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On the Kato Problem for Elliptic Operators in Non-Divergence Form

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Abstract

We consider the Kato square root problem for non-divergence second order elliptic operators \(L =- \sum _{i,j=1}^{n}a_{ij} D_iD_j\), and, especially, the normalized adjoints of such operators. In particular, our results are applicable to the case of real coefficients having sufficiently small BMO norm. We assume that the coefficients of the operator are smooth, but our quantitative estimates do not depend on the assumption of smoothness.

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Notes

  1. In fact, [23] precedes [24] chronologically; the result in [23] treats explicitly local versions of W, but the same arguments may be applied to the global W constructed in [24].

  2. We caution the reader that the proof of (1.5) is a somewhat non-trivial matter, and is based on the boundedness of the commutator [T, b], where T is a singular integral and \(b\in \) BMO [17], along with a suitable expansion in terms of spherical harmonics; see [13, 14] for related results in the unweighted case.

  3. (1.6) is used to show that (1.4) holds for all \(u \in \mathcal {D}(L)\).

  4. Thus, our results here are somewhat related to those of Cruz-Uribe and Rios [18].

  5. To clarify a possible point of confusion, we mention that in [8] and [18], the unit vectors were taken in \(\mathbb {C}^n\), because in the divergence form setting of those papers, one treats the case of complex coefficients; at present, our results in the non-divergence form case treat only the case of real coefficients, so we need only consider real unit vectors.

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Acknowledgements

The second author is supported by the grant CEX2019-000904-S-20-3, funded by MCIN/AEI/ 10.13039/501100011033, and acknowledges financial support from MCIN/AEI/ 10.13039/501100011033 grants CEX2019-000904-S and PID2019-107914GB-I00. The third author was supported by NSF grant DMS-2000048. Part of this work was carried out while the first and third authors were visiting ICMAT in Madrid, and part of this work was carried out while the second author was visiting the third author at the University of Missouri - Columbia. The authors express their gratitude to these institutions. The third author thanks Prof. X. T. Duong for an interesting conversation concerning the latter’s joint work with L. Yan [22], and in particular for pointing out to us the argument sketched in Remark 1.13.

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Authors and Affiliations

  1. Universidad del País Vasco, Apartado 644, 48080, Bilbao, Spain

    Luis Escauriaza

  2. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, E-28049, Madrid, Spain

    Pablo Hidalgo-Palencia

  3. Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, E-28040, Madrid, Spain

    Pablo Hidalgo-Palencia

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

    Steve Hofmann

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  1. Luis Escauriaza
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Correspondence to Steve Hofmann.

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Dedicated to Prof. Carlos Kenig on the occasion of his 70th birthday, and to the memory of Luis Escauriaza.

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Escauriaza, L., Hidalgo-Palencia, P. & Hofmann, S. On the Kato Problem for Elliptic Operators in Non-Divergence Form. Vietnam J. Math. (2024). https://doi.org/10.1007/s10013-024-00683-1

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