Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces (Q1187737)

The Fourier transformation on a symmetric space of the noncompact type is the composition of the Abel transformation and the Euclidean Fourier transformation. The main result of this paper is a pointwise estimate for the inverse Abel transform of a radial function, and pointwise estimates for the derivatives thereof. This is obtained by combining finite propagation speed ideas with \(L^ 2\) estimates. Using this remarkable result, the author proceeds to solve many open problems in harmonic analysis on noncompact symmetric spaces. These include: (i) finding pointwise estimates for the solution to the heat equation (these are precise enough to establish the weak type (1,1) boundedness of the associated maximal operator, and a striking asymptotic heat propagation result which shows that heat escapes exponentially to infinity); (ii) finding estimates for the complex powers of the resolvent of the Laplace- Beltrami operator, which lead to (nearly) optimal \(L^ p\)-\(L^ q\) boundedness results for these operators; (iii) proving boundedness results for Riesz transforms; (iv) establishing estimates for the Poisson kernels. The results obtained open up many new possibilities for research in the area.