Hodge decomposition for generalized Vekua spaces in higher dimensions
Briceyda B. Delgado
TL;DR
This work extends Hodge-type decompositions to $L^2$-solutions of the Vekua equation in higher dimensions within the Clifford algebra setting. By introducing the generalized Vekua spaces $A_{α,β}^p(Ω)$ and the transmutation operator $S_{α,β}$, the authors establish a Hilbert-space decomposition of $L^2(Ω,C\ell_{0,n})$ into $A_{α,β}^2(Ω,C\ell_{0,n})$ and its orthogonal complement, and derive a factorization linking the Vekua operator to Schrödinger-type operators. The Vekua projection $P_{α,β}$ is expressed via the Bergman projection and the isomorphism $S_{α,β}$, and componentwise reproductive Vekua kernels $K_{α,β}^A$ are constructed to obtain explicit kernel representations of the projection. Under a smallness condition on $α$ and $β$, reproducing kernels exist componentwise, enabling explicit integral representations that connect generalized monogenic functions with boundary-value and inverse problems in higher dimensions $Ω$.
Abstract
We introduce the spaces $A^p_{α, β}(Ω)$ of $L^p$-solutions to the Vekua equation (generalized monogenic functions) $D w=α\overline{w}+βw$ in a bounded domain in $\mathbb{R}^n$, where $D=\sum_{i=1}^n e_i \partial_i$ is the Moisil-Teodorescu operator, $α$ and $β$ are bounded functions on $Ω$. The main result of this work consists of a Hodge decomposition of the $L^2$ solutions of the Vekua equation, from this orthogonal decomposition arises an operator associated with the Vekua operator, which in turn factorizes certain Schrödinger operators. Moreover, we provide an explicit expression of the ortho-projection over $A^p_{α, β}(Ω)$ in terms of the well-known ortho-projection of $L^2$ monogenic functions and an isomorphism operator. Finally, we prove the existence of component-wise reproductive Vekua kernels and the interrelationship with the Vekua projection in Bergman's sense.