Abstract
Through a double-layer potential argument we define harmonic conjugates of the Cauchy type and prove their existence and uniqueness in Lipschitz domains. We further define inner and outer Hilbert transformations on Lipschitz surfaces and prove their boundedness in L p, where the range for the index p depends on the Lipschitz constant of the boundary surface. The inner and outer Poisson kernels, the Cauchy type conjugate inner and outer Poisson kernels, and the kernels of the inner and outer Hilbert transformations on the sphere are obtained. We also obtain Abel sum expansions of the kernels. The study serves as a justification of the methods in a series of papers of Brackx et al. based on their method for computation of a certain type of harmonic conjugates.
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© 2008 Birkhäuser Verlag Basel/Switzerland
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Qian, T. (2008). Hilbert Transforms on the Sphere and Lipschitz Surfaces. In: Sabadini, I., Shapiro, M., Sommen, F. (eds) Hypercomplex Analysis. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-9893-4_16
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DOI: https://doi.org/10.1007/978-3-7643-9893-4_16
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