\overline∂_{𝑏}-equations on certain unbounded weakly pseudo-convex domains

Kang, Hyeonbae

doi:10.1090/S0002-9947-1989-0989577-5

$\overline \partial _ b$-equations on certain unbounded weakly pseudo-convex domains
HTML articles powered by AMS MathViewer

by Hyeonbae Kang

Trans. Amer. Math. Soc. 315 (1989), 389-413

DOI: https://doi.org/10.1090/S0002-9947-1989-0989577-5

PDF | Request permission

Abstract:

We found an explicit closed formula for the relative fundamental solution of ${\bar \partial _b}$ on the surface ${H_k} = \{ ({z_1},{z_2}) \in {\mathbb {C}^2}:\operatorname {Im} {z_2} = |{z_1}{|^{2k}}\}$ . We then make estimates of the relative fundamental solution in terms of the nonisotropic metric associated with the surface. The estimates lead us to the regularity results. We also study the problem of finding weights $\omega$ so that ${\bar \partial _b}$ as an operator from $L_\omega ^2$ to ${L^2}$ has a closed range. We find the best possible weight among radial weights.

References

Steve Bloom, Solving weighted norm inequalities using the Rubio de Francia algorithm, Proc. Amer. Math. Soc. 101 (1987), no. 2, 306–312. MR 902547, DOI 10.1090/S0002-9939-1987-0902547-1
Harold P. Boas and Mei-Chi Shaw, Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries, Math. Ann. 274 (1986), no. 2, 221–231. MR 838466, DOI 10.1007/BF01457071
Michael Christ, Regularity properties of the $\overline \partial _b$ equation on weakly pseudoconvex CR manifolds of dimension $3$, J. Amer. Math. Soc. 1 (1988), no. 3, 587–646. MR 928903, DOI 10.1090/S0894-0347-1988-0928903-2
Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948, DOI 10.1007/BFb0058946
Katharine Perkins Diaz, The Szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains, Trans. Amer. Math. Soc. 304 (1987), no. 1, 141–170. MR 906810, DOI 10.1090/S0002-9947-1987-0906810-4
Charles L. Fefferman and Joseph J. Kohn, Hölder estimates on domains of complex dimension two and on three-dimensional CR manifolds, Adv. in Math. 69 (1988), no. 2, 223–303. MR 946264, DOI 10.1016/0001-8708(88)90002-3
G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Annals of Mathematics Studies, No. 75, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. MR 0461588
G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
P. C. Greiner, J. J. Kohn, and E. M. Stein, Necessary and sufficient conditions for solvability of the Lewy equation, Proc. Nat. Acad. Sci. U.S.A. 72 (1975), no. 9, 3287–3289. MR 380142, DOI 10.1073/pnas.72.9.3287
J. J. Kohn, The range of the tangential Cauchy-Riemann operator, Duke Math. J. 53 (1986), no. 2, 525–545. MR 850548, DOI 10.1215/S0012-7094-86-05330-5
Alexander Nagel, Vector fields and nonisotropic metrics, Beijing lectures in harmonic analysis (Beijing, 1984) Ann. of Math. Stud., vol. 112, Princeton Univ. Press, Princeton, NJ, 1986, pp. 241–306. MR 864374
Alexander Nagel, Jean-Pierre Rosay, Elias M. Stein, and Stephen Wainger, Estimates for the Bergman and Szegő kernels in certain weakly pseudoconvex domains, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 55–59. MR 919661, DOI 10.1090/S0273-0979-1988-15598-X
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Boundary behavior of functions holomorphic in domains of finite type, Proc. Nat. Acad. Sci. U.S.A. 78 (1981), no. 11, 6596–6599. MR 634936, DOI 10.1073/pnas.78.11.6596
Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
Jean-Pierre Rosay, Équation de Lewy—résolubilité globale de l’équation $\partial _{b}u=f$ sur la frontière de domaines faiblement pseudo-convexes de $\textbf {C}^{2}$ (ou $\textbf {C}^{n})$, Duke Math. J. 49 (1982), no. 1, 121–128 (French). MR 650372
Linda Preiss Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3-4, 247–320. MR 436223, DOI 10.1007/BF02392419
Mei-Chi Shaw, $L^2$-estimates and existence theorems for the tangential Cauchy-Riemann complex, Invent. Math. 82 (1985), no. 1, 133–150. MR 808113, DOI 10.1007/BF01394783