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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.

Fundamental solutions of Dirac type operators are introduced for a class of conformally flat manifolds. This class consists of manifolds obtained by factoring out the upper half-space of R n by arithmetic subgroups of generalized modular groups. Basic properties of these fundamental solutions are presented together with associated Eisenstein series.

SUMMARYIn this paper, we study the growth behaviour of entire Cli ord algebra-valued solutions to iterated Dirac and generalized Cauchy-Riemann equations in higher-dimensional Euclidean space. Solutions to this type of systems of partial di erential equations are often called k-monogenic functions or, more generically, polymonogenic functions. In the case dealing with the Dirac operator, the function classes of polyharmonic functions are included as particular subcases. These are important for a number of concrete problems in physics and engineering, such as, for example, in the case of the biharmonic equation for elasticity problems of surfaces and for the description of the stream function in the Stokes ow regime with high viscosity.Furthermore, these equations in turn are closely related to the polywave equation, the poly-heat equation and the poly-Klein-Gordon equation.In the ÿrst part we develop sharp Cauchy-type estimates for polymonogenic functions, for equations in the sense of Dirac as well as Cauchy-Riemann. Then we introduce generalizations of growth orders, of the maximum term and of the central index in this framework, which in turn then enable us to perform a quantitative asymptotic growth analysis of this function class. As concrete applications we develop some generalizations of some of Valiron's inequalities in this paper.

In this paper, we study the solutions to the Schrödinger equation on some conformally flat cylinders and on the n-torus. First, we apply an appropriate regularization procedure. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the regularized parabolic-type Dirac operator. We study their fundamental properties, give representation formulas of all these solutions in terms of multiperiodic generalizations of the elliptic functions in the context of the regularized parabolic-type Dirac operator. Furthermore, we also develop some integral representation formulas. In particular, we set up a Green type integral formula for the solutions to the homogeneous regularized Schrödinger equation on cylinders and n-tori. Then, we treat the inhomogeneous Schrödinger equation with prescribed boundary conditions in Lipschitz domains on these manifolds. We present an L p -decomposition where one of the components is the kernel of the first-order differential operator that factorizes the cylindrical (resp. toroidal) Schrödinger operator. Finally, we study the behavior of our results in the limit case where the regularization parameter tends to zero.