L^p Estimates for the ∂-Problem on Rational Hartogs Triangles

2026/07/29 by Tran Vu Khanh, Tu Nguyen
Mathematics · #math.CV

paper · pdf

arxiv created 2026/07/29 · arxiv updated 2026/07/30

Abstract

We investigate L^p estimates for the ∂-problem on rational Hartogs triangles ℍ_m/n = \ (z₁, z₂) ∈ ℂ² : |z₁|^m < |z₂|ⁿ < 1 \. For p ∈ (1, ∞), we establish the existence of a solution operator that is bounded on L^p(ℍ_m/n). Our approach avoid the need for any \it a priori condition on the data. We also show that the canonical solution K_ℍ_m/n is bounded on L^p(ℍ_m/n) for p ∈ (p₀, p₂), where p₀=\frac2m+2nm+n+1+min\m, n\ and p₂=(2m+2n)/(m+n-1). For classical Hartogs triangle, ℍ₁, this establishes boundedness for p ∈ (1, 4).

Citations

• Bergman-Toeplitz operators on fat Hartogs triangles
• L^p Mapping Properties of the Bergman Projection on the Hartogs Triangle
• Prolongement des fonctions holomorphes born�es et m�trique de Carath�odory

Related

• Optimal L^p regularity for ∂ on the Hartogs triangle
• A solution operator for ∂ on the Hartogs triangle and L^p estimates
• L^p boundedness of the Bergman projection on the generalized Hartogs triangles
• Optimal Sobolev regularity of ∂ on the Hartogs triangle
• L^p regularity of the Bergman projection on generalizations of the Hartogs triangle in ℂⁿ⁺¹