Dirac Operators with Gradient Potentials and Related Monogenic Functions

Abstract

We investigate some properties of solutions to Dirac operators with gradient potentials. Solutions to Dirac operators with gradient potentials are called monogenic functions with respect to the potential functions. We establish a Borel–Pompeiu formula, and obtain Cauchy integral formula and mean value formula about such functions. Based on integral formulas, we prove some geometric properties of monogenic functions with respect to the potential functions and construct some integral transforms. The boundedness of integral transforms in Holder space is given. We prove the Plemelj–Sokhotski formula and the Painleve theorem. As applications, we firstly prove that a kind of Riemann–Hilbert problem for monogenic functions with respect to the potential functions is solvable. Explicit representation formula of the solution is also given. We then establish solvability conditions of a kind of general Riemann–Hilbert problem for monogenic functions with respect to the potential functions. Finally, we give some examples about the above Riemann–Hilbert problem.