Relations between $H^ p_ u$ and $L^ p_ u$ in a product space

Abstract:

Relations between $L_u^p$ and $H_u^p$ are studied for the product space ${{\mathbf {R}}^1} \times {{\mathbf {R}}^1}$ in the case $1 < p < \infty$ and $u({x_1},{x_2}) = |{Q_1}({x_1}){|^p}|{Q_2}({x_2}){|^p}w({x_1},{x_2})$, where ${Q_1}$ and ${Q_2}$ are polynomials and $w$ satisfies the ${A_p}$ condition for rectangles. A description of the distributions in $H_u^p$ is given. Questions about boundary values and about the existence of dense subsets of smooth functions satisfying appropriate moment conditions are also considered.