Hermitian Clifford Analysis

• Hermitian Clifford analysis is a modern theory that extends classical Clifford analysis by incorporating a Hermitian complex structure to study null solutions called H-monogenic functions.
• It integrates harmonic analysis, representation theory, and complex geometry to develop specialized integral formulas, special functions, and spectral projections in even-dimensional spaces.
• The theory employs advanced tools like Hermitian Dirac operators, Hardy and Szegő spaces, and explicit orthogonal bases, offering practical insights for mathematical physics and operator theory.

Hermitian Clifford analysis is a modern function theory in several complex variables that extends classical Clifford analysis through the introduction of a Hermitian complex structure. It focuses on the study of null solutions—Hermitian monogenic functions—of a pair of conjugate Hermitian Dirac operators acting on spinor-valued or matrix-valued function spaces in even-dimensional Euclidean space. The theory uniquely intertwines harmonic analysis, representation theory, and complex geometry, and provides a comprehensive framework for developing analogues of integral formulas, special functions, and spectral projections in higher-dimensional and operator-valued settings.

1. Algebraic and Analytic Framework

Let $m=2n$ and $\{e_1, ..., e_{2n}\}$ be the orthonormal basis of $\mathbb{R}^{2n}$, generating the real Clifford algebra $\mathbb{R}_{0,2n}$. The essential refinement in Hermitian Clifford analysis is the introduction of a distinguished complex structure $J$: $$J(e_j) = -e_{n+j},\quad J(e_{n+j})=e_j,\quad j=1,...,n.$$ One forms the Witt basis of the complexified Clifford algebra $$\mathbb{C}_{2n} = \mathbb{R}_{0,2n} \otimes \mathbb{C}$$: $$f_j = \frac12(e_j - i e_{n+j}),\qquad f_j^\dagger = -\frac12(e_j + i e_{n+j}),\quad j=1,...,n,$$ with Clifford relations

$$f_jf_k+f_kf_j=0,\,\,\, f_j^\dagger f_k^\dagger + f_k^\dagger f_j^\dagger=0,\,\,\, f_j f_k^\dagger + f_k^\dagger f_j = \delta_{jk}.$$

Complex coordinates are defined by $z_j = x_j + i y_j$, $\bar{z}_j = x_j - i y_j$, and Hermitian Clifford variables as

$$Z = \sum_{j=1}^n f_j z_j,\qquad Z^\dagger = \sum_{j=1}^n f_j^\dagger \bar{z}_j.$$

2. Hermitian Dirac Operators and Monogenic Functions

The Hermitian Clifford refinement splits the standard Dirac operator into two conjugate first-order systems: $$D = \sum_{j=1}^n f_j \frac{\partial}{\partial z_j},\qquad \widetilde{D} = \sum_{j=1}^n f_j^\dagger \frac{\partial}{\partial \bar{z}_j}.$$ A function $F(z, \bar{z})$ (spinor- or matrix-valued) is called Hermitian monogenic ($H$-monogenic) if

$D F = 0,\qquad \widetilde{D} F = 0$

over the domain $\Omega \subset \mathbb{R}^{2n}$. The symmetry group of this system is $U(n)$, the centralizer of $J$ in $SO(2n)$, as both operators commute with the $U(n)$ representation on spinor space. The decomposition of spinor space under $U(n)$ yields

$$S = \bigoplus_{r=0}^n S^r,\qquad \dim S^r = \binom{n}{r},$$

with $D$ and $\widetilde{D}$ shifting $r \mapsto r+1$ and $r \mapsto r-1$, respectively. Hermitian monogenicity is strictly stronger than Euclidean monogenicity, leading to a strictly smaller but more structurally rich function class (Brackx et al., 2019).

3. Hardy and Szegő Spaces, Matrix Hilbert Transform

In domains $\Omega$ with smooth boundary $\partial \Omega$, the space $L^2(\partial \Omega, \mathbb{C}_{2n}^{2 \times 2})$ of square-integrable circulant matrix-valued functions is equipped with the scalar-valued inner product. The Hardy space $H^2(\Omega)$ comprises $L^2$-closure of boundary values of $H$-monogenic functions. Canonical Cauchy kernels,

$$E(Z)=\frac{2}{\omega_{2n}}\,\frac{Z^\dagger}{|Z|^{2n}},\qquad E^\dagger(Z) = \frac{2}{\omega_{2n}}\,\frac{Z}{|Z|^{2n}},$$

define the Hermitian Cauchy transform. The boundary Plemelj–Sokhotski formulas lead to the matrix Hilbert transform $H$, which is bounded, involutive ($H^2=I$), satisfies $H^* = v H v$ for an explicit dipole-flip matrix $v(X)$, and commutes with $D$ and $\widetilde{D}$. The Hardy (Cauchy) projection $P$ is given by

$P = \frac{1}{2}(I + H),$

and projects $L^2$ data onto $H^2(\Omega)$ (Ku et al., 2010).

