## A robust approach to sharp multiplier theorems for Grushin operators

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- by Gian Maria Dall’Ara and Alessio Martini PDF
- Trans. Amer. Math. Soc.
**373**(2020), 7533-7574 Request permission

## Abstract:

We prove a multiplier theorem of Mihlin–Hörmander-type for operators of the form $-\Delta _x - V(x) \Delta _y$ on $\mathbb {R}^{d_1}_x \times \mathbb {R}^{d_2}_y$, where $V(x) = \sum _{j=1}^{d_1} V_j(x_j)$, the $V_j$ are perturbations of the power law $t \mapsto |t|^{2\sigma }$, and $\sigma \in (1/2,\infty )$. The result is sharp whenever ${d_1} \geq \sigma {d_2}$. The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schrödinger operators, which are stable under perturbations of the potential.## References

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## Additional Information

**Gian Maria Dall’Ara**- Affiliation: Fakultät für Mathematik, Oskar–Morgenstern–Platz 1, 1090 Vienna, Austria
- Address at time of publication: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
- MR Author ID: 1105845
- Email: dallara@altamatematica.it
**Alessio Martini**- Affiliation: School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
- MR Author ID: 903052
- Email: a.martini@bham.ac.uk
- Received by editor(s): February 6, 2018
- Received by editor(s) in revised form: January 22, 2019
- Published electronically: September 9, 2020
- Additional Notes: The first-named author was supported by the FWF-project P28154.

The second-named author was supported in part by the EPSRC Grant “Sub-Elliptic Harmonic Analysis” (EP/P002447/1). Part of this work was developed during a visit of the second-named author to the Mathematisches Seminar of the Christian–Albrechts–Universität zu Kiel (Germany), made possible by the university’s kind hospitality and the financial support of the Alexander von Humboldt Foundation. - © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 7533-7574 - MSC (2010): Primary 34L20, 35J70, 35H20, 42B15
- DOI: https://doi.org/10.1090/tran/7844
- MathSciNet review: 4169667