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In this paper we study Clifford and harmonic analysis on some examples of conformal flat manifolds that have a spinor structure, or more generally, at least a pin structure. The examples treated here are manifolds that can be parametrized by U/Γ where U is a subdomain of either S n or R n and Γ is a Kleinian group acting discontinuously on U . The examples studied here include RP n and the Hopf manifolds S 1 × S n−1 . Also some hyperbolic manifolds will be treated. Special kinds of Clifford-analytic automorphic forms associated to the different choices of Γ are used to construct explicit Cauchy kernels, Cauchy integral formulas, Green's kernels and formulas together with Hardy spaces and Plemelj projection operators for L p spaces of hypersurfaces lying in these manifolds.

Cotangent type functions in R n are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds R n /Z k where 1 ≤ k ≤ n. Basic properties of these kernels are discussed including introducing a Cauchy formula, Green's formula, Cauchy transform, Poisson kernel, Szegö kernel and Bergman kernel for certain types of domains. Singular Cauchy integrals are also introduced as are associated Plemelj projection operators. These in turn are used to study Hardy spaces in this context. Also the analogues of Calderón-Zygmund type operators are introduced in this context, together with singular Clifford holomorphic, or monogenic, kernels defined on sector domains in the context of cylinders. Fundamental differences in the context of the n-torus arising from a double singularity for the generalized Cauchy kernel on the torus are also discussed.

A. is paper deals with some special integral transforms of Bargmann-Fock type in the se ing of quaternionic valued slice hyperholomorphic and Cauchy-Fueter regular functions. e construction is based on the well-known Fueter mapping theorem. In particular, starting with the normalized Hermite functions we can construct an Appell system of quaternionic regular polynomials. e ranges of such integral transforms are quaternionic reproducing kernel Hilbert spaces of regular functions. New integral representations and generating functions in this quaternionic se ing are obtained in both the Fock and Bergman cases.Kamal Diki : Marie Sklodowska-Curie fellow of the Istituto Nazionale di Alta Matematica . 1 20 which is defined making use of the slice hyperholomorphic extension operator, i.e.e next result relates the slice Bergman kernel on the quaternionic half ball to the slice Bergman kernels in the case of the quaternionic unit ball and of the half space.is a right quaternionic reproducing kernel Hilbert space. Moreover, for all (q, r) ∈ B + × B + we have:where K B and K H + are, respectively, the slice Bergman kernels of the quaternionic unit ball and half space.Proof.e first assertion follows from the general theory. en, let us fix r ∈ B + such that r belongs to the slice C J with J ∈ S. en, we consider the function ψ r defined byClearly ψ r belongs to A Slice (B + ) since B + is contained in both B and H + and since by definition K B and K H + are the slice Bergman kernels of the quaternionic unit ball and half space. en, we only need to prove the reproducing kernel property. Indeed, let f ∈ A Slice (B + ). In particular, by the Spli ing Lemma we can write f Jus, by applying the results from the classical complex se ing we getSo, it follows that the function ψ r belongs and reproduces any element of the space A Slice (B + ) for any r ∈ B + . Hence, by the uniqueness of the reproducing kernel we getis completes the proof. e explicit expression of the slice Bergman kernel of the quaternionic half-ball is given by the following eorem 5.3. For all (q, r) ∈ B + × B + , we have: K B + (q, r) = (1 + q 2 ) [(1 − qr) * (q + r)] − * 2 (1 + r 2 ),where the * -product is taken with respect to the variable q. 21

In this paper we analyze the behavior of growth of entire monogenic functions in higher dimensional Euclidean spaces. Generalizations of growth orders, the maximum term and of the central index to Clifford analysis provide the basic tools for our analysis. We obtain generalizations of some Valiron's inequalities for transcendental entire monogenic functions. Further to this an asymptotic relation between the growth of a monogenic function and their iterated radial derivatives is established.