### Abstract

In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □_{b} -heat equation on quadric submanifolds M ⊂ ℂ^{n} ×^{m} . As a consequence, we can also compute the heat kernel associated with the weighted ∂̄ -equation in ℂ^{n} when the weight is given by exp(-φ(z,z) λ) where φ:ℂ^{n} ×ℂ ^{n} →ℂ^{m} is a quadratic, sesquilinear form and λ ∈ ℝ^{m} . Our method involves the representation theory of the Lie group M and the group Fourier transform.

Original language | English (US) |
---|---|

Pages (from-to) | 256-275 |

Number of pages | 20 |

Journal | Journal of Geometric Analysis |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2011 |

Externally published | Yes |

### Fingerprint

### Keywords

- Fundamental solution
- Heat equation
- Heat kernel
- Heisenberg group
- Kohn Laplacian
- Lie group
- Quadric manifold

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

_{b}-heat equation on quadric manifolds.

*Journal of Geometric Analysis*,

*21*(2), 256-275. https://doi.org/10.1007/s12220-010-9146-z

**The □ _{b}-heat equation on quadric manifolds.** / Boggess, Albert; Raich, Andrew.

Research output: Contribution to journal › Article

_{b}-heat equation on quadric manifolds',

*Journal of Geometric Analysis*, vol. 21, no. 2, pp. 256-275. https://doi.org/10.1007/s12220-010-9146-z

_{b}-heat equation on quadric manifolds. Journal of Geometric Analysis. 2011 Apr;21(2):256-275. https://doi.org/10.1007/s12220-010-9146-z

}

TY - JOUR

T1 - The □b-heat equation on quadric manifolds

AU - Boggess, Albert

AU - Raich, Andrew

PY - 2011/4

Y1 - 2011/4

N2 - In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □b -heat equation on quadric submanifolds M ⊂ ℂn ×m . As a consequence, we can also compute the heat kernel associated with the weighted ∂̄ -equation in ℂn when the weight is given by exp(-φ(z,z) λ) where φ:ℂn ×ℂ n →ℂm is a quadratic, sesquilinear form and λ ∈ ℝm . Our method involves the representation theory of the Lie group M and the group Fourier transform.

AB - In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the □b -heat equation on quadric submanifolds M ⊂ ℂn ×m . As a consequence, we can also compute the heat kernel associated with the weighted ∂̄ -equation in ℂn when the weight is given by exp(-φ(z,z) λ) where φ:ℂn ×ℂ n →ℂm is a quadratic, sesquilinear form and λ ∈ ℝm . Our method involves the representation theory of the Lie group M and the group Fourier transform.

KW - Fundamental solution

KW - Heat equation

KW - Heat kernel

KW - Heisenberg group

KW - Kohn Laplacian

KW - Lie group

KW - Quadric manifold

UR - http://www.scopus.com/inward/record.url?scp=79952185453&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79952185453&partnerID=8YFLogxK

U2 - 10.1007/s12220-010-9146-z

DO - 10.1007/s12220-010-9146-z

M3 - Article

VL - 21

SP - 256

EP - 275

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 2

ER -