A robust approach to sharp multiplier theorems for Grushin operators
| Autor: | Gian Maria Dall'Ara, Alessio Martini |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
34L20
35J70 35H20 42B15 General Mathematics 01 natural sciences Power law Multiplier (Fourier analysis) Mathematics - Analysis of PDEs Spectral multiplier Classical Analysis and ODEs (math.CA) FOS: Mathematics 0101 mathematics Eigenvalues and eigenvectors Mathematical physics Mathematics Schrödinger operator Applied Mathematics 010102 general mathematics Sigma Bochner-Riesz mean Eigenfunction 16. Peace & justice Functional Analysis (math.FA) Mathematics - Functional Analysis Grushin operator Mathematics - Classical Analysis and ODEs Analysis of PDEs (math.AP) |
| Popis: | We prove a multiplier theorem of Mihlin-H\"ormander type for operators of the form $-\Delta_x - V(x) \Delta_y$ on $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2}_y$, where $V(x) = \sum_{j=1}^{d_1} V_j(x_j)$, the $V_j$ are perturbations of the power law $t \mapsto |t|^{2\sigma}$, and $\sigma \in (1/2,\infty)$. The result is sharp whenever $d_1 \geq \sigma d_2$. 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 Schr\"odinger operators, which are stable under perturbations of the potential. Comment: 38 pages, accepted for publication in Transactions of the American Mathematical Society |
| Databáze: | OpenAIRE |
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