Connection between the harmonic analysis on the sphere and the harmonic analysis on the one-sheeted hyperboloid: an analytic continuation viewpoint

In a previous paper $[$B,V-1$]$, an algebra of holomorphic ``perikernels'' on a complexified hyperboloid $X^{(c)}_{d-1}$ (in $\Bbb C^d)$ has been introduced; each perikernel ${\cal K}$ can be seen as the analytic continuation of a kernel ${\bf K}$ on the unit sphere $\Bbb S^{d-1}$ in an appropriate ``cut-domain'' , while the jump of ${\cal K}$ across the corresponding ``cut'' defines a Volterra kernel $K$ (in the sense of J. Faraut $\lbrack$Fa-1$\rbrack$) on the one-sheeted hyperboloid $X_{d-1}$ (in $\Bbb R^d).$ \par In the present paper, we obtain results of harmonic analysis for classes of perikernels which are invariant under the group ${\rm SO}(d,\Bbb C)$ and of moderate growth at infinity. For each perikernel ${\cal K}$ in such a class, the Fourier-Legendre coefficients of the corresponding kernel ${\bf K}$ on $\Bbb S^{d-1}$ admit a carlsonian analytic interpolation $\tilde F(\lambda)$ in a half-plane, which is the ``spherical Laplace transform''\ of the associated Volterra kernel $K$ on $X_{d-1}.$ Moreover, the composition law ${\cal K }= {\cal K}_1\ast^{( c)}{\cal K}_2$ for perikernels (interpreted in terms of convolutions for the