Sharp estimates of the Jacobi heat kernel

Adam Nowak¹ ; Peter Sjögren²

¹Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-956 Warszawa, Poland
²Mathematical Sciences University of Gothenburg and Mathematical Sciences Chalmers University of Technology SE-412 96 Göteborg, Sweden

Studia Mathematica, Tome 218 (2013) no. 3, pp. 219-244

Résumé

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.

Détail

DOI : 10.4064/sm218-3-2

Keywords: heat kernel associated setting classical jacobi polynomials defined oscillatory sum which cannot computed explicitly contrast situation other classical systems orthogonal polynomials deduce sharp estimates giving order magnitude kernel type parameters alpha beta using quite different methods coulhon kerkyacharian petrushev recently obtained estimates application bounds maximal operator multi dimensional jacobi heat semigroup satisfies weak type inequality obtain sharp estimates poisson jacobi kernel