Spectral summability for the quartic oscillator with applications to the Engel group

In this article, we investigate spectral properties of the sublaplacian −ΔG on the Engel group, which is the main example of a Carnot group of step 3. We develop a new approach to the Fourier analysis on the Engel group in terms of a frequency set. This enables us to give fine estimates on the convolution kernel satisfying F (−ΔG)u = u * kF , for suitable scalar functions F, and in turn to obtain proofs of classical functional embeddings, via Fourier techniques. This analysis requires a summability property on the spectrum of the quartic oscillator, which we obtain by means of semiclassical techniques and which is of independent interest.