JournalsrmiVol. 30, No. 4pp. 1265–1280

A sharp multiplier theorem for Grushin operators in arbitrary dimensions

  • Alessio Martini

    University of Birmingham, UK
  • Detlef Müller

    Christian-Albrechts-Universität zu Kiel, Germany
A sharp multiplier theorem for Grushin operators in arbitrary dimensions cover
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Abstract

In a recent work by A. Martini and A. Sikora, sharp LpL^p spectral multiplier theorems for the Grushin operators acting on Rxd1×Rxd2\mathbb R^{d_1}_{x'} \times \mathbb R^{d_2}_{x''} and defined by the formula

L=j=1d1xj2(j=1d1xj2)k=1d2xk2L=-\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 d1d2d_1 \geq d_2. Here we complete the picture by proving sharp results in the case d1<d2d_1 < d_2. Our approach exploits L2L^2 weighted estimates with "extra weights" depending essentially on the second factor of Rd1×Rd2\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.

Cite this article

Alessio Martini, Detlef Müller, A sharp multiplier theorem for Grushin operators in arbitrary dimensions. Rev. Mat. Iberoam. 30 (2014), no. 4, pp. 1265–1280

DOI 10.4171/RMI/814