The Bergman projection and weighted 𝐶^{𝑘} estimates for the canonical solution to the ∂̄ problem on non-smooth domains

Ehsani, Dariush

doi:10.1090/S0002-9947-2011-05277-3

The Bergman projection and weighted $C^k$ estimates for the canonical solution to the $\bar {\partial }$ problem on non-smooth domains
HTML articles powered by AMS MathViewer

by Dariush Ehsani

Trans. Amer. Math. Soc. 363 (2011), 3959-3975

DOI: https://doi.org/10.1090/S0002-9947-2011-05277-3

Published electronically: March 10, 2011

PDF | Request permission

Abstract:

We apply integral representations for functions on non-smooth strictly pseudoconvex domains, the Henkin-Leiterer domains, to derive weighted $C^k$-estimates for the component of a given function, $f$, which is orthogonal to holomorphic functions in terms of $C^k$-norms of $\bar {\partial } f$. The weights are powers of the gradient of the defining function of the domain.

References

D. Ehsani. Integral representations on non-smooth domains. Illinois J. Math., 53(4):1127–1156, 2010.
D. Ehsani. Weighted $C^k$ estimates for a class of integral operators on non-smooth domains. Mich. Math. Jornal, 59(3):589–620.
Dariush Ehsani and Ingo Lieb, $L^p$-estimates for the Bergman projection on strictly pseudoconvex non-smooth domains, Math. Nachr. 281 (2008), no. 7, 916–929. MR 2431567, DOI 10.1002/mana.200710649
Gennadi Henkin and Jürgen Leiterer, Theory of functions on complex manifolds, Monographs in Mathematics, vol. 79, Birkhäuser Verlag, Basel, 1984. MR 774049
Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
Ingo Lieb and Joachim Michel, The Cauchy-Riemann complex, Aspects of Mathematics, E34, Friedr. Vieweg & Sohn, Braunschweig, 2002. Integral formulae and Neumann problem. MR 1900133, DOI 10.1007/978-3-322-91608-2
Ingo Lieb and R. Michael Range, Integral representations and estimates in the theory of the $\overline \partial$-Neumann problem, Ann. of Math. (2) 123 (1986), no. 2, 265–301. MR 835763, DOI 10.2307/1971272
R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923, DOI 10.1007/978-1-4757-1918-5
Gerd Schmalz, Solution of the $\overline \partial$-equation with uniform estimates on strictly $q$-convex domains with nonsmooth boundary, Math. Z. 202 (1989), no. 3, 409–430. MR 1017581, DOI 10.1007/BF01159970