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A Variant of Hörmander's L2 Theorem for Dirac Operator in Clifford Analysis

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A Variant of Hörmander's L2 Theorem for Dirac Operator in Clifford Analysis Noname manuscript No. (will be inserted by the editor) A variant of H¨ormander's L2 theorem for Dirac operator in Clifford analysis Yang Liu · Zhihua Chen · Yifei Pan Received: date / Accepted: date Abstract In this paper, we give the H¨ormander's L2 theorem for Dirac operator over open subset Ω 2 Rn+1 with Clifford algebra. Some sufficient conditions on the existence of weak solutions for Dirac operators have been found in the sense of Clifford analysis. In particular, if Ω is bounded, then we prove that for any f in L2 space with value in Clifford algebra, we can find a weak solution of Dirac operator such that Du = f with the solution u in the L2 space as well. The method is based on H¨ormander's L2 existence theorem in complex analysis and the L2 weighted space is utilised. Keywords H¨ormander's L2 theorem · Clifford analysis · weak solution · Dirac operator Mathematics Subject Classification (2000) 32W50 · 15A66 1 Introduction The development of function theories over Clifford algebras has proved a useful set- ting for generalizing many aspects of one variable complex function theory to high- er dimensions. The study of these function theories is referred to as Clifford analysis [Brackx et al(1982),Huang et al(2006),Gong et al(2009),Ryan(2000)]. This analysis is closely related to a number of studies made in mathematical physics, and many appli- cations have been found in this area in recent years. In [Ryan(1995)], Ryan considered solutions of the polynomial Dirac operator, which affords an integral representation. Furthermore, the author gave a Pompeiu representation for C1-functions in a Lipschitz bounded domain. In [Ryan(1990)], the author presented a classification of linear, confor- mally invariant, Clifford-algebra-valued differential operators over Cn, which comprise the Dirac operator and its iterates. In [Qian and Ryan(1996)], Qian and Ryan used Vahlen matrices to study the conformal covariance of various types of Hardy spaces over hypersurfaces in Rn. In [De Ridder et al(2012)], the discrete Fueter polynomial- s was introduced, which formed a basis of the space of discrete spherical monogenics. Moreover, the explicit construction for this discrete Fueter basis, in arbitrary dimension m and for arbitrary homogeneity degree k was presented as well. Yang Liu Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Fax: +86-57982298897 E-mail: [email protected] Zhihua Chen Department of Mathematics, Tongji University, Shanghai 200092, China Yifei Pan Department of Mathematical Sciences, Indiana University-Purdue University Fort Wayne, Fort Wayne, Indiana 46805, USA 2 Yang Liu et al. In [H¨ormander(1965)],the famous H¨ormander's L2 existence and approximation theorems was given for the @¯ operator in pseudo-convex domains in Cn. When n = 1, the existence theorem of complex variable can be deduced. The aim of this paper is to establish a H¨ormander's L2 theorem in Rn+1 with Clifford analysis, and provide sufficient conditions on the existence of the weak solutions for Dirac operator in the sense of Clifford algebra. Let A be an real Clifford algebra over an (n+1)-dimensionalp real vector space Rn+1 and the corresponding norm on A is given by jλj0 = (λ, λ)0 (see subsection 2.1). Let Ω be an open subset of Rn+1, L2(Ω; A;') be a right Hilbert A-module for a given function ' 2 C2(Ω; R) with the norm given by Definition 29. (see subsection 2.3). D ∗ denotes the Dirac differential operator and the operator D' is given by (9). Then we can present the main results of the paper as follows. Theorem 11 Given f 2 L2(Ω; A;'), there exists u 2 L2(Ω; A;') such that Du = f (1) with Z k k2 j j2 −' ≤ n u = u 0e dx 2 c (2) Ω if Z ∗ ∗ j j2 ≤ k k2 j j2 −' 8 2 1 A (f; α)' 0 c D'α = c D'α 0e ; α C0 (Ω; ): (3) Ω Conversely, if there exists u 2 L2(Ω; A;') such that (1) is satisfied with kuk2 ≤ c Then we can get (3). The factor 2n in (2) comes from the definition of the norm in Clifford analysis. If n = 1, then the factor would disappear which gives the necessary and sufficient condition in the theorem. From the above theorem, we give the following sufficient conditions on the existence of weak solutions for Dirac operator. Theorem 12 Given ' 2 C2(Ω; R) with Ω being an open subset of Rn+1 and n > 1; ≥ @2' 6 ≤ ≤ @2' ≤ ≤ ≤ ∆φ 0, and @x @x = 0; i = j; 1 i; j n and @x2 0; 1 i n. Then for all j i R i jfj2 − 2 2 A 0 ' 1 2 2 A f L (Ω; ;') with Ω ∆φ e dx = c < , there exists a u L (Ω; ;') such that Du = f with Z Z j j2 k k2 j j2 −' ≤ n f 0 −' u = u 0e dx 2 e dx: Ω Ω ∆φ 2 Rn+1 2 Remark 13 Assuming x = (x0; x1; :::; xn) , it is easy to see that '(x) = x0 satisfies the conditions in Theorem 12. Another simple example would be Xn 2 − 2 '(x) = (n + 1)x0 xi : i=1 @2' @2' It is obvious that ∆φ(x) = 2, 2 = −2, and = 0; i =6 j; 1 ≤ i; j ≤ n. @xi @xj @xi 2 2 R 00 ≥ Corollary 14 GivenR' C (Ω; ); and '(x) = '(x0) with ' (x0) 0. Then for all jfj2 2 2 A 0 −' 1 2 2 A f L (Ω; ;') with Ω '00 e dx = c < , there exists a u L (Ω; ;') such that Du = f with Z Z j j2 k k2 j j2 −' ≤ n f 0 −' u = u 0e dx 2 00 e dx: Ω Ω ' A variant of H¨ormander's L2 theorem for Dirac operator in Clifford analysis 3 Furthermore, there is nothing to do with the boundary conditions of Ω in the above results. This phenomenon is totally different with the famous H¨ormander's L2 existence theorems of several complex variables in [H¨ormander(1965)].Then we can also have the following corollary on global solutions. 2 2 Rn+1 R Theorem 15 Given ' C ( ; ) with allR derivative conditions in Theorem 11 jfj2 − 2 2 Rn+1 A 0 ' 1 satisfied. Then for all f L ( ; ;') with Rn+1 ∆φ e dx = c < , there exists a u 2 L2(Rn+1; A;') satisfying Du = f with Z Z j j2 k k2 j j2 −' ≤ n f 0 −' u = u 0e dx 2 e dx: Rn+1 Rn+1 ∆φ On the other hand, if the boundary of Ω is concerned, we consider a special kind n+1 of Ω0 = fx 2 R : a ≤ x0 ≤ bg for any a; b 2 R with a < b, then we can get the following theorem within L2 space instead of L2 weighted space. 2 2 Theorem 16 Let f 2 L (Ω0; A). Then there exists a u 2 L (Ω0; A) such that Du = f with Z Z j j2 ≤ n j j2 u 0dx 2 c(a; b) f 0dx Ω0 Ω0 and c(a; b) is a factor depending on a; b. 2 2 A 2 A Proof Let '(x) = x0. It can be obtained that L (Ω0; ) = L (Ω0; ;') for the bound- ary of x0. Then the theorem is proved with Theorem 12. Remark 17 In particular, any bounded domain Ω in Rn+1 can be regarded as one type 2 of Ω0. Therefore, it comes from Theorem 16 that for any f 2 L (Ω; A), we can find a weak solution of Dirac operator Du = f with u 2 L2(Ω; A). 2 Preliminaries To make the paper self-contained, some basic notations and results used in this paper are included. 2.1 The Clifford algebra A Let A be an real Clifford algebra over an (n+1)-dimensional real vector space Rn+1 n+1 with orthogonal basis e := fe0; e1; :::; eng, where e0 = 1 is a unit element in R . Furthermore, ( eiej + ejei = 0; i =6 j (4) 2 − ei = 1; i = 1; :::; n: Then A has its basis f ··· ≤ ≤ ≤ ≤ g eA = eh1···hr = eh1 ehr : 1 h1 < ::: < hr n; 1 r n : If i 2 fh1; :::; hrg, we denote i 2 A and if i 62 fh1; :::; hrg, we denote i 62 A. A − i f g n f g f g [ f g means h1; :::; hr i and A + i means Ph1; :::; hr i . So real Clifford algebra is composed of elements having the type a = xAeA, in which xA 2 R are real numbers. A P 2 A ∗ ∗ For a , we give the inversion in the Clifford algebra as follows: a = xAeA where A ∗ jAj + e = (−1) e and jAj = n(A) is the r 2 Z as e = e ··· . When A = ;, jAj = 0. A A A h1 hr P y y Next, we define the reversion in the Clifford algebra, which is given by a = xAeA A 4 Yang Liu et al. y − (jA|−1)jAj=2 where eA = ( 1) eA: Now we present the involution which is a combination of the inversion and the reversion introduced above. X a¯ = xAe¯A A ∗† − (jAj+1)jAj=2 wheree ¯A = eA = ( 1) eA: From the definition, one can easily deduce that eAe¯A =e ¯AeA = 1: Furthermore, we have λµ =µ ¯λ,¯ 8λ, µ 2 A: P Let a = xAeA be a Clifford number. The coefficient xA of the eA-component will A also be denoted by [a]A.
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