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Spaces of homogeneous spherical monogenics in dimension 3 can be considered naturally as -modules. As finite-dimensional irreducible -modules, they have canonical bases which are, by construction, orthogonal. In this note, we show that these orthogonal bases form the Appell system and coincide with those constructed recently by S. Bock and K. Gürlebeck in [3]. Moreover, we obtain simple expressions of elements of these bases in terms of the Legendre polynomials.
We present necessary and sufficient conditions for a measure to be a p-Carleson measure, based on the Poisson and Poisson-Szegő kernels of the n-dimensional unit ball.
In this note, based on a natural isomorphism between the spaces of differential forms and Clifford algebra-valued multi-vector functions, the Cauchy type integral for self-conjugate differential forms in ℝⁿ is considered.
Cotangent type functions in Rn are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds Rn/Zk where 1 < = k ≤ M. 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...
In this paper we consider operators acting on a subspace of the space of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace is defined as the orthogonal sum of spaces of specific Clifford basis functions of . Every Clifford endomorphism of can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they...
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