Duality between $H^{p}$ and $H^{q}$ and associated projections

Abstract:

If $U/G$ represents a Riemann surface as the disk $U$ modulo a discontinuous group $G$ and if ${L^p}/G$ denotes the ${L^p}$ functions on the circle which are $G$ invariant, then it is shown that ${L^p}/G = {N_p} \oplus {K_p}$ if and only if ${H^p}/G$ and ${\bar H^q}/G$ are naturally dual. Here ${K_p}$ is the subset of ${L^p}/G$ consisting of those functions which are invariant and whose conjugates are invariant; ${N_p}$ is $E({H^p}) \cap E(\bar H_0^p)$ where $E$ is the conditional expectation operator. ${H^p}$ is the space of boundary values of holomorphic functions and $1 < p < \infty$.