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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.

We study discrete fixed point sets of holomorphic self-maps of complex manifolds. The main attention is focused on the cardinality of this set and its configuration. As a consequence of one of our observations, a bounded domain in ${\Bbb C}^n$ with no non-trivial holomorphic retractions is constructed., 12 pages

We study the fixed point sets of holomorphic self- maps of a bounded domain in C n . Specifically we investigate the least number of fixed points in general position in the domain that forces any automorphism (or endomorphism) to be the identity. We have discovered that in terms of this number one can give the necessary and sufficient condition for the domain to be biholomor- phic to the unit ball. Other theorems and examples generalize and complete previous results in this area, especially the recent work of Jean-Pierre Vigue.

Results are reported of a numerical implementation of the hypcrbolic Fourier transform and the geodesic and horocyclic Radon transforms on the hyperbolic plane, and of their inverses. The study is motivated by the hyperbolic geometry approach to the linearized inverse conductivity problem, suggested by C. A. Berenstein and E. Casadio Tarabusi.

In its proof, a bijective holomorphic mapping σ :Ω → B with σ(q)= 0 has been constructed. However, no explicit arguments showing the complex analyticity of σ−1 had been presented. Indeed, the main purpose of this article is to give it a concise proof. On the other hand, we present it as a separate article since we believe that the arguments here are of a separate interest, in regard to the following unsolved problem

In this paper we prove the best possible Lp estimates for the ∂-b -equation on the boundaries of real ellipsoids in Cn, and provide examples to show why they cannot be improved.

The disc-hull of a set is the union of the set arid all HiX discs whose boundaries lie in the set. We give an example of a, compact set in C2 whose disc-hull is not compact, answering a question posed by P. Ahern and W. Rudin. The polynomial hull of a compact set X c C'" is the set X of all points x E C"' at which the inequality IP(x)l : U --* C,7 is a non-constant map whose components are in H' (U), its range d>(U) is called an H?-disc, parametrized by d>. If lim.,/1(4>(rei0)) E X for almost all e'O on the unit circle T, then 4)(U) is an H'-disc whose boundary lies in X." They further define the disc-hull D(X) to be the union of X and all Hc-discs whose boundaries lie in X. Because of the maximum principle, D(X) C X. One of the questions posed in [1] (see p. 25) is whether the disc-hull D(X) is always compact for a compact set X C C". Below we answer this question negatively by constructing a counter-exaimple in c2 1. Define w {z E U: Rez > 4 Let Ro: U -> W be the Riemann map satisfying o(+i) = 1 + 23i and ,o(1) = 1. Therefore Re,o(e&0) = 2 for Ree0o ,cn(0) = 0. 2. Let X {((, ?7) E C2 ( E T, q E F(}, where the fiber F, is defined as follows. 1( = T for Re( > 0, 1T( U for ( ?i, and 1( {f01(() 'n E N} U {0} for ReX D? (0) (0, 0); therefore (0, 0) E D(X). Received by the editors August 31, 1999. 2000 Mathematics Subject Classification. Primary 32E20.