On the boundary complex of the k-Cauchy–Fueter complex

Abstract

The k-Cauchy–Fueter complex, $k=0,1,\ldots$, in quaternionic analysis are the counterpart of the Dolbeault complex in the theory of several complex variables. In this paper, we construct explicitly boundary complexes of these complexes on boundaries of domains, corresponding to the tangential Cauchy–Riemann complex in complex analysis. They are only known boundary complexes outside of complex analysis that have interesting applications to the function theory. As an application, we establish the Hartogs–Bochner extension for k-regular functions, the quaternionic counterpart of holomorphic functions. These boundary complexes have a very simple form on a kind of quadratic hypersurfaces, which have the structure of right-type nilpotent Lie groups of step two. They allow us to introduce the quaternionic Monge–Ampère operator and open the door to investigate pluripotential theory on such groups. We also apply abstract duality theorem to boundary complexes to obtain the generalization of Malgrange’s vanishing theorem and the Hartogs–Bochner extension for k-CF functions, the quaternionic counterpart of CR functions, on this kind of groups.

8. Appendix

The $\sigma$-th symmetric power $\odot ^{\sigma }{\mathbb {C}}^{2 }$ is a subspace of $\otimes ^{\sigma }{\mathbb {C}}^{2 }$, and an element of $\odot ^{\sigma } {\mathbb {C}}^2$ is given by a $2^\sigma$-tuple $( f_{\textbf{A} ' })\in \otimes ^{\sigma } {\mathbb {C}}^2$ with $\textbf{A} '=A_1' \ldots A_\sigma '$ ($A_1',\ldots , A_\sigma ' =0',1'$) such that $f_{\textbf{A} '}$ is invariant under permutations of subscripts, i.e.

$$\begin{aligned} f_{A_1' \ldots A_\sigma '} = f_{ A_{\pi (1)}'\ldots A_{\pi (\sigma )} '} \end{aligned}$$

for any $\pi \in {S}_\sigma$, the group of permutations on $\sigma$ letters. The symmetrization of indices is defined as

$$\begin{aligned} f_{\cdots (A_1'\ldots A_\sigma ')\cdots }: =\frac{1}{\sigma !}\sum _{\pi \in {S}_\sigma } f_{\cdots A_{\pi (1)}'\ldots A_{\pi (\sigma )}' \cdots }. \end{aligned}$$

(8.1)

In particular, if $(f_{A_1'\ldots A_\sigma '})\in \otimes ^\sigma {\mathbb {C}}^{2 }$ is symmetric in $A_2'\ldots A_\sigma '$, then we have

$$\begin{aligned} f_{(A_1'\ldots A_\sigma ')}=\frac{1}{\sigma }\left( f_{A_1'A_2'\ldots A_\sigma '}+\cdots + f_{A_s'A_1'\ldots \widehat{A_s'}\ldots A_\sigma '}+\cdots + f_{A_\sigma 'A_1'\ldots A_{\sigma -1}' } \right) . \end{aligned}$$

(8.2)

For $j=0,1,\cdots ,k-1$, we write an element of $\Gamma ({\mathbb {H}}^{n+1}, \mathcal {{V}}_j)$ as a tuple $f=(f_{\textbf{A}'})$ with symmetric indices $\textbf{A}'=A_1'\cdots A_{ \sigma _j }'$ and $f_{ \textbf{A}'}\in \Gamma ({\mathbb {H}}^{n+1},\wedge ^{j}{\mathbb {C}}^{2(n+1)})$. $\mathcal { {D}}_j$ is a differential operators of first order given by

$$\begin{aligned} \left( \mathcal {{D}}_jf\right) _{\textbf{A}'}= \sum _{A '=0',1'}d^{A '}f_{ A '\textbf{A}'}; \end{aligned}$$

(8.3)

while for $j=k+1\cdots ,2n+1$, we write an element of $\Gamma ({\mathbb {H}}^{n+1}, \mathcal {{V}}_j)$ as a tuple $f=(f^{ \textbf{A}'})$ with symmetric indices $\textbf{A}'$ and $f^{ \textbf{A}'}\in \Gamma ({\mathbb {H}}^{n+1},\wedge ^{j+1}{\mathbb {C}}^{2(n+1)})$. Then,

$$\begin{aligned} \left( \mathcal { {D}}_{j}f\right) ^{A '\textbf{A}'}= d^{(A '}f^{\textbf{A}')} , \end{aligned}$$

(8.4)

where $(\cdots )$ is the symmetrization of indices.

