Abstract
This chapter concentrates on the averages of functions along polynomial sequences in discrete nilpotent groups, illustrating the problems that arise from studying these averages. Though special polynomial sequences can still use the Fourier transform in the central variables to analyze the operators, it appears that one needs to proceed in an entirely different way in the case of general polynomial maps, when the Fourier transform method is not available. This chapter is the first attempt to treat discrete Radon transforms along general polynomial sequences in the non-commutative nilpotent settings. It does so by analyzing the problem of L² boundedness of singular Radon transforms.
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