On sharp heat kernel estimates in the context of Fourier-Dini expansions (Q6632948)

In this contribution, the authors focus the attention on Fourier-Dini expansions on \((0, 1)\) equipped with Lebesgue measure. Let us remind that the Fourier -Dini system is constituted by eigenfunctions of the Bessel differential operator \(- D^2 - \frac{\frac{1}{4}- \nu^2}{x^2}, \nu >-1.\) It is a complete orthonormal system in \(L^{2} ((0.1), dx)\), see [\textit{G. N. Watson}, A treatise on the theory of Bessel functions. 2nd ed. Cambridge: Cambridge University Press (1966; Zbl 0174.36202)].\N\NSharp global estimates of the Fourier-Dini heat kernel viz. the integral kernel of the Fourier-Dini semigroup are given. This kernel is given only implicitly by a heavily oscillating series involving Bessel functions and zeros of some related functions. To prove the short-time bounds, an indirect method having its roots in [\textit{A. Nowak} and \textit{L. Roncal}, Acta Math. Sin., Engl. Ser. 30, No. 3, 437--444 (2014; Zbl 1372.42022)], is used. This method is connected with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel. As an auxiliary result, the applicability of the method within the general framework of Fourier-Bessel/Fourier-Dini expansions is shown.\N\NThe motivation for investigating the Fourier-Dini heat kernel comes from an harmonic analysis related to Fourier-Dini expansions. The sharp result the authors prove has far reaching consequences. Some of them are pointed out as applications. These include sharp estimates for the related Poisson kernels and potential kernels. The last one yields a complete characterization of \(L^{p}\)-\(L^{q}\) mapping properties of Fourier-Dini potential operators. Further consequences pertain to (optimal) \(L^{p}\) mapping properties of the Fourier-Dini semigroup maximal operators hence the a.e. boundary convergence of the semigroup.