Abstract:

We characterize a sufficient and necessary condition which ensures that the operator
H_φf(x)=...f(x₁ t₁,…,x_n t_nφ(t₁,…,t_n)dt₁…dt_n
is bounded on L^p(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.

Key words: Hardy operator, product space, L^p