ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

10. Partial Differential Equations

1991 MS Classification: Ratnakumar, P. K., Indian Statistical Institute, Bangalore, India Wave Equation for the Hermite Operator In this lecture we talk about the Cauchy problem for the wave equation for the Hermite operator H = −Δ + |x|^{2} where Δ = ∑_{j=1}^{n} \frac{∂^{2}}{∂x_{j}^{2}} is the Laplacian on IR^{n}. We show that the solution u(x, t) of the wave equation converges to the L^{p} initial data as t → 0. This is done by expressing the solution in terms of the Weyl transform W (μ_{t}) of the normalised surface measure μ_{t} on the sphere |z| = t inC^{n} and proving the L^{p} boundedness of the associated maximal operator Mf(z) = sup_{t>0}|W (μ_{t})f(z)|.