Define isomorphisms

$$\begin{aligned} \dot{{\varvec{\Pi }}}_j: \Gamma ({\mathbb {H}}^{n+1}, \mathcal {{V}}_j) \longrightarrow \Gamma \left( {\mathbb {H}}^{n+1}, {\mathcal {P}}_{\sigma _j } ({\mathbb {C}}^2) \otimes \wedge ^{\tau _j}{\mathbb {C}}^{2n +2} \right) . \end{aligned}$$

(8.5)

For $j=0,1,\cdots ,k$, the isomorphism $\dot{{\varvec{\Pi }}}_j$ is given by

$$\begin{aligned} \dot{{\varvec{\Pi }}}_j\left( f_{ \textbf{A} ' }\right) = f_{ a } \textbf{S}^{a}_{\sigma _j } , \end{aligned}$$

(8.6)

where $f_{ a }:=f_{ \textbf{A} ' }$ with $o( \textbf{A} ')=a$, and the summation is taken over $a=0,1,\ldots ,\sigma _{j }$. If $j=k+1,\cdots ,2n+1$, the isomorphism $\dot{{\varvec{\Pi }}}_j$ is given by

$$\begin{aligned} \dot{{\varvec{\Pi }}}_j\left( f ^{\textbf{A} ' }\right) = f^{ \textbf{A} ' }s_{\textbf{A}'}=f ^{a} \widetilde{ \textbf{S}}^{a}_{\sigma _j }\left( {\begin{array}{c}\sigma _j \\ a\end{array}}\right) \end{aligned}$$

(8.7)

where $f ^{a } =f^{ \textbf{A} ' }$ with $o( \textbf{A} ')=a$. Under this realization, operators ${\mathcal {D}}_j$’s in (1.6) in the k-Cauchy–Fueter complex is the same as (8.6)-(8.7) by the following proposition.

Proposition 8.1

For $f\in \Gamma ({\mathbb {H}}^{n+1}, \mathcal {{V}}_j )$, we have

$$\begin{aligned} \dot{{\varvec{\Pi }}}_{j+1}( {\mathcal {D}}_jf) = \left\{ \begin{array}{ll} \partial _{A'} d^{ {A} '}\dot{{\varvec{\Pi }}}_j( f),\quad &{}\textrm{if}\quad j=0, \ldots , k-1,\\ s_{A'} d^{ {A} '}\dot{{\varvec{\Pi }}}_j( f) ,\quad &{}\textrm{if}\quad j=k+1,\ldots ,2n+1. \end{array}\right. \end{aligned}$$

Proof

Note that for $j=0, \ldots , k-1$,

$$\begin{aligned} \begin{aligned} \partial _{A'} d^{A'}\left( f_{ a } \textbf{S}^{a}_{\sigma _j } \right)&= d^{0'} f_{ a } \textbf{S}^{a}_{\sigma _{j+1 } }+ d^{1'} f_{ a } \textbf{S}^{a-1}_{\sigma _{j+1 } } = \left( d^{0'} f_{ b }+d^{1'} f_{b+1}\right) \textbf{S}^{b}_{\sigma _{j+1 }} , \end{aligned} \end{aligned}$$

by (2.5), where b is taken over $0,1,\ldots ,\sigma _{j+1}=\sigma _{j }-1$. Apply the mapping $\dot{{\varvec{\Pi }}}_{j+1}^{-1}$ to get

$$\begin{aligned} \begin{aligned} \left[ \dot{{\varvec{\Pi }}}^{-1}_{j+1}\left( \partial _{A'} d^{ {A} '}\dot{{\varvec{\Pi }}}_j( f)\right) \right] _{ \textbf{A} ' }&= d^{0'} f_{0' \textbf{A} ' }+d^{1'} f_{1' \textbf{A} ' } . \end{aligned} \end{aligned}$$

For $j=k+1,\cdots ,2n+1$, noting that $\sigma _{j+1 }=\sigma _{j }+1$, we get

$$\begin{aligned} \begin{aligned} s_{A'} d^{A'}\left( f ^{ a } \widetilde{ \textbf{S}}^{a}_{\sigma _j } \left( {\begin{array}{c}\sigma _j \\ a\end{array}}\right) \right)&= \left( d^{0'} f ^{ a } \widetilde{ \textbf{S}}^{a}_{\sigma _j+1 }+ d^{1'} f ^{ a } \widetilde{ \textbf{S}}^{a+1}_{\sigma _j+1 }\right) \left( {\begin{array}{c}\sigma _j \\ a\end{array}}\right) \\&=\left( \frac{ \sigma _{j+1 }-(a+1)}{\sigma _{j+1 }}d^{0'} f ^{ a+1 } + \frac{a+1}{\sigma _{j+1 }}d^{1'} f ^{ a} \right) \widetilde{ \textbf{S}}^{a+1}_{\sigma _{j+1 } } \left( {\begin{array}{c} {\sigma _{j+1 } }\\ a+1\end{array}}\right) . \end{aligned} \end{aligned}$$

Thus, for $A'\textbf{A} '$ with $|A'\textbf{A} '|=\sigma _{j+1 }$ and $o(A'\textbf{A} ')=a+1$, we have

$$\begin{aligned} \begin{aligned} \left[ \dot{{\varvec{\Pi }}}^{-1}_{j+1}\left( s_{A'} d^{ {A} '} \dot{{\varvec{\Pi }}}_j( f)\right) \right] ^{ A'\textbf{A} ' }&=\frac{ \sigma _{j+1 }-(a+1)}{\sigma _{j+1 }} d^{0'}f^{\scriptstyle 0'\ldots 0'\overbrace{\scriptstyle 1' \ldots 1'}^{a+1} }+ \frac{a+1}{\sigma _{j+1 }} d^{1'} f^{\scriptstyle 0'\ldots 0'\overbrace{\scriptstyle 1' \ldots 1' }^{a } } \\ {}&= d^{(A'}f^{ \textbf{A} ') } \end{aligned} \end{aligned}$$

by (8.2). $\square$