The aim of this article is to establish two-sided Gaussian bounds for the heat kernels on the unit ball and simplex in $${{\mathbb {R}}}^n$$ R n , and in particular on the interval, generated by classical differential operators whose eigenfunctions are algebraic polynomials. To this end we develop a general method that employs the natural relation of such operators with weighted Laplace operators on suitable subsets of Riemannian manifolds and the existing general results on heat kernels. Our general scheme allows us to consider heat kernels in the weighted cases on the interval, ball, and simplex with parameters in the full range.

中文翻译：

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区间、球和单纯形上加权热核的高斯边界

本文的目的是在 $${{\mathbb {R}}}^n$$ R n 中为单位球和单纯形上的热核建立两侧高斯边界，特别是在生成的区间上通过其特征函数是代数多项式的经典微分算子。为此，我们开发了一种通用方法，该方法在黎曼流形的合适子集上使用此类算子与加权拉普拉斯算子的自然关系，以及现有的关于热核的通用结果。我们的一般方案允许我们在参数在全范围内的区间、球和单纯形的加权情况下考虑热核。