Fractional type operators on the Heisenberg group

Let () be the Koranyi norm on the Heisenberg group H^n (R^2n R, \, \, ) defined by \ (x, t) = (|x|^4 + 16 t^2) ^1/4, \, \, \, \, (x, t) H^n. \ For 0 < Q: =2n+2, m N (1 - Q, ), and m positive constants ₁,. . . , ₘ such that ₁ + + ₘ = Q -, we consider the following generalization of the Riesz potential on H^n \ T, \, ₌f (x, t) = ₇^₍ f (y, s) ₉=₁^m ( (Aⱼ y, rⱼ^-2 s) ^-1 (x, t) ) ^-ⱼ \, dy \, ds, \ where, in the case 0 < < Q, the Aⱼ's are matrices belonging to Sp (2n, R) SO (2n) and rⱼ = 1 for every j=1,. . . , m; for = 0, we consider Aⱼ = rⱼ^-1 \, I₂₍ ₂₍ for every j=1,. . . , m, where the rⱼ's are positive constants such that r₈^2 - r₉^2 0 if i j. In this note we study the behavior of these operators on variable Hardy spaces in H^n.