## Injectivity sets for spherical means on the Heisenberg group.(English)Zbl 0931.43007

Let $$\mu _{r}$$, $$r>0$$, be the normalized surface measure on the sphere $$\{w\in {\mathbb C^n} : |w|=r\}$$ in $${\mathbb C^n}$$. For $$f\in C({\mathbb C^n})$$, the twisted spherical means of $$f$$ are defined as follows: $f\times \mu _r(z)=\int_{|w|=r} f(z-w)e^{i {1\over 2} \operatorname{Im} (z\cdot \overline w)} d\mu_r(w),\quad z\in {\mathbb C}^n.$ A set $$S\subset {\mathbb C}^n$$ is said to be a set of injectivity for the twisted spherical means in a subclass $$\mathcal C$$ of the continuous functions on $${\mathbb C}^n$$ if $$f\in \mathcal C$$ and $$f\times \mu_r(z)=0$$, for all $$r\geq 0$$ and $$z\in S$$, imply that $$f=0$$. The authors consider functions that satisfy $$f(z)e^{({1\over {4}}+\varepsilon)|z|^2}\in L^p({\mathbb C}^n)$$ for some $$\varepsilon >0$$ and $$1\leq p \leq \infty$$. They show that in this class of functions, the boundary of a bounded domain in $${\mathbb C}^n$$ is a set of injectivity for the twisted spherical means. As a consequence some results about injectivity of the spherical mean operator in the Heisenberg group and the complex Radon transform are given.

### MSC:

 43A80 Analysis on other specific Lie groups 44A12 Radon transform
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### References:

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