A restriction theorem for Grushin operators

Abstract

We study the Grushin operators acting on {ℝ}_x^{d\mathrm{1}}\times {ℝ}_t^{d\mathrm{2}} and defined by the formula L=-{\sum }_{j=1}^{d_{\mathrm{1}}}{\partial }_{x_j}^{\mathrm{2}}-{\sum }_{j=1}^{d_{\mathrm{1}}}{\left|x_j\right|}^2{\sum }_{k=1}^{d_{\mathrm{2}}}{\partial }_{t_k}^{\mathrm{2}}. We establish a restriction theorem associated with the considered operators. Our result is an analogue of the restriction theorem on the Heisenberg group obtained by D. Müller [Ann. of Math., 1990, 131: 567–587].

Keywords

Grushin operator / scaled Hermite operator / restriction operator

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Higher Education Press and Springer-Verlag Berlin Heidelberg