Boundedness of the Bergman projection on LpLp-spaces with exponential weights

Let v(r)=exp⁡(−α1−r) with α>0α>0, and let DD be the unit disc in the complex plane. Denote by Avp the subspace of analytic functions of Lp(D,v)Lp(D,v) and let PvPv be the orthogonal projection from L2(D,v)L2(D,v) onto Av2. In 2004, Dostanic revealed the intriguing fact that PvPv is bounded from Lp(D,v)Lp(D,v) to Avp only for p=2p=2, and he posed the related problem of identifying the duals of Avp for p≥1p≥1, p≠2p≠2. In this paper we propose a solution to this problem by proving that PvPv is bounded from Lp(D,vp/2)Lp(D,vp/2) to Avp/2p whenever 1≤p<∞1≤p<∞, and, consequently, the dual of Avp/2p for p≥1p≥1 can be identified with Avq/2q, where 1/p+1/q=11/p+1/q=1. In addition, we also address a similar question on some classes of weighted Fock spaces.