Sharp estimates of the Jacobi heat kernel
Tom 218 / 2013
Studia Mathematica 218 (2013), 219-244 MSC: Primary 42C05; Secondary 35K08. DOI: 10.4064/sm218-3-2
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 other two classical systems of orthogonal polynomials. We deduce sharp estimates giving the order of magnitude of this kernel, for type parameters $\alpha ,\beta \ge -1/2$. Using quite different methods, Coulhon, Kerkyacharian and Petrushev recently also obtained such estimates. As an application of the bounds, 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.