The inverse of Radon-Penrose transformation \\ and ${\bm k}$-Cauchy-Fueter equations

Abstract

It is known that there is a one-one correspondence between the first cohomology of the sheaf $\mathcal{O}(-k-2)$ over the projective space and the solutions to the $k$-Cauchy-Fueter equations on the quaternionic space $\Bbb H^{n}$ and there exists an explicit Radon-Penrose type integral formula $\mathcal{P}$ realizing this correspondence. In this paper, we find the inverse formula of this transformation $\mathcal{P}$, i.e., given a solution $\widetilde{\phi}$ to the quaternionic $k$-Cauchy-Fueter equations, there exists a $\overline{\partial}$-closed $(0,1)$-form $f$ with coefficients in the $(-k-2)$-th power of the hyperplane section bundle $H^{-k-2}$, such that $\iota^{*}(\mathcal{P}f)=\widetilde{\phi}$, where $\iota$ is an embedding $\Bbb H^{n}\simeq \Bbb R^{4n} \hookrightarrow \Bbb C^{2n\times2}$.