Complex Analysis and Operator Theory (2020) 14:53 Complex Analysis https://doi.org/10.1007/s11785-020-01010-5 and Operator Theory Dirac Operators with Gradient Potentials and Related Monogenic Functions Longfei Gu1 · Daowei Ma2 Received: 25 June 2019 / Accepted: 28 May 2020 © Springer Nature Switzerland AG 2020 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 Hölder space is given. We prove the Plemelj– Sokhotski formula and the Painlevé theorem. As applications, we firstly prove that a kind of Riemann–Hilbert problem for monogenic functions with respect to the poten- tial functions is solvable. Explicit representation formula of the solution is also given. We then establish solvability conditions of a kind of general Riemann–Hilbert prob- lem for monogenic functions with respect to the potential functions. Finally, we give some examples about the above Riemann–Hilbert problem. Keywords Dirac operators · Gradient potentials · Clifford analysis Mathematics Subject Classification 30G35 Communicated by Irene Sabadini, Michael Shapiro and Daniele Struppa. This article is part of the topical collection “Higher Dimensional Geometric Function Theory and Hypercomplex Analysis” edited by Irene Sabadini, Michael Shapiro and Daniele Struppa. This work was partially supported by NSF of China (Grant Nos. 11401287, 11771195) and AMEP of Linyi University. B Longfei Gu [email protected] Daowei Ma [email protected] 1 Department of Mathematics, Linyi University, Linyi 276005, Shandong, People’s Republic of China 2 Department of Mathematics, Wichita State University, Wichita, KS 67260-0033, USA 0123456789().: V,-vol 53 Page 2 of 19 L. Gu, D. Ma 1 Introduction In books, [4,10], R. Delanghe, F, Brackx, F. Sommen and V. Soucek use Clifford algebra to establish a function theory for the Dirac operator. This theory is also called Clifford analysis. Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis and generalize classical complex analysis in plane to a higher dimension in a natural and elegant way. The theory is centered around the concept of monogenic functions, i.e., null solutions of the Dirac operator. Under the framework of Clifford analysis, in [1–3,13,15,16,18–20,30], many interesting results about boundary value problems for monogenic functions and poly-monogenic func- tions, singular integral equations, and partial differential equations in Clifford analysis and quaternionic analysis are presented. These results in higher dimensions are sim- ilar to classical boundary value problems for analytic functions and singular integral equations in complex plane (see, [5,6,21,26]). In [31], Xu obtain the general solution of the equation (D − λ)w = 0 in SO(n)-invariant open subsets of Rn, whereby D is the Dirac operator and λ is a complex constant, and construct the fundamental solution of D − λ and integral representation formulas. From then on, many scholars in mathematics and physics have researched theories of integral representation for- mulas about the equation (D − λ)kw = 0 where k is a positive integer and λ is a real, complex or quaternionic constant by Clifford analytic methods. Moreover, based on integral representation formulas, some Dirichlet, Riemann–Hilbert boundary value problems are considered. We refer to [12,15,17,18,25,28,29,31]. However, to the best of our knowledge, about function theories for the operator D − λ(x) where λ(x) is a real functions in Rn have not been considered. In this paper, we mainly study a func- tion theory for the operator D − λ(x) and associate Riemann–Hilbert problems in the universal Clifford algebra Cl(Vn,n). The main motivations is that a large number of interesting physical applications (see [3,22–24]), for instance, Maxwell’s equations in physics, which are the fundamental equations of electromagnetism, can be expressed in terms of Clifford analysis as (D − λ(x))[u]=0. Its boundary value problems reduce to the Riemann–Hilbert problems. This article is organized as follows: In Sect. 2, we recall some basic facts about uni- versal Clifford algebras and Clifford analysis which will be needed in the article. In the section, we also introduce the Dirac operators with gradient potentials. In Sect. 3,we establish some integral formulas, including Borel–Pompeiu formula, Cauchy integral formula, and the mean value formula. Using the Cauchy integral formula, we investi- gate some geometric properties of the “monogenic functions” related. In Sect. 4,we discuss some integral transforms in Hölder space. Moreover we obtain Plemelj– for- mula and Painlevé theorem. Finally, in Sects. 