Abstract
In this paper, we first show that the canonical solution operator \(S_1\) to \({\bar{\partial }}\) restricted to (0,1)-forms with holomorphic function coefficients can be expressed by an integral operator using the Dirichlet kernel. Then we prove that operator \(S_k\,(k\ge 1)\) is a Hilbert–Schmidt operator on the Dirichlet space of \({\mathbb {D}}\), but fails to be the Hilbert–Schmidt operator on the Dirichlet space of \({\mathbb {D}}^{2}\). Finally we show that the concomitant operator \(P_{k}\) of \(S_{k}\,(k\ge 2)\) is similar to the direct sum of k copies of the concomitant operator \(P_{1}\) of \(S_{1}\).
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The authors are partially supported by the NNSF of China (Grant No. 11371119). The authors would like to express their sincere appreciation to the referee for his/her valuable suggestions and comments.
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Li, Y., Zhang, Y. On the properties of canonical solution operator to \({\bar{\partial }}\) restricted to Dirichlet space. Complex Anal Synerg 7, 17 (2021). https://doi.org/10.1007/s40627-021-00081-0
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DOI: https://doi.org/10.1007/s40627-021-00081-0