We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by N α,A ((z, t), for α ≥ 2 and A > 0. This natural family includes the canonical Cygan-Korányi norm, corresponding to α = 4. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius R. The exponent we establish for the error in the case α = 2 is the best possible, in all dimensions.

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Considering functions f on R n for which both f and f are bounded by the Gaussian e − 1 2 a|x| 2 , 0 < a < 1 we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for O(n)−finite functions thus extending the one dimensional result of Vemuri [11].

We consider the Rubio de Francia's Littlewood-Paley square function associated with an arbitrary family of intervals in R with finite overlapping. Quantitative weighted estimates are obtained for this operator. The linear dependence on the characteristic of the weight [w] A p/2 turns out to be sharp for 3 ≤ p < ∞, whereas the sharpness in the range 2 < p < 3 remains as an open question. The results arise as a consequence of a sparse domination shown for these operators, obtained by suitably adapting the ideas coming from [11] and [16]. Finally, some conjectures are stated in relation with the problem under study.

This article deals with the structure of analytic and entire vectors for the Schrödinger representations of the Heisenberg group. Using refined versions of Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.