5 and 6, two kinds of Riemann–Hilbert problems for such monogenic functions are considered. Dirac Operators with Gradient Potentials and Related … Page 3 of 19 53 2 Preliminaries Let A := R(e1,...,en) denote the free R-algebra with n indeterminants {e1,...,en}. Let J be the two-sided ideal in A generated by the elements { 2 − , = ,..., ; 2 + , = + ,..., ; + , ≤ < ≤ }. ei 1 i 1 s ei 1 i s 1 n ei e j e j ei 1 i j n The quotient algebra Cl(Vn,s) := A/J is called the Clifford algebra with parameters n, s. Without risk of ambiguity, we take the usual practice of using the same symbol to denote an indeterminant ei in A and its equivalent class in A/J. Therefore, e1,...,en considered as elements of A/J have the following relations: ⎧ 2 = , = ,..., , ⎨ ei 1 i 1 s 2 =− , = + ,..., , (2.1) ⎩ ei 1 i s 1 n ei e j + e j ei = 0, i = j. Set := ··· , ≤ < ···< ≤ . el1...lr el1 elr while 1 l1 lr n For more information on Cl(Vn,s), we refer to [4,10]. In this article, we only consider s = n. Thus Cl(Vn,n) is a real linear non-commutative algebra. An involution is defined by ⎧ ( )( ( )+ ) ⎨ = (− ) n A n A 3 , ∈ P , eA 1 2 eA if A N (2.2) ⎩ λ = λAeA, if λ = λAeA, A∈P N A∈P N where {eA, A ={l1,...,lr }∈P N, 1 ≤ l1 < ···< lr ≤ n}, n(A) is the cardinal number of the set A, N stands for the set {1, 2,...,n} and P N denotes the family of all order-preserving subsets of N in the above way. The Cl(Vn,n)- value n-1-differential form n σ = (− )i−1 N d 1 ei dxi i=1 is exact, where N = ∧···∧ ∧ ∧···∧ . dxi dx1 dxi−1 dxi+1 dxn 53 Page 4 of 19 L. Gu, D. Ma If dS stands for the classical surface element and n n = ei ni , i=1 where ni is the i-th component of the outward pointing normal, which yields the Clifford-valued surface element dσ can be written as dσ = ndS. (2.3) λ λ=( |λ |2) 1 R (λ) The norm of is defined by A∈P N A 2 .If e denotes the scalar portion of λ ∈ Cl(Vn,n), then it follows 2 2 Re(λλ) = Re(λλ) = |λA| =λ . A∈P N n Suppose be an open bounded non-empty subset of R (n ≥ 3). We introduce = n ∂ . = the Dirac operator D i=1 ei ∂x In particular, we obtain that DD where n i is the Laplacian over R . A function f : → Cl(Vn,n) is said to be left monogenic if it satisfies the equation D[ f ](x) = 0 for each x ∈ . A similar definition can be given for right monogenic functions. Denote z j = x j − x1e1e j , j = 2, 3,...,n and 1 V ,..., (x) = z ···z l1 l p p! l1 l p π(lr ,...,l p) p where (l1,...,l p) ∈{1, 2,...,n} , the sum is taken over all permutations with rep- ( ,..., ) ( ) = = etition of the sequence l1 l p . In particular we define Vl1,...,l p x 1forp 0 ( ) = < and Vl1,...,l p x 0forp 0. In what follows, M(r, f ) denotes the maximum modulus of a monogenic function f in a ball centered at 0, with radius r. Based on the maximum principle holds for monogenic functions, we obtain M(r, f ) = max { f (x)}, x=r which is well defined and strictly monotonic increasing whenever f is not constant. ( , ) M(r, f ) Moreover, M r f is a continuous function about r. When the expression rm remains bounded for r →∞, f is a monogenic polynomial. As the general version of Liouville’s theorem in Clifford analysis, we have the following result: Dirac Operators with Gradient Potentials and Related … Page 5 of 19 53 n M(r, f ) Lemma 2.1 [7,16] Let D[ f ]=0 in R and lim inf m = L < ∞,m∈ N\{0}. r→∞ r Then m ( ) = ( ) . f x Vl1,...,l p x Cl1,...,l p (2.4) p=0 (l1,...,l p) Elementary properties of the Dirac operators and left monogenic functions can be found in References [4,8–10,15,27,32,34]. n In this article, assume that B : R → R is a bounded continuous differentiable 1 n ∂B function, i.e., C -function and b := = e . We now consider the following i 1 i ∂xi inhomogeneous Dirac operator: Db(x)[ f ](x) = D[ f ](x) + b(x) f (x). 1 Definition 2.2 A C -function f : → Cl(Vn,n) is said to be (left) monogenic with respect to the potential B if Db(x)[ f ](x) = 0 for all x ∈ .