×

zbMATH — the first resource for mathematics

Global analyticity for \(\square_ b\) on three dimensional pseudoconvex CR manifolds. (English) Zbl 0791.35087
The authors study global real analytic regularity for the canonical solution to the equation \(\overline {\partial}_ b u=f\) on a compact, real analytic, three (real) dimensional pseudoconvex CR manifold \(M\). The hypotheses imposed are these:
1) \(u\) is in the range of \(\overline {\partial}_ b^*\), which is always the case if \(\overline {\partial}_ b\) has closed range;
2) there exists a globally defined purely imaginary real analytic vector field \(T\), complementary to \(T^{1,0}+ T^{0,1}\), such that if \(L\) is any smooth section of \(T^{1,0}\) then \([T,L]\) is a section of \(T^{1,0}+ \overline {T^{1,0}}\).
Both of these conditions are known to hold when \(M\) is the boundary of a strongly pseudoconvex domain in space, or when \(M\) is the boundary of a Reinhardt domain.

MSC:
35N15 \(\overline\partial\)-Neumann problems and formal complexes in context of PDEs
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32T99 Pseudoconvex domains
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Boutet de Monvel L., Ann Inst. Fourier 22 pp 229– (1972) · Zbl 0235.47029
[2] Boutet de Monvel L., Ann Inst. Fourier 17 pp 295– (1967) · Zbl 0195.14403
[3] DOI: 10.1007/BF01393998 · Zbl 0621.35067
[4] DOI: 10.1512/iumj.1988.37.37022 · Zbl 0637.32016
[5] Chen S.-C., Pacific Journal of Math. 148 pp 225– (1991)
[6] Christ, M. ”Some nonanalytichypoelliptic sums of squares of vector fields”. Amer: Bull. to appear Math. Soc.
[7] DOI: 10.1155/S1073792891000053 · Zbl 0747.35005
[8] DOI: 10.2307/2946576 · Zbl 0758.35024
[9] Derridj M., Journal of Differential Geometry 13 pp 559– (1978)
[10] DOI: 10.1007/BF01245093 · Zbl 0733.35081
[11] Derridj M., Aspects of Mathematics (1991) · Zbl 0749.32008
[12] DOI: 10.1215/S0012-7094-91-06419-7 · Zbl 0790.35078
[13] DOI: 10.2307/1971120 · Zbl 0378.32014
[14] Geller, D. D., Analytic Pseudodifferential Operators for the Heisenberg Group and Local Solvability (1990) · Zbl 0695.47051
[15] Grigis A., CRAS 293 pp 397– (1981)
[16] Kohn J.J., J. Diff. Geom. 6 pp 523– (1972)
[17] Kohn J.J., Trans. Amer. Math. Soc. 181 pp 272– (1973)
[18] DOI: 10.1007/BF02395058 · Zbl 0395.35069
[19] Kohn J.J., Proc. Symp. in Pure Math. 43 pp 207– (1985)
[20] DOI: 10.1215/S0012-7094-86-05330-5 · Zbl 0609.32015
[21] DOI: 10.2748/tmj/1178240889 · Zbl 0319.35056
[22] DOI: 10.1080/03605308108820170 · Zbl 0455.35040
[23] Sjöstrand J., Astérisque 95 (1982)
[24] Sjöstrand J., Hokkaido Math. J. 12 pp 392– (1983)
[25] DOI: 10.1080/03605307608820013
[26] DOI: 10.1073/pnas.75.7.3027 · Zbl 0384.35020
[27] DOI: 10.1007/BF02414189 · Zbl 0456.35019
[28] Tartakoff D.S., Analytic Solutions of Partial Differential Equations 89 pp 85– (1981) · Zbl 0497.35023
[29] Tartakoff D.S., Conference on Linear Partial and Pseudodifferential Operators pp 251– (1983)
[30] DOI: 10.1080/03605307808820074 · Zbl 0384.35055
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.