Relations between $H^{p}_{u}$ and $L^{p}_{u}$ with polynomial weights

Abstract:

Relations between $L_u^p$ and $H_u^p$ of the real line are studied in the case when $p > 1$ and $u(x) = |q(x){|^p}w(x)$, where $q(x)$ is a polynomial and $w(x)$ satisfies the ${A_p}$ condition. It turns out that $H_u^p$ and $L_u^p$ can be identified when all the zeros of $q$ are real, and that otherwise $H_u^p$ can be identified with a certain proper subspace of $L_u^p$. In either case, a complete description of the distributions in $H_u^p$ is given. The questions of boundary values and of the existence of dense subsets of smooth functions are also considered.