Riesz Transforms Associated With Heisenberg Groups And Grushin Operators

• Main Author: Sanjay, P K
• Other Authors: Thangavelu, Sundaram
• Language: en_US
• Published: 2015
• Subjects: Riesz Transforms; Heisenberg Groups; Grushin Operators; Differential Operators; Heisenberg Group; Hermite Polynomials; Hermite Functions; Laguerre Functions; Hermite Expansion; Laguerre Expansion; Mathematics
• Online Access: http://etd.iisc.ernet.in/handle/2005/2496; http://etd.ncsi.iisc.ernet.in/abstracts/3223/G25451-Abs.pdf

We characterise the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions. Next we study the Riesz transforms associated to the Grushin operator G = - Δ - |x|2@t2 on Rn+1. We prove that both the first order and higher order Riesz transforms are bounded on Lp(Rn+1): We also prove that norms of the first order Riesz transforms are independent of the dimension n.