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Suppose that $f$ is a holomorphic function on a domain $D$ in $\mathbb{C}^n$, $\partial D$ is smooth, and $f$ is $C^1$ on $\partial D$. Then, the Bochner-Martinelli formula states that $f(z) = \int_{\partial D} f(\zeta) \omega(\zeta, z)$, where $\omega(\zeta, z)$ is the Bochner-Martinelli kernel. I am wondering if a $C^1$ function satisfying this formula is necessarily holomorphic.

This question arose when I was trying to prove that if a sequence of $C^1$ functions $f_n:D\rightarrow\mathbb{C}$ satisfies $f_n\rightarrow f$ uniformly and $\partial\bar{f}_n\rightarrow0$ uniformly, then $f$ is holomorphic. Is this statement correct?

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    $\begingroup$ A try would be to use distribution theory. Your assumptions imply that $f$ is a distribution with the corresponding derivative vanishing and this implies that it is a holomorphic function. $\endgroup$
    – terceira
    Jul 4, 2023 at 17:11

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The result you need is a now classical result of Aronov and Kytmanov [see for example [1] chapter 4 §15.1, p. 161, theorem 15.1]: if $D$ is a bounded domain in $\Bbb C^n$ with piecewise smooth boundary and $f\in C^1(\bar D)$ then $$ f\in \mathscr{O}(D) \iff f(z) = \int_{\partial D} f(\zeta) \omega(\zeta, z) $$ Therefore if you have a sequence of $C^1$ functions $\{f_n\}_{n\in\Bbb N}$ which are representable as Bochner-Martinelli integrals converging uniformly to a function $f$ representable in the same way, both each single function in the sequence and their limit function are holomorphic in $D$.

Note

Aizenberg and Kytmanov proved that the same result is true also for $f\in C^0(\bar{D})$, provided the boundary of $D$ satisfies a higher regularity requirement (namely $\partial D\in C^2$, see [1] chapter 4 §15.2, p. 162, theorem 15.4 for the details).

Reference

[1] Alexander M. Kytmanov (1995) [1992], The Bochner-Martinelli integral and its applications, Birkhäuser Verlag, pp. xii+305, doi:10.1007/978-3-0348-9094-6, ISBN 978-3-7643-5240-0, MR1409816, Zbl 0834.32001.

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    $\begingroup$ Off-topic, but is all of this MathJaxing of old posts and answers really necessary? We have had debates on meta.MO about this before... $\endgroup$
    – Yemon Choi
    Jul 5, 2023 at 14:12
  • $\begingroup$ @YemonChoi thanks, I was not aware of this. I'll stop immediately. $\endgroup$ Jul 5, 2023 at 14:16
  • $\begingroup$ @YemonChoi could you provide some link? I do not visit meta.MO very often... $\endgroup$ Jul 5, 2023 at 14:17
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    $\begingroup$ The most recent seems to be meta.mathoverflow.net/questions/3631/… I have my own personal views on this, which are probably not representative of the majority consensus, but it seems that most people think some occasional editing is acceptable, it is just potentially annoying when done in one large batch in a short period of time. See also meta.mathoverflow.net/questions/591/… $\endgroup$
    – Yemon Choi
    Jul 5, 2023 at 14:21
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    $\begingroup$ @YemonChoi thank you. I naively did some batch edit since I thought it was an encouraged practice in order to update old, non Math Jax posts. I'll read the discussion on meta to gain some information. $\endgroup$ Jul 5, 2023 at 14:28

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