Details
| Original language | English |
|---|---|
| Pages (from-to) | 283-337 |
| Number of pages | 55 |
| Journal | Advances in mathematics |
| Volume | 277 |
| Publication status | Published - 28 Mar 2015 |
Abstract
By employing a new reduction procedure we derive explicit expressions for the fundamental solutions of a family Pk,λ of degenerate second order differential operators on RN+ℓ. Here λ is a complex parameter located in the strip |Re(λ)|<N+k-1. As is pointed out in [2] Pk,0 has a geometric background and arises as a Grushin-type operator induced by a sub-Riemannian structure on a k + 1-step nilpotent Lie group. Our method leads to new formulas for the inverse of the Kohn-Laplacian δλ which has been widely studied before in the framework of pseudo-convex domains and CR geometry. As an application we show that in all cases the fundamental solutions have a meromorphic extension in the parameter λ to C\Q. All poles are simple and Q⊂R is an explicitly given discrete set. We recover the invertibility of δ1 modulo the classical Szegö projection. This phenomenon had been observed before in [11].
Keywords
- Bessel function, CR-structure, Fundamental solution, Grushin type operator, Kohn-Laplacian, Modified Bessel function, Nilpotent Lie group, Sub-Laplacian, Szegö kernel
ASJC Scopus subject areas
- Mathematics(all)
- General Mathematics
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In: Advances in mathematics, Vol. 277, 28.03.2015, p. 283-337.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The inverse of a parameter family of degenerate operators and applications to the Kohn-Laplacian
AU - Bauer, Wolfram
AU - Furutani, Kenro
AU - Iwasaki, Chisato
N1 - Publisher Copyright: © 2015 Elsevier Inc. Copyright: Copyright 2015 Elsevier B.V., All rights reserved.
PY - 2015/3/28
Y1 - 2015/3/28
N2 - By employing a new reduction procedure we derive explicit expressions for the fundamental solutions of a family Pk,λ of degenerate second order differential operators on RN+ℓ. Here λ is a complex parameter located in the strip |Re(λ)|k,0 has a geometric background and arises as a Grushin-type operator induced by a sub-Riemannian structure on a k + 1-step nilpotent Lie group. Our method leads to new formulas for the inverse of the Kohn-Laplacian δλ which has been widely studied before in the framework of pseudo-convex domains and CR geometry. As an application we show that in all cases the fundamental solutions have a meromorphic extension in the parameter λ to C\Q. All poles are simple and Q⊂R is an explicitly given discrete set. We recover the invertibility of δ1 modulo the classical Szegö projection. This phenomenon had been observed before in [11].
AB - By employing a new reduction procedure we derive explicit expressions for the fundamental solutions of a family Pk,λ of degenerate second order differential operators on RN+ℓ. Here λ is a complex parameter located in the strip |Re(λ)|k,0 has a geometric background and arises as a Grushin-type operator induced by a sub-Riemannian structure on a k + 1-step nilpotent Lie group. Our method leads to new formulas for the inverse of the Kohn-Laplacian δλ which has been widely studied before in the framework of pseudo-convex domains and CR geometry. As an application we show that in all cases the fundamental solutions have a meromorphic extension in the parameter λ to C\Q. All poles are simple and Q⊂R is an explicitly given discrete set. We recover the invertibility of δ1 modulo the classical Szegö projection. This phenomenon had been observed before in [11].
KW - Bessel function
KW - CR-structure
KW - Fundamental solution
KW - Grushin type operator
KW - Kohn-Laplacian
KW - Modified Bessel function
KW - Nilpotent Lie group
KW - Sub-Laplacian
KW - Szegö kernel
UR - http://www.scopus.com/inward/record.url?scp=84925615861&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2014.12.041
DO - 10.1016/j.aim.2014.12.041
M3 - Article
AN - SCOPUS:84925615861
VL - 277
SP - 283
EP - 337
JO - Advances in mathematics
JF - Advances in mathematics
SN - 0001-8708
ER -