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Given bounded pseudoconvex domains in 2-dimensional complex Euclidean space, we derive analytical and geometric conditions which guarantee the Diederich-Forn\ae ss index is 1. The analytical condition is independent of strongly pseudoconvex points and extends Forn\ae ss--Herbig's theorem in 2007. The geometric condition reveals the index reflects topological properties of boundary. The proof uses an idea including differential equations and geometric analysis to find the optimal defining function. We also give a precise domain of which the Diederich--Forn\ae ss index is 1. The index of this domain can not be verified by formerly known theorems.
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참고문헌 (28)
참고문헌 신청
M. Adachi / 2015 / A global estimate for the Diederich-Fornaess index of weakly pseudoconvex domains / Nagoya Math. J. 220:67~80
D. E. Barrett / 1992 / Behavior of the Bergman projection on the Diederich-Fornaess worm / Acta Math. 168(1-2):1~10
M. Behrens / 1985 / Plurisubharmonic defining functions of weakly pseudoconvex domains in $C^2$ / Math. Ann. 270(2):285~296
B. Berndtsson / 2000 / A Sobolev mapping property of the Bergman kernel / Math. Z. 235(1):1~10
H. P. Boas / 1991 / Sobolev estimates for the ${\bar{\partial}}$ -Neumann operator on domains in $C^n$ admitting a defining function that is plurisubharmonic on the boundary / Math. Z. 206(1):81~88
D. E. Barrett / 1992 / Behavior of the Bergman projection on the Diederich-Fornaess worm / Acta Math. 168(1-2):1~10
M. Behrens / 1985 / Plurisubharmonic defining functions of weakly pseudoconvex domains in
B. Berndtsson / 2000 / A Sobolev mapping property of the Bergman kernel / Math. Z. 235(1):1~10
H. P. Boas / 1991 / Sobolev estimates for the
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