Cauchy transform and Poisson's equation

2010/03/19 by David Kalaj, Kalaj, David
Mathematics · #Advanced Harmonic Analysis Research #Algebraic and Geometric Analysis #Complex Variables (math.CV) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1003.3822

openalex publication_date 2010/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let u∈ W^2,p₀, 1≤ p≤ ∞ be a solution of the Poisson equation Δu = h, h∈ L^p, in the unit disk. It is proved that ‖∇ u‖_L^p ≤ a_p‖h‖_L^p with sharp constant a_p for p=1 and p=∞ and that ‖∂ u‖_L^p ≤ b_p‖h‖_L^p with sharp constant b_p for p=1, p=2 and p=∞. In addition is proved that for p>2 ||∂ u||_L^∞≤ c_p\Vert h\Vert_L^p , and ||∇ u||_L^∞≤ C_p\Vert h\Vert_L^p, with sharp constants c_p and C_p. An extension to smooth Jordan domains is given. These problems are equivalent to determining the precise value of L^p norm of \it Cauchy transform of Dirichlet's problem.

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