Sharp estimates of the Jacobi heat kernel (Q2859336)

Given the type parameters \(\alpha,\beta>-1\) let \(P^{\alpha,\beta}_n\), \(n=0,1,2,\ldots\), be the classical Jacobi polynomials. The Jacobi heat kernel is then defined by NEWLINE\[NEWLINE G_t^{\alpha,\beta}(x,y)=\sum_{n=0}^\infty \exp\big(-tn(n+\alpha+\beta+1)\big)\frac{P^{\alpha,\beta}_n(x)P^{\alpha,\beta}_n(y)}{h^{\alpha,\beta}_n}, NEWLINE\]NEWLINE where \(t>0\), \(x,y\in[-1,1]\), and \(h^{\alpha,\beta}_n=\int_{-1}^1P^{\alpha,\beta}_n(x)^2(1-x)^\alpha(1+x)^\beta\,dx\) are normalizing constants. The numbers \(n(n+\alpha+\beta+1)\) are the eigenvalues of the Jacobi differential operator.NEWLINENEWLINEThe explicit expressions of heat kernels associated with two other families of classical orthogonal polynomials, the Hermite and Laguerre polynomials, are easily derived by using the Mehler and the Hille-Hardy formulas. An analogue of these formulas in the Jacobi setting is Bailey's formula. However, in contrast with the Hermite and Laguerre cases, Bayley's formula is not helpful in computing a closed formula for the Jacobi heat kernel. This is because the eigenvalues \(n(n+\alpha+\beta+1)\) occuring in the defining series are not linear in \(n\).NEWLINENEWLINEThe main result of the paper provides a qualitatively sharp description of the behavior of \(G_t^{\alpha,\beta}(x,y)\) for \(\alpha,\beta\geq-1/2\); the restriction on \(\alpha\) and \(\beta\) is caused by the used method. Since the multi-dimensional Jacobi heat kernel is a tensor product of one-dimensional kernels, the estimates of \(G_t^{\alpha,\beta}(x,y)\) provide similar bounds also in the multi-dimensional setting. It is worth mentioning that recently analogous estimates of \(G_t^{\alpha,\beta}(x,y)\) have been independently obtained for \(\alpha,\beta>-1\) by \textit{T. Coulhon} et al. [J. Fourier Anal. Appl. 18, No. 5, 995--1066 (2012; Zbl 1270.58015)] by means of Dirichlet spaces and other tools.NEWLINENEWLINEThe proof of the main result is insightful and quite involved and uses several ingredients. These are, among others, the product formula due to \textit{A. Dijksma} and \textit{T. H. Koornwinder} [Indag. Math. 33, 191--196 (1971; Zbl 0209.09302)] and a resulting double integral representation of \(G_t^{\alpha,\beta}(\cos\theta,\cos\varphi)\), \(\theta,\varphi\in[0,\pi]\), a transference of heat kernel estimates from a sphere, a comparison principle relating heat kernels for different type parameters and, finally, a rough estimate of the series defining \(G_t^{\alpha,\beta}(x,y)\).NEWLINENEWLINEAs a natural application of the estimates of \(G_t^{\alpha,\beta}(x,y)\), a weak type \((1,1)\) estimate for the maximal operator associated to the multi-dimensional Jacobi semigroup is proved. Finally, sharp estimates for the Poisson-Jacobi kernel are derived. Although the Poisson-Jacobi kernel is essentially the sum occurring in Bailey's formula, the representation of this sum in terms of Appell's function \(F_4\) is not used. Instead, the already mentioned double integral representation formula is again employed.