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

Abstract

If U / G U/G represents a Riemann surface as the disk U U modulo a discontinuous group G G and if L p / G {L^p}/G denotes the L p {L^p} functions on the circle which are G G invariant, then it is shown that L p / G = N p ⊕ K p {L^p}/G = {N_p} \oplus {K_p} if and only if H p / G {H^p}/G and H ¯ q / G {\bar H^q}/G are naturally dual. Here K p {K_p} is the subset of L p / G {L^p}/G consisting of those functions which are invariant and whose conjugates are invariant; N p {N_p} is E ( H p ) ∩ E ( H ¯ 0 p ) E({H^p}) \cap E(\bar H_0^p) where E E is the conditional expectation operator. H p {H^p} is the space of boundary values of holomorphic functions and 1 > p > ∞ 1 > p > \infty .