# Extension of a Szegő Kernel to the boundary

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.

• 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? Commented Jun 17, 2022 at 7:36

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.

• 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. Commented Jun 15, 2022 at 7:18
• 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 Commented Jun 15, 2022 at 10:12
• 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 $\partial\Omega.$ Commented Jun 24, 2022 at 13:48