A robust approach to sharp multiplier theorems for Grushin operators

Abstract

We prove a multiplier theorem of Mihlin–Hörmander-type for operators of the form − Δ x − V ( x ) Δ y -\Delta _x - V(x) \Delta _y on R x d 1 × R y d 2 \mathbb {R}^{d_1}_x \times \mathbb {R}^{d_2}_y , where V ( x ) = ∑ j = 1 d 1 V j ( x j ) V(x) = \sum _{j=1}^{d_1} V_j(x_j) , the V j V_j are perturbations of the power law t ↦ | t | 2 σ t \mapsto |t|^{2\sigma } , and σ ∈ ( 1 / 2 , ∞ ) \sigma \in (1/2,\infty ) . The result is sharp whenever d 1 ≥ σ d 2 {d_1} \geq \sigma {d_2} . 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.