Abstract
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.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.