arXiv subjects
Publications and source records attributed to Chiara Alba Taranto.
In this paper, we define and study Gevrey spaces associated with a Hörmander family of (globally defined) vector fields and its corresponding sub-Laplacian. We show some natural relations between the various Gevrey spaces in this setting on general manifolds, and more particular properties on Lie groups with polynomial growth of the volume. In the case of the Heisenberg group and of $SU(2)$, we show that all our descriptions coincide.
The overall goal of this dissertation is to investigate certain classical results from harmonic analysis, replacing the Euclidean setting, the abelian structure and the elliptic Laplace operator with a non-commutative environment and hypoelliptic operators. More specifically, we consider wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent non-negative propagation speeds that are Hölder-regular or even more so. The corresponding Euclidean problem has been extensively studied in the `80s and some additional results have been recently obtained by Garetto and Ruzhansky in the case of a compact Lie group. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In this investigation, a structure reminiscent of Gevrey regularity appears, inspiring deeper investigation of certain classes of functions and a comparison with Gevrey classes. In the latter part of this thesis we discuss such Gevrey spaces associated to the sums of squares of vector fields satisfying the Hörmander condition on manifolds. This provides a deeper understanding of the Gevrey hypoellipticity of sub-Laplacians. It is known that if $\mathcal L$ is a Laplacian on a closed manifold $M$ then the standard Gevrey space $γ^s$ on $M$ defined in local coordinates can be characterised by the condition that $\|e^{D\mathcal L^{\frac{1}{2s}}}ϕ\|_{L^2(M)}<\infty$ for some $D>0$. The aim in this part is to discuss the conjecture that a similar characterisation holds true if $\mathcal L$ is Hörmander's sum of squares of vector fields, with a sub-Laplacian version of the Gevrey spaces involving these vector fields only. We prove this conjecture in one direction, while in the other we show it holds for sub-Laplacians on $SU(2)$ and on the Heisenberg group $\mathbb{H}_n$.