Weak-Type Estimates for the Bergman Projection on Simply Connected Domains

Abstract: The Bergman projection is a fundamental operator in complex analysis and PDE, defined as the orthogonal projection from L²(Ω) onto the Bergman space A²(Ω) of square-integrable holomorphic functions on a simply connected domain Ω ⊂ ℂ. Its Lᵖ mapping properties are closely tied to the geometry of the underlying domain. In this work, we investigate weak-type bounds for the Bergman projection Π_Ω via the transformation law that reduces these estimates to weighted estimates for Π_𝔻 on the unit disc, with weights determined by the modulus |ψ'| of the derivative of a conformal map ψ: 𝔻 → Ω. We show that Π_Ω is of weak type (1,1) whenever |ψ'| belongs to the Békollé–Bonami class B₁, thereby sharpening a result of Békollé. We also establish a necessary condition for weak-type (p,p) bounds for 1 ≤ p < ∞.