Adv Appl Clifford Algebras 21 (2011), 591–605 © 2011 Springer Basel AG 0188-7009/030591-15 published online January 4, 2011 DOI 10.1007/s00006-010-0272-2 Advances in Applied Clifford Algebras Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra Thanh Van Nguyen Abstract This paper deals with the initial value problem of the type ∂w ∂w = L t, x, w, ∂t ∂xi w(0, x) = ϕ(x) (1) (2) where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ϕ is a regular function taking values in the Quaternionic Algebra The article proves necessary and sufficient conditions on the coefficients of operator L under which L is associated to the Cauchy-Fueter operator of Quarternionic Analysis This criterion makes it possible to construct the operator L for which the initial problem (1), (2) is solvable for an arbitrary initial regular function ϕ and the solution is also regular for each t Mathematics Subject Classification (2010) 35B45; 35F10; 47H10 Keywords Initial value problem, associated space, interior estimate Preliminaries and Notations Let H be a Quaternion algebra with the basis formed by e0 , e1 , e2 , e3 where e0 = 1, e3 = e1 e2 = e12 Suppose that Ω is a bounded domain of R4 A function f defines in Ω and takes values in the Quaternionic Algebra H which can be presented as f= fj ej , j=0 where fj (x) are real-valued functions We introduce the Cauchy-Fueter operator D= ek k=0 ∂ ∂xk 592 T.V Nguyen Adv Appl Clifford Algebras Definition A function f ∈ C (Ω, H) is said to be regular in Ω if f satisfies Df = Necessary and Sufficient Conditions for Associated Pairs Suppose that f = j=0 fj ej is a twice continuously differentiable function with respect to the space-like x0 , x1 , x2 , x3 Now assume that f is regular This means that Df = It is easy to verify that the condition Df = is equivalent to ∂f Ai = 0, ∂x i i=0 where ⎡ ⎢0 A0 = ⎢ ⎣0 0 0 0 ⎤ 0⎥ ⎥, 0⎦ ⎡ −1 ⎢1 A1 = ⎢ ⎣0 0 ⎡ ⎢0 A3 = ⎢ ⎣0 We define an operator 0 ⎤ 0 0⎥ ⎥, −1⎦ ⎤ −1 −1 ⎥ ⎥, 0⎦ 0 ⎡ ⎢0 A2 = ⎢ ⎣1 ⎛ ∂f0 ⎞ 0 −1 −1 0 ⎤ 1⎥ ⎥ 0⎦ ∂x ⎜ ∂f1i ⎟ ⎜ ∂xi ⎟ ∂f ⎟ =⎜ ⎜ ∂f2 ⎟ ∂xi ⎝ ∂xi ⎠ ∂f3 ∂xi as follows, f= Ai i=0 ∂f ∂xi (3) It is clear that ⎛Df⎞= if and only if f = Next, we identify the function f0 ⎜f1 ⎟ ⎟ f with f := ⎜ ⎝f2 ⎠ and introduce a differential operator L as follows, f3 ∂f + Cf + K, (4) ∂xj j=0 ⎛ ⎞ d0 ⎜d1 ⎟ (j) (j) ⎟ where Bj = [bαβ ], C = [cαβ ], K = ⎜ ⎝d2 ⎠, bαβ , cαβ , dα , (α, β = 0, 1, 2, 3) are d3 real-valued functions which are supposed to depend at least continuously on the time t and the space-like x0 , x1 , x2 , x3 A pair of operators , L is said to be associated (see [9]) if f = implies (Lf ) = (for each t in case the coefficients of L depend on t) Now we formulate necessary and sufficient conditions on the coefficients of operator L under which L is associated to Lf = Bj Vol 21 (2011) Differential Associated Operators and Their Applications 593 the operator (in other words, L is associated to the Cauchy-Fueter operator (j) of Quaternionic Analysis) Assume that the functions bαβ , cαβ , dα (j, α, β = 0, 1, 2, 3) are continuously differentiable with respect to the space-like variable x0 , x1 , x2 , x3 and differentiable on t Put (j) Pj = [pαβ ] = Aj Bj , Qij = (ij) [qαβ ] (j) j = 0, 1, 2, = Ai B j + A j B i , Rj = [rαβ ] = Ai i=0 (5) 0≤i