and by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement 101018680).

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama, Tuscaloosa, AL, 35487, USA

    Simon Bortz

  2. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Steve Hofmann

  3. Department of Mathematics and Statistics, McMaster University, Ontario, L8S 4K1, Canada

    José Luis Luna Garcia

  4. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Svitlana Mayboroda

  5. Department of Mathematics, Universitat Autònoma de Barcelona, Barcelona, Catalonia, Spain

    Bruno Poggi

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Bortz, S., Hofmann, S., Luna Garcia, J.L. et al. Critical Perturbations for Second Order Elliptic Operators—Part II: Non-tangential Maximal Function Estimates. Arch Rational Mech Anal 248, 31 (2024). https://doi.org/10.1007/s00205-024-01977-x

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