The Clifford Hilbert transform and Riemann–Hilbert problems associated to the Dirac operator in Lebesgue spaces

Abstract

In this paper, we establish some properties of the Hilbert transform in Clifford analysis setting, and mainly show the weak type (1,1) inequality for the Clifford Hilbert transform in Euclidean space Rm (m ≥ 3) using the Clifford algebra and analysis generalizations of the one dimensional proofs. Furthermore, we prove Kolmogorov’s inequality for Clifford Hilbert transform. With the help of the properties for the Hilbert transform and Clifford analytic techniques, we establish the Riemann–Hilbert problems in Clifford value Lebesgue p-integrable spaces. Based on the Newton embedding method, we prove an existence and uniqueness for the nonlinear Riemann–Hilbert problem and also give an error estimation for the approximate solutions in the Newton embedding procedure in Lebesgue spaces.