A robust approach to sharp multiplier theorems for Grushin operators

Gian Maria Dall'Ara, Alessio Martini

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Abstract

We prove a multiplier theorem of Mihlin-Hörmander-type for operators of the form -Δx–V(x) Δy on ℝd1x ×ℝd2, where V(x) = ∑d1 j=1 Vj(xj) the Vj are perturbations of the power law t ⟼|t| , and σ∈(1/2, ∞). The result is sharp whenever d1 ≥ σd2. The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenvalues and eigenfunctions of one-dimensional Schrödinger operators, which are stable under perturbations of the potential.
Original languageEnglish
Pages (from-to)7533-7574
Number of pages42
JournalTransactions of the American Mathematical Society
Volume373
Issue number11
Early online date9 Sep 2020
DOIs
Publication statusPublished - Nov 2020

Keywords

  • Bochner-Riesz mean
  • Grushin operator
  • Schrödinger operator
  • Spectral multiplier

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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