Fractional type operators on the Heisenberg group

Let ρ(·) be the Koranyi norm on the Heisenberg group Hⁿ ≡ (R²ⁿ × R, \, · \, ) defined by \[ ρ(x,t) = ( |x|⁴ + 16 t² )1/4, \,\,\,\, (x,t) ∈ Hⁿ. \] For 0 ≤ α < Q:=2n+2, m ∈ N ∩ (1 - α/Q, ∞ ), and m positive constants α₁, ..., αₘ such that α₁ + · · · + αₘ = Q - α, we consider the following generalization of the Riesz potential on Hⁿ \[ Tα, \, mf(x,t) = ∫_{Hⁿ} f(y,s) ∏ⱼ₌₁ᵐ ρ((A_j y, r_j⁻² s)⁻¹ · ( x, t))-α_j \, dy \, ds, \] where, in the case 0 < α < Q, the Aⱼ's are matrices belonging to Sp (2n, R) ∩ SO(2n) and rⱼ = 1 for every $j=1, ..., m$; for α = 0, we consider Aⱼ = rⱼ⁻¹ \, I2n × 2n for every $j=1, ..., m$, where the rⱼ's are positive constants such that rᵢ² - rⱼ² ≠ 0 if i ≠ j. In this note we study the behavior of these operators on variable Hardy spaces in Hⁿ.