Spherical Means, Wave Equations, and Hermite–Laguerre Expansions

✍ Scribed by P.K Ratnakumar; S Thangavelu

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 419 KB
• Volume: 154
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3135

✦ Synopsis

In this paper we study the maximal function associated to the Weyl transform W(+ r ) of the normalised surface measure + r on the sphere |z|=r in C n . This operator is given by the expansion

where . k are Laguerre functions of type (n&1) and P k are Hermite projection operators. We show that when p>2nÂ(2n&1), the maximal operator

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(+ r ) motivates the study of operators of the form

where : k are Laguerre functions of type :. We study various mapping properties of these operators with applications to Hermite expansions and solutions of Darboux type equations.