Hilbert Transforms on the Sphere and Lipschitz Surfaces
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.
KeywordsPoisson kernel Conjugate Poisson kernel Schwarz kernel Hilbert transformation Cauchy integral double-layer potential Clifford algebra
Mathematics Subject Classification (2000)Primary 62D05, 30D10 Secondary 42B35
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