Let ω and ν be radial weights on the unit disc of the complex
plane, and denote σ = ωp′
ν− p′
p and ωx = ∫ 1
0 sxω(s) ds for all 1 ≤ x < ∞.
Consider the one-weight inequality
‖Pω (f )‖Lp
ν ≤ C‖f ‖Lp
ν , 1 < p < ∞, (†)
for the Bergman projection Pω induced by ω. It is shown that the moment
condition
Dp(ω, ν) = sup
n∈N∪{0}
(νnp+1) 1
p (σnp′+1) 1
p′
ω2n+1
< ∞
is necessary for (†) to hold. Further, Dp(ω, ν) < ∞ is also sufficient for
(†) if ν admits the doubling properties sup0≤r<1
∫ 1
r ν(s)s ds
∫ 1
1+r
2
ν(s)s ds < ∞ and
sup0≤r<1
∫ 1
r ν(s)s ds
∫ 1− 1−r
K
r ν(s)s ds
< ∞ for some K > 1. In addition, an analogous
result for the one weight inequality ‖Pω (f )‖Dp
ν,k ≤ C‖f ‖Lp
ν , where
‖f ‖p
Dp
ν,k
=
k−1∑
j=0
|f (j)(0)|p +
∫
D
|f (k)(z)|p(1 − |z|)kpν(z) dA(z) < ∞, k ∈ N,
is established. The inequality (†) is further studied by using the necessary
condition Dp(ω, ν) < ∞ in the case of the exponential type weights ν(r) =
exp
(
− α
(1−rl)β
)
and ω(r) = exp
(
− ̃α
(1−r ̃l) ̃β
)
, where 0 < α, ̃α, l, ̃l < ∞
and 0 < β, ̃β ≤ 1
