# zbMATH — the first resource for mathematics

Fundamentals of harmonic analysis on domains in complex space. (English) Zbl 0831.31005
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 195-218 (1995).
This is an up-to-date survey of selected topics in harmonic analysis in the complex domain.
After a brief description of the role of singular integrals, Riesz transforms, holomorphic Hardy spaces, maximal functions, BMO, the author gets to the more recent results in several variables that are due mainly to E. Stein, G. Weiss, J. P. d’Angelo, J. McNeal, S. G. Krantz and co- workers. This involves homogeneous type structures on the (smooth) boundary of domains in $$\mathbb{C}^n$$ and brings in the complex geometry (Levi form, stratification of pseudoconvex points, etc.) on the boundary, which determines the shape of the canonical balls there. Here atomic Hardy spaces can be defined and studied. Several recent theorems are stated, but there are no proofs and many essential details are necessarily left out.
For the entire collection see [Zbl 0819.00022].
##### MSC:
 31B20 Boundary value and inverse problems for harmonic functions in higher dimensions 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 31-02 Research exposition (monographs, survey articles) pertaining to potential theory 32A35 $$H^p$$-spaces, Nevanlinna spaces of functions in several complex variables