Sublinear elliptic problems under radiality. Harmonic NA groups and Euclidean spaces

Let {\mathcal{L}} be the Laplace operator on {\mathbb{R}^{d}} , {d\geq 3} , or the Laplace–Beltrami operator on the harmonic NA group (in particular, on a rank one noncompact symmetric space). For the equation {\mathcal{L}u-\varphi(\,\cdot\,,u)=0} we give necessary and sufficient conditions for the existence of entire bounded or large solutions under the hypothesis of radiality of φ with respect to the first variable. A Harnack-type inequality for positive continuous solutions is also proved.

CONVOLUTION OF ORBITAL MEASURES IN SYMMETRIC SPACES

Let G/K be a noncompact symmetric space, Gc/K its compact dual, 𝔤=𝔨⊕𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian subspace of 𝔭, H be an element of 𝔞, a=exp (H) , and ac =exp (iH) . In this paper, we prove that if for some positive integer r, νrac is absolutely continuous with respect to the Haar measure on Gc, then νra is absolutely continuous with respect to the left Haar measure on G, where νac (respectively νa) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare [‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

PATH INTEGRALS AND PERTURBATIVE EXPANSIONS FOR NONCOMPACT SYMMETRIC SPACES

We show how to construct path integrals for quantum-mechanical systems where the space of configurations is a general noncompact symmetric space. Associated with this path integral is a perturbation theory which respects the global structure of the system. This perturbation expansion is evaluated for a simple example and leads to a new exactly soluble model. This work is a step towards the construction of a strong coupling perturbation theory for quantum gravity.

Zeta functions of Selberg’s type for some noncompact quotients of symmetric spaces of rank one

In a previous paper [5], one of the present authors has worked out a theory of zeta functions of Selberg’s type for compact quotients of symmetric spaces of rank one. In the present paper, we consider the analogues of those results when G/K is a noncompact symmetric space of rank one and Γ is a discrete subgroup of G such that G/Γ is not compact but such that vol(G/Γ)<∞. Thus, Γ is a non-uniform lattice. Certain mild restrictions, which are fulfilled in many arithmetic cases, will be put on Γ, and we shall consider how one can define a zeta function ZΓ of Selberg’s type attached to the data (G, K, Γ).