We characterize a sufficient and necessary condition which ensures that the operator
Hφf(x)=...
f(x1 t1,…,xn tnφ(t1,…,tn)dt1…dtn
is bounded on Lp(Gn) with 1≤p≤∞. The condition deeply depends on the nonnegative function φ defined on [0,1]×…×[0,1]. Furthermore, the corresponding operator norms are worked out.