A Multiplier theorem on the Heisenberg group

Abstract

The question of boundedness of operators is of importance in analysis. We look into the operators which act on the frequency space of functions, that is those operators which act on the Fourier transform of an Lp function, 1 <p< →. Our operators are given by multiplication with a certain function called the multiplier. The boundedness conditions of these operators, on the Euclidean space have been solved. This thesis asks the boundedness question for the case of a non-abelian group called the Heisenberg group (Hn). Our main theorem gives a smoothness condition of the multiplier, constrained to the dimension of Hn. This is smoothness condition is similar to Mikhil’s condition for boundedness of multipliers on Rn. We use the tools developed by Stein, improving the theory given by Littlewood and Paley. These tools are the g↑functions, which we can bound and so get the bounds for our multiplier. To use this on Hn we have to move all the required ideas and machinery from Rn to Hn. To move all the required machinery we need some understanding of certain family of functions like the Hermite functions and the Laguerre functions. In essence, this thesis uses the tools and ideas developed for the Euclidean space, to tackle the problem of boundedness of multipliers and then builds analogues of it for the Heisenberg group. These tools are then put to use to show that under certain conditions, the boundedness of a multiplier can be established. We can further expand on this study by looking into the unsolved problems in Rn and ask the same in Hn.