Estimates for singular convolution operators on the Heisenberg group

1984 ◽  
Vol 267 (1) ◽  
pp. 1-15 ◽  
Author(s):  
D. Geller ◽  
E. M. Stein
Keyword(s):  
Heisenberg Group ◽  
Convolution Operators

scholarly journals A CLASS OF STRONGLY SINGULAR RADON TRANSFORMS ON THE HEISENBERG GROUP

2007 ◽  
Vol 50 (2) ◽  
pp. 429-457 ◽  
Author(s):  
Neil Lyall
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
Singular Integrals ◽  
Radon Transforms ◽  
Laguerre Functions ◽  
Convolution Operators ◽  
Fourier Transform Methods ◽  
Group Fourier Transform ◽  
Asymptotic Forms ◽  
Singular Radon Transforms

AbstractWe primarily consider here the $L^2$ mapping properties of a class of strongly singular Radon transforms on the Heisenberg group $\mathbb{H}^n$; these are convolution operators on $\mathbb{H}^n$ with kernels of the form $M(z,t)=K(z)\delta_0(t)$, where $K$ is a strongly singular kernel on $\mathbb{C}^n$. Our results are obtained by using the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi.We also discuss the behaviour of related twisted strongly singular operators on $L^2(\mathbb{C}^n)$ and obtain results in this context independently of group Fourier transform methods. Key to this argument is a generalization of the results for classical strongly singular integrals on $L^2(\mathbb{R}^d)$.

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