Cauchy transform and Poisson’s equation

Let u∈W1,p∩W01,p, 1⩽p⩽∞1⩽p⩽∞ be a solution of the Poisson equation Δu=hΔu=h, h∈Lph∈Lp, in the unit disk. We prove ‖∇u‖Lp⩽ap‖h‖Lp‖∇u‖Lp⩽ap‖h‖Lp and ‖∂u‖Lp⩽bp‖h‖Lp‖∂u‖Lp⩽bp‖h‖Lp with sharp constants apap and bpbp, for p=1p=1, p=2p=2, and p=∞p=∞. In addition, for p>2p>2, with sharp constants cpcp and CpCp, we show ‖∂u‖L∞⩽cp‖h‖Lp‖∂u‖L∞⩽cp‖h‖Lp and ‖∇u‖L∞⩽Cp‖h‖Lp‖∇u‖L∞⩽Cp‖h‖Lp. We also give an extension to smooth Jordan domains.These problems are equivalent to determining a precise value of the LpLp norm of the Cauchy transform of Dirichlet’s problem.