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Let $\Omega\subset\mathbb{C}^n$ be any smooth bounded pseudoconvex domain. Let $S$ denote the Szegő kernel of $\Omega$.

Recall: the Szegő kernel is a kernel of the Szegő projection $P: L^{2}(\partial\Omega)\to H^{2}(\partial\Omega)$ where $H^{2}(\partial\Omega)$ denotes a holomorphic Hardy space contained in the harmonic Hardy Space that contains holomorphic functions on $\Omega$.

What can we say for $w\in\Omega$, about the extension of $S(\cdot,w)$ to the $\partial\Omega$ from $\Omega$. Does it extend as a continuous function to $\partial\Omega$ from $\Omega$? It is very difficult to find material about the Szegő kernel for even smooth bounded pseudoconvex domain. Please help me in regards if you know some references.

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  • $\begingroup$ if this is not true always then there should exist a smooth bounded pseudoconvex domain for which $S(.,w)$ does not extend smoothly to the $\partial\Omega$. Can you give me such example? $\endgroup$
    – Naruto
    Commented Jun 17, 2022 at 7:36

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Disclaimer: more a long comment than an answer. I can only say the Szegő kernel $S$ exists even if $\Omega$ is "only" a piecewise smooth bounded domain (no need for pseudoconvexity), as an old result by Lutz Bungart ([2] §5 p. 1157-1158, theorem 5.1) also found in Aizenberg and Yuzhakov's monograph ([1] chapter 1, §6 p.48, theorem 6.7) (this reference is possibly the most complete reference on Szegő kernel I'm aware of): the result is precisely the following

Theorem. Let $f\in A_C(\Omega)$ where $\Omega$ is a piecewise smooth bounded domain in $\Bbb C^n$, $n>1$. Then there exist an integral representation $$ f(w) =\int_{\partial \Omega} f(\bar\zeta) S(\bar\zeta, w) \operatorname{d\!}\sigma_\zeta $$ whose kernel $S(\bar\zeta, w)\in \mathscr{H}\big(\{(\bar\zeta, w): w\in\Omega,\;\zeta\in\Omega\}\big)$ is in $H^2(\operatorname{d\!}\sigma_\zeta)\equiv H^2(\partial\Omega)$ (here we have used $\mathscr{H} (D)$ tho denote the vector space of holomorphic function on an open domain $D$).

Thus, in general, the only thing you can expect is that $S(\cdot,w)$ has a radial limit a.e. on $\partial\Omega$, and not that it extends continuously from $\Omega$ to its boundary: for example, a(n almost) classical result ([3] chapter II, §10 pp. 38-40 theorem 10) says that this happens if $\Omega\in C^2$.

References

[1] L. A. Aizenberg and A. P. 3, Integral representations and residues in multidimensional complex analysis, translated from the Russian by H. H. , ed. by Lev J. Leifman, (English), Translations of Mathematical Monographs, 58. Providence, R.I.: American Mathematical Society (AMS), pp. X+283 (1983), ISBN: 0-8218-4511-X, MR0735793, Zbl 0537.32002.

[2] Lutz Bungart, "Boundary kernel functions for domains on complex manifolds" (English), Pacific Journal of Mathematics 14, 1151-1164 (1964), MR0174783, Zbl 0144.08001.

[3] Elias M. Stein, Boundary behavior of holomorphic functions of several complex variables (English), Mathematical Notes, Princeton, N. J.: Princeton University Press, pp. IX+72 (1972), MR0473215, Zbl 0242.32005.

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    $\begingroup$ There is an article Regularity of the Szego Projection in Weakly Pseudoconvex Domains of Harold P. Boas in which Theorem 1(c) implies $S(.,w)\in C^{\infty}({\overline{\Omega}})$ for $w\in\Omega$ but I don't know converse holds or not. $\endgroup$
    – Naruto
    Commented Jun 15, 2022 at 7:18
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    $\begingroup$ In general for a $C^{\infty}$ bounded domain Ω, Is $H^2(\partial\Omega)$ is the $L^2(\partial\Omega)$-closure of the restrictions of $A^{\infty}(\Omega)$ functions to ∂Ω? I have seen this definition in a book of Several complex variables of Steven G Krantz but in an article Boundary behaviour of holomorphic functions of Several complex variables by Stein , pg no.-19 the author said in general it is not true $\endgroup$
    – Naruto
    Commented Jun 15, 2022 at 10:12
  • $\begingroup$ Can you please provide an example of a $C^{\infty}$ bounded domain $\Omega\subset\mathbb{C}^n$ for which $z \to S(z.,w)\;\;for\;w\in\Omega$ do not extends as a continous function to the $\partial\Omega.$ $\endgroup$
    – Naruto
    Commented Jun 24, 2022 at 13:48

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