Riesz potential on weighted product Hardy spaces and inequalities

In this paper, we investigate the weighted product Hardy spaces Hwᵖ(Rd₁× Rd₂). Under some conditions on the weight, we prove that the Riesz potential operator I_α is bounded from Lwᵖ(Rd₁× Rd₂) to L_wq/pq(Rd₁× Rd₂) when α=(α₁, α₂) and 1/p- 1/q = α₁/d₁ =α₂/d₂. We also verify the boundedness of I_α from Hwᵖ(Rd₁× Rd₂) to H_wq/pq(Rd₁× Rd₂) and from Hwᵖ(Rd₁× Rd₂) to L_wq/pq(Rd₁× Rd₂). We consider similar questions for the maximal fractional operator, too.