The Szegő projection $S$ is the orthogonal projection onto $H^2(\Omega)$, with a unique matrix-valued Szegő kernel $S(z, w)$ giving the reproducing property: $$\Phi(w) = \int_{\partial \Omega} S(z, w) \Phi(z) \, dS(z).$$ For the unit ball, $S(z, w)$ coincides with the Cauchy kernel. The Kerzman–Stein formula relates $S$ and $P$: $S(I + A) = P$ with $A = P - P^*$, yielding $S = P - A H$.

4. Orthogonal Bases and Embedding Factors

Spaces $\mathcal{M}_{a,b}^{(r)}(\mathbb{C}^n)$ of homogeneous $H$-monogenic polynomials (bidegree $(a,b)$, spinor degree $r$) correspond to irreducible $U(n)$-modules of highest weight $[a+1,\underbrace{1,\dots,1}_r,0,\dots,0,-b]$. The Fischer inner product

$$(P, Q)_F = \left[\overline{P(\partial_{\underline{z}},\partial_{\underline{z}^\dagger})} Q(\underline{z},\underline{z}^\dagger)\right]_{\underline{z} = \underline{z}^\dagger = 0}$$

is used for orthogonalization (Brackx et al., 2011). The Gel'fand–Tsetlin (GT) construction provides explicit, algorithmic orthogonal bases via Cauchy–Kovalevskaya extension and induction on dimension. Embedding factors between lower-dimensional spaces and higher-dimensional monogenics are given in terms of generalized Jacobi polynomials, yielding an explicit branching decomposition along $U(n-1) \subset U(n)$ (Brackx et al., 2013).

5. Special Functions and the Cauchy-Kovalevskaya Extension

Key families of special functions arise from the Hermitian Dirac system: Hermite polynomials, Bessel functions, and generalized powers. The Cauchy–Kovalevskaya extension theorem provides an explicit isomorphism between initial data on a hyperplane (subject to compatibility conditions) and their unique $H$-monogenic extension. Explicit recurrences and Rodrigues-type formulas yield the structure of Hermite and Bessel systems in the Clifford–Hermitean context, and lead to explicit power series and integral representation for axially symmetric solutions (Schepper et al., 2012, Brackx et al., 2011).

6. Representation-Theoretic Structure and Further Refinements

The $H$-monogenic system is invariant under $U(n)$ but does not admit a natural analogue of the orthogonal Dirac operator with conformal symmetry. The underlying representation theory shows that the natural Clifford module for $U(n)$ is reducible, preventing a Stein–Weiss construction of a single "Hermitian Dirac" operator with larger symmetry (Shirrell et al., 2016). The theory can be further refined: symplectic Clifford analysis introduces a Kähler structure on $\mathbb{R}^{2n}$ and develops metaplectic counterparts of monogenic systems, revealing new Fischer decompositions and Howe dual pairs ($U(n) \times \mathrm{su}(1,2)$) (Eelbode et al., 2023). Extensions to quaternionic Clifford analysis and "osp(4|2)"-monogenicity underscore the versatility and depth of the Hermitian approach (Brackx et al., 2019).

7. Geometric and Operator-Theoretic Results

A fundamental result is the characterization of domains for which the matrix Hilbert transform $H$ is unitary. The following are equivalent: (i) $H$ is unitary; (ii) the Hardy projection is self-adjoint, $P = P^*$; (iii) the Szegő kernel coincides with the Cauchy kernel; (iv) the domain is a ball. The proof combines harmonic analysis (Calderón–Zygmund theory), jump relations, and direct geometric analysis of surface operators (Ku et al., 2010). These results cement the deep connection between operator theory, function theory in several complex variables, and the geometry of domains in Hermitian Clifford analysis.

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References:

(Ku et al., 2010, Brackx et al., 2019, Brackx et al., 2011, Brackx et al., 2011, Shirrell et al., 2016, Schepper et al., 2012, Brackx et al., 2013, Eelbode et al., 2023)