Revista Matemática Iberoamericana

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Volume 30, Issue 4, 2014, pp. 1265–1280
DOI: 10.4171/RMI/814

Published online: 2014-12-15

A sharp multiplier theorem for Grushin operators in arbitrary dimensions

Alessio Martini[1] and Detlef Müller[2]

(1) University of Birmingham, UK
(2) Christian-Albrechts-Universität zu Kiel, Germany

In a recent work by A. Martini and A. Sikora, sharp $L^p$ spectral multiplier theorems for the Grushin operators acting on $\mathbb R^{d_1}_{x'} \times \mathbb R^{d_2}_{x''}$ and defined by the formula $$L=-\sum_{j=1}^{d_1}\partial_{x'_j}^2 - \Big(\sum_{j=1}^{d_1}|x'_j|^2\Big) \sum_{k=1}^{d_2}\partial_{x''_k}^2$$ are obtained in the case $d_1 \geq d_2$. Here we complete the picture by proving sharp results in the case $d_1 < d_2$. Our approach exploits $L^2$ weighted estimates with "extra weights" depending essentially on the second factor of $\mathbb R^{d_1} \times \mathbb R^{d_2}$ (in contrast to the mentioned work, where the "extra weights" depend only on the first factor) and gives a new unified proof of the sharp results without restrictions on the dimensions.

Keywords: Grushin operator, spectral multiplier, Mihlin–Hörmander multiplier, Bochner–Riesz mean, singular integral operator

Martini Alessio, Müller Detlef: A sharp multiplier theorem for Grushin operators in arbitrary dimensions. Rev. Mat. Iberoam. 30 (2014), 1265-1280. doi: 10.4171/RMI/814