Abstract
In a previous work we proved a spectral multiplier theorem of Mihlin–Hörmander type for two-dimensional Grushin operators \(-\partial _x^2 - V(x) \partial _y^2\), where V is a doubling single-well potential, yielding the surprising result that the optimal smoothness requirement on the multiplier is independent of V. Here we refine this result, by replacing the \(L^\infty \)-Sobolev condition on the multiplier with a sharper \(L^2\)-Sobolev condition. As a consequence, we obtain the sharp range of \(L^1\)-boundedness for the associated Bochner–Riesz means. The key new ingredient of the proof is a precise pointwise estimate in the transition region for eigenfunctions of one-dimensional Schrödinger operators with doubling single-well potentials.
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25 April 2022
A Correction to this paper has been published: https://doi.org/10.1007/s00041-022-09937-3
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Communicated by Krzysztof Stempak.
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The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)
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Dall’Ara, G.M., Martini, A. An Optimal Multiplier Theorem for Grushin Operators in the Plane, II. J Fourier Anal Appl 28, 32 (2022). https://doi.org/10.1007/s00041-022-09931-9
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DOI: https://doi.org/10.1007/s00041-022-09931-9