English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $\mathscr{L}$ be a smooth second-order real differential operator indivergence form on a manifold of dimension $n$. Under a bracket-generatingcondition, we show that the ranges of validity of spectral multiplier estimatesof Mihlin--H\"ormander type and wave propagator estimates of Miyachi--Peraltype for $\mathscr{L}$ cannot be wider than the corresponding ranges for theLaplace operator on $\mathbb{R}^n$. The result applies to all sub-Laplacians onCarnot groups and more general sub-Riemannian manifolds, without restrictionson the step. The proof hinges on a Fourier integral representation for the wavepropagator associated with $\mathscr{L}$ and nondegeneracy properties of thesub-Riemannian geodesic flow.

Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type