We define the weighted Bergman space b β p ( S T ) consisting of temperature functions on the cylinder S T = S 1 × ( 0 , T ) and belonging to L p ( Ω T , t β d x d t ) , where Ω T = ( 0 , 2 ) × ( 0 , T ) . For -1$"> β > − 1 we construct a family of bounded projections of L p ( Ω T , t β d x d t ) onto b β p ( S T ) . We use this to get, for 1 < p < ∞ and 1 / p + 1 / p ′ = 1 , a duality b β p ( S T ) ∗ = b β ′ p ′ ( S T ) , where β ′ depends on p and β .