Spherical means, wave equations, and Hermite-Laguerre expansions

Abstract

In this paper we study the maximal function associated to the Weyl transformW(µr) of the normalised surface measureµron the sphere |z|=rin n. This operator is given by the expansion W(μ_r) ƒ = Σ^∞_k-0 k!(n-1)!/(k+n-1)! phivk (r) P_kƒ where φ_k are Laguerre functions of type (n-1) and P_k are Hermite projection operators. We show that whenp>2n/(2n-1), the maximal operator supr>0 |W(µ_r) f(x)| is bounded on L^p(Rⁿ). Using this we study almost everywhere convergence to initial data of solutions of the wave equation associated to the Hermite operator. The above expansion for W(µ_r) motivates the study of operators of the form S_t^αƒ = Σ_k-0^∞ phiv_k^α (t) P_kƒ, where phiv;^α_k are Laguerre functions of type α. We study various mapping properties of these operators with applications to Hermite expansions and solutions of Darboux type equations.