Sharp estimates of the Jacobi heat kernel

Adam Nowak, and Peter Sjögren. "Sharp estimates of the Jacobi heat kernel." Studia Mathematica 218.3 (2013): 219-244. <http://eudml.org/doc/285650>.

@article{AdamNowak2013,
abstract = {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 α, β ≥ -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.},
author = {Adam Nowak, Peter Sjögren},
journal = {Studia Mathematica},
keywords = {Jacobi polynomial; Jacobi expansion; Jacobi heat kernel; Poisson-Jacobi kernel; Jacobi semigroup; maximal operator},
language = {eng},
number = {3},
pages = {219-244},
title = {Sharp estimates of the Jacobi heat kernel},
url = {http://eudml.org/doc/285650},
volume = {218},
year = {2013},
}

TY - JOUR
AU - Adam Nowak
AU - Peter Sjögren
TI - Sharp estimates of the Jacobi heat kernel
JO - Studia Mathematica
PY - 2013
VL - 218
IS - 3
SP - 219
EP - 244
AB - 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 α, β ≥ -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.
LA - eng
KW - Jacobi polynomial; Jacobi expansion; Jacobi heat kernel; Poisson-Jacobi kernel; Jacobi semigroup; maximal operator
UR - http://eudml.org/doc/285650
ER -