Sharp estimates of the Jacobi heat kernel

Details Sharp estimates of the Jacobi heat kernel Sharp estimates of the Jacobi heat kernel

2011-11-14

arxiv.org/abs/1111.3145v1

Mathematics

Classical Analysis and ODEs

21 pages

Scientific paper

The heat kernel associated with the setting of the classical Jacobi polynomials is defined by an oscillatory sum which cannot be computed explicitly, in contrast to the situation for the two other classical systems of orthogonal polynomials. We deduce sharp estimates giving the order of magnitude of this kernel, for type parameters $\alpha,\beta \ge -1/2$. As an application of the upper bound obtained, we show that the maximal operator of the multi-dimensional Jacobi heat semigroup satisfies a weak type $(1,1)$ inequality. We also obtain sharp estimates of the Poisson-Jacobi kernel.

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