Weighted Plancherel estimates and sharp spectral multipliers for the Grushin operators

Alessio Martini, Adam Sikora

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    13 Citations (Scopus)

    Abstract

    We study the Grushin operators acting on ℝd 1x' × ℝd 2 x″ and defined by the formula L = -Σd1 j=12 x'j- (Σd1 j=1 |x'j|2) Σd2 k=12 x″ k. We obtain weighted Plancherel estimates for the considered operators. As a consequence we prove Lp spectral multiplier results and Bochner-Riesz summability for the Grushin operators. These results are sharp if d1 ≥ d2. We discuss also an interesting phenomenon for weighted Plancherel estimates for d1 < d2. The described spectral multiplier theorem is the analogue of the result for the sublaplacian on the Heisenberg group obtained by Müller and Stein and by Hebisch.

    Original languageEnglish
    Pages (from-to)1075-1088
    Number of pages14
    JournalMathematical Research Letters
    Volume19
    Issue number5
    Publication statusPublished - 2012

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