Spherical Means, Wave Equations, and Hermite–Laguerre Expansions

In this paper we study the maximal function associated to the Weyl transformW(μr) of the normalised surface measureμron the sphere |z|=rin Cn. This operator is given by the expansionW(μr) f=∑k=0∞ k!(n−1)!(k+n−1)! phiv;k(r) Pkf,whereϕkare Laguerre functions of type (n−1) andPkare Hermite projection operators. We show that whenp>2n/(2n−1), the maximal operator supr>0 |W(μr) f(x)| is bounded onLp(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 forW(μr) motivates the study of operators of the formSαtf=∑k=0∞ phiv;αk(t) Pkf,wherephiv;αkare Laguerre functions of typeα. We study various mapping properties of these operators with applications to Hermite expansions and solutions of Darboux type equations.