Polynomial sequences in discrete nilpotent groups of step 2. (English) Zbl 1543.37009
Summary: We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the noncommutative nilpotent setting. In particular, we present what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.
MSC:
| 37A46 | Relations between ergodic theory and harmonic analysis |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 45P05 | Integral operators |
Keywords:
discrete nilpotent groups; pointwise ergodic theorems; nilpotent circle method; Weyl inequalityReferences:
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