On the H p -L q boundedness of some fractional integral operators

Let A ₁, …, A _m be n × n real matrices such that for each 1 ? i ? m, A _i is invertible and A _i ? A _j is invertible for i ≠ j. In this paper we study integral operators of the form $$Tf(x) = int {{k_1}(x - {A_{1y}}){k_2}(x - {A_{2y}}) ldots {k_m}(x - {A_{my}})f(y){rm{d}}y}$$ ${k_i}(y) = sumlimits_{j in z} {{2^{jn/{q_i}}}} varphi i,j({2^j}y),1 le {q_i} < infty ,1/{q_1} + 1/q + ... + 1/q = 1 - r,0 le r < 1, and varphi i,j$ satisfying suitable regularity conditions. We obtain the boundedness of T: H ^p (? ⁿ ) → L ^q (? ⁿ ) for 0 < p < 1/r and 1/q = 1/p-r. We also show that we can not expect the H ^p -H ^q boundedness of this kind of operators.