Bergman projection on Lebesgue space Induced by doubling weight

Abstract

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