Let G be a semisimple Lie group with Iwasawa decomposition G = KAN. Let X = G/K be the associated symmetric space and assume that X is of rank one. Let M be the centraliser of A in K and consider an orthonormal basis $Y_{\delta ,j}:\delta \in K̂₀,1\le j\le d_{\delta }$ of L²(K/M) consisting of K-finite functions of type δ on K/M. For a function f on X let f̃(λ,b), λ ∈ ℂ, be the Helgason Fourier transform. Let $h_t$ be the heat kernel associated to the Laplace-Beltrami operator and let $Q_{\delta }(i\lambda +\varrho )$ be the Kostant polynomials. We establish the following version...

We prove some Hardy-type inequalities related to quasilinear second-order degenerate elliptic differential operators $L_{p}u:=-{\nabla }_L^{*}\left({\left|{\nabla }_{L}u\right|}^{p-2}{\nabla }_{L}u\right)$. If $\phi$ is a positive weight such that $-L_p\phi \ge 0$, then the Hardy-type inequalityholds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups.

We develop the L² harmonic analysis for (Dirac) spinors on the real hyperbolic space Hⁿ(ℝ) and give the analogue of the classical notions and results known for functions and differential forms: we investigate the Poisson transform, spherical function theory, spherical Fourier transform and Fourier transform. Very explicit expressions and statements are obtained by reduction to Jacobi analysis on L²(ℝ). As applications, we describe the exact spectrum of the Dirac operator, study the Abel transform...

Let G be a Lie group. The main new result of this paper is an estimate in L2 (G) for the Davies perturbation of the semigroup generated by a centered sublaplacian H on G. When G is amenable, such estimates hold only for sublaplacians which are centered. Our semigroup estimate enables us to give new proofs of Gaussian heat kernel estimates established by Varopoulos on amenable Lie groups and by Alexopoulos on Lie groups of polynomial growth.

Let G be a Lie group of polynomial volume growth. Consider a differential operator H of order 2m on G which is a sum of even powers of a generating list $A₁,...,A_{d^{\text{'}}}$ of right invariant vector fields. When G is solvable, we obtain an algebraic condition on the list $A₁,...,A_{d^{\text{'}}}$ which is sufficient to ensure that the semigroup kernel of H satisfies global Gaussian estimates for all times. For G not necessarily solvable, we state an analytic condition on the list which is necessary and sufficient for global Gaussian estimates....

Let G be a Lie group of polynomial volume growth, with Lie algebra g. Consider a second-order, right-invariant, subelliptic differential operator H on G, and the associated semigroup St = e-tH. We identify an ideal n' of g such that H satisfies global regularity estimates for spatial derivatives of all orders, when the derivatives are taken in the direction of n'. The regularity is expressed as L2 estimates for derivatives of the semigroup, and as Gaussian bounds for derivatives of the heat kernel....

MSC 2010: 30C60A more general version of Hilbert's fifth problem, called the Hilbert-Smith conjecture, asserts that among all locally compact topological groups only Lie groups can act effectively on finite-dimensional manifolds. We give a solution of the Hilbert-Smith Conjecture for K - quasiconformal groups acting on domains in the extended n - dimensional Euclidean space.

On a real hypersurface $M$ in ${ℂ}^{n+1}$ of class $C^{2,\alpha }$ we consider a local CR structure by choosing $n$ complex vector fields $W_j$ in the complex tangent space. Their real and imaginary parts span a $2n$-dimensional subspace of the real tangent space, which has dimension $2n+1.$ If the Levi matrix of $M$ is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations...

We prove that, for a distinguished laplacian on an Iwasawa AN group corresponding to a complex semisimple Lie group, a Hörmander type multiplier theorem holds. Our argument is based on Littlewood-Paley theory.

Let L be the full laplacian on the Heisenberg group ${ℍ}^n$ of arbitrary dimension n. Then for $f\in L^2\left({ℍ}^n\right)$ such that ${(I-L)}^{s/2}f\in L^2\left({ℍ}^n\right)$, s > 3/4, for a $\varphi \in C_c\left({ℍ}^n\right)$ we have ${ʃ}_{{ℍ}^n}\left|\varphi \left(x\right)\right|su{p}_{0<t\le 1}{\left|e^{(\surd -1)tL}f\left(x\right)\right|}^{2}dx\le C_{\varphi }\parallel f{\parallel }_{W^s}^2$. On the other hand, the above maximal estimate fails for s < 1/4. If Δ is the sublaplacian on the Heisenberg group ${ℍ}^n$, then for every s < 1 there exists a sequence $f_n\in L^2\left({ℍ}^n\right)$ and $C_n>0$ such that ${(I-L)}^{s/2}{f}_n\in L^2\left({ℍ}^n\right)$ and for a $\varphi \in C_c\left({ℍ}^n\right)$ we have ${ʃ}_{{ℍ}^n}\left|\varphi \left(x\right)\right|su{p}_{0<t\le 1}{\left|e^{(\surd -1)t\Delta }{f}_n\left(x\right)\right|}^{2}dx\ge C_n\parallel f_n{\parallel }_{W^s}^2,li{m}_{n\to \infty }{C}_n=+\infty$.