In this memoir we extend the theory of global pseudo-differential operators to the setting of arbitrary sub-Riemannian structures on a compact Lie group. More precisely, given a compact Lie group $G$, and the sub-Laplacian $\mathcal{L}$ associated to a system of vector fields $X=\{X_1,\cdots,X_k\}$ satisfying the Hörmander condition, we introduce a (subelliptic) pseudo-differential calculus associated to $\mathcal{L},$ based on the matrix-valued quantisation process developed in [138]. This theory will be developed as follows. First, we will investigate the singular kernels of this calculus, estimates of $L^p$-$L^p$, $H^1$-$L^1$, $L^\infty$-$BMO$ type and also the weak (1,1) boundedness of these subelliptic Hörmander classes. Between the obtained estimates we prove subelliptic versions of the celebrated sharp Fefferman $L^p$-theorem and the Calderón-Vaillancourt theorem. The obtained estimates will be used to establish the boundedness of subelliptic operators on subelliptic Sobolev and Besov spaces. We will investigate the ellipticity, the construction of parametrices, the heat traces and the regularisation of traces for the developed subelliptic calculus. A subelliptic global functional calculus will be established as well as a subelliptic version of Hulanicki theorem. The approach established in characterising our subelliptic Hörmander classes (by proving that the definition of these classes is independent of certain parameters) will be also applied in order to characterise the global Hörmander classes on arbitrary graded Lie groups developed in [90].
In this work we investigate the well-posedness for difussion equations associated to subelliptic pseudo-differential operators on compact Lie groups. The diffusion by strongly elliptic operators is considered as a special case and in particular the fractional diffusion with respect to the Laplacian. The general case is studied within the Hörmander classes associated to a sub-Riemannian structure on the group (encoded by a Hörmander system of vector fields). Applications to diffusion equations for fractional sub-Laplacians, fractional powers of more general subelliptic operators, and the corres…
We obtain improved bounds for pseudodifferential operators with rough symbols on Hardy spaces for Fourier integral operators. The symbols $a(x,η)$ are elements of $C^{r}_{*}S^{m}_{1,δ}$ classes that have limited regularity in the $x$ variable. We show that the associated pseudodifferential operator $a(x,D)$ maps between Sobolev spaces $\mathcal{H}^{s,p}_{FIO}(\mathbb{R}^{n})$ and $\mathcal{H}^{t,p}_{FIO}(\mathbb{R}^{n})$ over the Hardy space for Fourier integral operators $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$. Our main result is that for all $r>0$, $m=0$ and $δ=1/2$, there exists an interval…
In this work we obtain sharp $L^p$-estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to Hörmander on $\mathbb{R}^n$. The main result extends the classical Fefferman's sharp theorem on the $L^p$-boundedness of pseudo-differential operators for Hörmander classes on $\mathbb{R}^n$ to general graded Lie groups, also adding the borderline $ρ=δ$ case.
