In this paper, the properties of the heat diffusion semigroup {e^(t△)}_(t≥0) generated by the Hodge-deRham operator in a Riemannian manifold are discussed.
The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n, C) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of th...The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n, C) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n + 2, C). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a^→∈ C^n, we prove that the space forms an irreducible o(n + 2, C)-module for any c ∈ C if a^→ is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o(n+2, C) on the polynomial algebra in n variables. Moreover, we prove that l forms an infinite-dimensional irreducible weight o(n +2, C)-module with finite-dimensional weight subspaces if c Z/2.展开更多
By using the solution to the Helmholtz equation u-λu = 0(λ≥ 0),the explicit forms of the so-called kernel functions and the higher order kernel functions are given.Then by the generalized Stokes formula,the integra...By using the solution to the Helmholtz equation u-λu = 0(λ≥ 0),the explicit forms of the so-called kernel functions and the higher order kernel functions are given.Then by the generalized Stokes formula,the integral representation formulas related with the Helmholtz operator for functions with values in C(V3,3) are obtained.As application of the integral representations,the maximum modulus theorem for function u which satisfies Hu = 0 is given.展开更多
文摘In this paper, the properties of the heat diffusion semigroup {e^(t△)}_(t≥0) generated by the Hodge-deRham operator in a Riemannian manifold are discussed.
基金supported by National Natural Science Foundation of China(Grant Nos.11171324 and 11321101)
文摘The conformal transformations with respect to the metric defining the orthogonal Lie algebra o(n, C) give rise to a one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebra o(n + 2, C). Letting these operators act on the space of exponential-polynomial functions that depend on a parametric vector a^→∈ C^n, we prove that the space forms an irreducible o(n + 2, C)-module for any c ∈ C if a^→ is not on a certain hypersurface. By partially swapping differential operators and multiplication operators, we obtain more general differential operator representations of o(n+2, C) on the polynomial algebra in n variables. Moreover, we prove that l forms an infinite-dimensional irreducible weight o(n +2, C)-module with finite-dimensional weight subspaces if c Z/2.
基金Project supported by Deutscher Akademischer Austausch Dienst (German Academic Exchange Service)the National Natural Science Foundation of China (No.10471107)
文摘By using the solution to the Helmholtz equation u-λu = 0(λ≥ 0),the explicit forms of the so-called kernel functions and the higher order kernel functions are given.Then by the generalized Stokes formula,the integral representation formulas related with the Helmholtz operator for functions with values in C(V3,3) are obtained.As application of the integral representations,the maximum modulus theorem for function u which satisfies Hu = 0 is given.