The Cauchy-Szegő projection for domains in \(\mathbb{C}^n\) with minimal smoothness. (English) Zbl 1367.32005

Lanzani, Loredana; Stein, Elias M.

Duke Math. J. 166, No. 1, 125-176 (2017).

When \(D\) is a bounded, strongly pseudoconvex domain in \(\mathbb{C}^{n}\) with smooth boundary of class \(C^{\infty}\), a fundamental work of D. H. Phong and E. M. Stein [Duke Math. J. 44, 695–704 (1977; Zbl 0392.32014)] contains a theorem stating that the Szegő projection (the orthogonal projection from \(L^{2}(bD)\) onto the subspace of boundary values of holomorphic functions) induces a bounded operator on \(L^{p}(bD)\) when \(1<p<\infty\). The authors overcome substantial technical obstacles to extend this result to strongly pseudoconvex domains with minimally smooth boundary of class \(C^{2}\). The \(L^{p}\) boundedness holds also for the weighted Szegő projection when the Lebesgue measure on the boundary is multiplied by a continuous, strictly positive weight function. The authors have discussed elsewhere the \(L^{p}\) regularity on minimally smooth domains for the Bergman projection [Ill. J. Math. 56, No. 1, 127–154 (2012; Zbl 1282.32001)] and for the Cauchy integral [Adv. Math. 264, 776–830 (2014; Zbl 1317.32007)]; see also their survey article [Bull. Math. Sci. 3, No. 2, 241–285 (2013; Zbl 1277.32002)].

Reviewer: Harold P. Boas (College Station)

MSC:

32A25	Integral representations; canonical kernels (Szegő, Bergman, etc.)
32A26	Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
32A50	Harmonic analysis of several complex variables
32A55	Singular integrals of functions in several complex variables
42B20	Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E22	Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B34	Kernel operators

Keywords:

Cauchy integral; T(1) theorem; space of homogeneous type; Leray-Levi measure; Hardy space; pseudoconvex domain; minimal smoothness

Citations:

Zbl 0392.32014; Zbl 1282.32001; Zbl 1317.32007; Zbl 1277.32002