Duality of holomorphic function spaces and smoothing properties of the Bergman projection

Herbig, A.-K.; McNeal, J.; Straube, E.

doi:10.1090/S0002-9947-2013-05827-8

Duality of holomorphic function spaces and smoothing properties of the Bergman projection
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by A.-K. Herbig, J. D. McNeal and E. J. Straube PDF

Trans. Amer. Math. Soc. 366 (2014), 647-665 Request permission

Abstract:

Let $\Omega \Subset \mathbb {C}^{n}$ be a domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_{1}}(\Omega )$ (continuously) into $H^{k_{2}}(\Omega )$. We establish two smoothing results: (i) the full Sobolev norm $\|Bf\|_{k_{2}}$ is controlled by $L^2$ derivatives of $f$ taken along a single, distinguished direction (of order $\leq k_{1}$), and (ii) the projection of a conjugate holomorphic function in $L^{2}(\Omega )$ is automatically in $H^{k_{2}}(\Omega )$. There are obvious corollaries for when $B$ is globally regular.

References

David E. Barrett, Regularity of the Bergman projection and local geometry of domains, Duke Math. J. 53 (1986), no. 2, 333–343. MR 850539, DOI 10.1215/S0012-7094-86-05321-4
David E. Barrett, Duality between $A^\infty$ and $A^{-\infty }$ on domains with nondegenerate corners, Multivariable operator theory (Seattle, WA, 1993) Contemp. Math., vol. 185, Amer. Math. Soc., Providence, RI, 1995, pp. 77–87. MR 1332055, DOI 10.1090/conm/185/02149
Steve Bell and Ewa Ligocka, A simplification and extension of Fefferman’s theorem on biholomorphic mappings, Invent. Math. 57 (1980), no. 3, 283–289. MR 568937, DOI 10.1007/BF01418930
Steven R. Bell, Biholomorphic mappings and the $\bar \partial$-problem, Ann. of Math. (2) 114 (1981), no. 1, 103–113. MR 625347, DOI 10.2307/1971379
Steven R. Bell, A duality theorem for harmonic functions, Michigan Math. J. 29 (1982), no. 1, 123–128. MR 646379
Steven R. Bell, A representation theorem in strictly pseudoconvex domains, Illinois J. Math. 26 (1982), no. 1, 19–26. MR 638551
Steven R. Bell and Harold P. Boas, Regularity of the Bergman projection and duality of holomorphic function spaces, Math. Ann. 267 (1984), no. 4, 473–478. MR 742893, DOI 10.1007/BF01455965
Harold P. Boas, The Szegő projection: Sobolev estimates in regular domains, Trans. Amer. Math. Soc. 300 (1987), no. 1, 109–132. MR 871667, DOI 10.1090/S0002-9947-1987-0871667-7
Harold P. Boas and Emil J. Straube, Global regularity of the $\overline \partial$-Neumann problem: a survey of the $L^2$-Sobolev theory, Several complex variables (Berkeley, CA, 1995–1996) Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 79–111. MR 1748601
So-Chin Chen and Mei-Chi Shaw, Partial differential equations in several complex variables, AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001. MR 1800297, DOI 10.1090/amsip/019
Maklouf Derridj and David S. Tartakoff, On the global real analyticity of solutions to the $\overline \partial$-Neumann problem, Comm. Partial Differential Equations 1 (1976), no. 5, 401–435. MR 413193, DOI 10.1080/03605307608820016
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. MR 944909
A.-K. Herbig and J. D. McNeal, A smoothing property of the Bergman projection, Math. Ann. 354 (2012), no. 2, 427–449. MR 2965249, DOI 10.1007/s00208-011-0734-4
Norberto Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195 (1972), 149–158. MR 294694, DOI 10.1007/BF01419622
Gen Komatsu, Boundedness of the Bergman projector and Bell’s duality theorem, Tohoku Math. J. (2) 36 (1984), no. 3, 453–467. MR 756028, DOI 10.2748/tmj/1178228810
Ewa Ligocka, The Sobolev spaces of harmonic functions, Studia Math. 84 (1986), no. 1, 79–87. MR 871847, DOI 10.4064/sm-84-1-79-87
J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications. Vol. I, Die Grundlehren der mathematischen Wissenschaften, Band 181, Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth. MR 0350177
Ja. A. Roĭtberg, The values on the boundary of the domain of generalized solutions of elliptic equations, Mat. Sb. (N.S.) 86(128) (1971), 248–267 (Russian). MR 0315264
Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
Emil J. Straube, Harmonic and analytic functions admitting a distribution boundary value, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 4, 559–591. MR 808424
Emil J. Straube, Lectures on the $\scr L^2$-Sobolev theory of the $\overline {\partial }$-Neumann problem, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zürich, 2010. MR 2603659, DOI 10.4171/076