Conjugate Harmonic Functions and Clifford Algebras

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1 Conjugate Harmonic Functions and Clifford Algebras Craig A. Nolder Department of Mathematics Florida State University Tallahassee, FL , USA Abstract We generalize a Hardy-Littlewood inequality and a Privalov inequality for conjugate harmonic functions in the plane to components of Clifford-valued monogenic functions. Introduction Throughout this paper a domain Ω R n is a connected open set. Given u : Ω R we write u p,ω = u p We denote the Lipschitz norm of u over Ω by u L k,ω = Ω p, p > 0. u(x ) u(x 2 ) sup x,x 2 Ω x x 2 k x x 2 for 0 < k. In [3], Hardy and Littlewood proved the following result. Theorem. If u+iv is analytic in a disk D centered at z 0, then there exists a constant C, depending only on p, such that u u(z 0 ) p,d C v p,d. (.) Similarly, Theorem 2. is given in [9] by Privalov. Preprint submitted to Elsevier Science 4 April 2004

2 Theorem.2 If u + iv is analytic in a disk D, then there exists a constant C, depending only on k, such that u L k,d C v L k,d. (.2) In fact Theorem.2 follows from Theorem. [7]. We prove versions of these theorems for components of monogenic functions valued in the universal Clifford algebras over R n. See Theorem 3.. In [2], Stein and Weiss studied systems of conjugate harmonic functions in R n. These are vectors of harmonic functions (u, u 2,..., u n ), which satisfy n i= u i = 0 and (.3) u i x j = u j for all i and j. (.4) Notice for x = x, x 2 = y, u 2 = u and u = v these are the usual Cauchy- Riemann equations. The results of this paper hold in the special case of such Stein-Weiss systems. In this special case the results appear in [6] as well as versions for quasiregular mappings. The quasiregular theory was published in [4] and subsequently developed in [7] and [8]. The quasiregular case is a one- dimensional analytic theory in the sense that the properties of one component often determine those of the rest. The theory we present here is a one- codimensional analytic theory. The results for Stein-Weiss systems that appear in [6] have never been published. We fix an orthonormal basis of R n, (e, e 2,..., e n ), and denote by U n the (real) Clifford algebra spanned by the reduced multi-indexed products α <... < α k n, with the rule e α = e α e αk, e j e k + e k e j = 2δ jk. Here δ jk = 0 if j k, if j = k. We have an increasing chain R C H U 3... U n.... 2

3 Here H is the quaternions. As such a function F : Ω U n, can be represented as F = α F α e α where each F α : Ω R. We define a norm F = ( α F α 2 ) 2. We consider here a Dirac operator D, defined as follows. If F = α F α e α, then DF = α ( n i= ) F α e i e α. Definition.3 A Clifford-valued function F : Ω U n is monogenic if DF = 0. We also define the Clifford Laplacian F = α F α e α. Since D 2 =, it follows that the coefficients of any monogenic function are harmonic in the usual sense. We denote the length of a multi-index α by α and decompose any Cliffordvalued function into its even and odd parts. We write F = F even + F odd = α even F α e α + α odd F α e α. Notice that the Dirac operator D maps even parts to odd parts and odd parts to even parts: D (F even ) = (DF ) odd and D (F odd ) = (DF ) even. As such, F is monogenic if and only if D(F even ) = 0 and D(F odd ) = 0. Definition.4 A system of conjugate harmonic functions in a Clifford algebra consists of the coefficients of F even or F odd for some monogenic function F. A way to realize the Stein-Weiss systems in this context is through the natural embedding of R n into U n, namely Here (x, x 2,..., x n ) x e x n e n. ( n ) n u i D u i e i = + i= n n i= i j u i x j e j e i. As such D(F odd ) = 0 is equivalent to (.3) and (.4) where F = u e u n e n. We will use the notation u = ( u,..., u ) = u e u e n x x n x x n 3

4 for u : Ω R. Notice that DF = 0 is equivalent to 2 n linear equations involving the components of the vectors F α. Because the operator D intertwines parity, 2 n of these equations is a system for the even coefficients and the rest for the odd. Example.5 Let F = u 0 + u e + u 2 e 2 + u 3 e 3 + u 4 e e 2 + u 5 e e 3 + u 6 e 2 e 3 + u 7 e e 2 e 3. If DF = 0, then ( u 0 = u 4 u 5 u ) 5 e + x 2 x 2 x 3 ( u4 + u 5 u ) 6 e 2 + x x x 3 ( ) u6 and ( u = u 2 u 3, u 2 u 7, u 3 + u ) 7. x 2 x 3 x x 3 x x 2 Notice that the second equation is a part of the Stein-Weiss system (.3) and (.4) when u 7 = 0. x 2 e 3 Clearly such representations hold generally and we have the following simple estimate. Lemma.6 If {u α } is a system of conjugate harmonic functions in the Clifford algebra U n, defined in Ω R n, then for each α, u α 2 C(n) u β 2 (.5) β α in Ω. Here C(n) is a constant that depends only on n. We mention [], [0] and [] as references for Clifford analysis. 2 Notations and Domains We assume throughout that w is a Muckenhoupt weight and write w A q M(Ω), < q <, M <, when w 0 a.e. and Q Q w M Q Q w ( q) q for all cubes Q Ω. Here Q is the volume of Q. For u : Ω R we write for 0 < p <, p u p,ωµ = inf u a p dµ a R Ω 4

5 where dµ = wdx is weighted Lebesgue measure. We define the Hardy-Littlewood sharp maximal function for 0 < p <, Also the sharp BMO norm is Mp(u, µ)(x) = sup µ(q) p u p,q,µ. Q Ω x Q u BMO Ω,µ = sup M(u, µ)(x). x Ω Definition 2. A domain Ω is a δ-john domain, 0 < δ, if there exists a point x 0 Ω which can be joined with any other point x Ω by a continuous curve γ Ω which satisfies for all ξ γ. δ ξ x d(ξ, Ω) John domains do not have external cusps. Using the geometry of John domains, weak local weighted L p -estimates patch together to form global estimates. It is in this way that Theorem 3.2 is obtained. See [4] and [7]. Definition 2.2 For 0 < k, Ω is a Lip k -extension domain if there is a constant N such that every pair of points x, x 2 Ω can be joined by a continuous curve γ Ω for which γ d(γ(s), Ω) k ds N x x 2 k. Theorem 2.3 appears in [2]. Theorem 2.3 Suppose that Ω is a Lip k -extension domain. If there are constants C and C 2, C 2 <, so that f(x ) f(x 2 ) C x x 2 k, for all x, x 2 Ω with x x 2 C 2 d(x, Ω), then there is a constant C 3, depending only on C, C 2,N and k, so that for all x, x 2 Ω. f(x ) f(x 2 ) C 3 x x 2 k, 5

6 3 Main Results Theorem 3. Assume that {u α } is a system of conjugate harmonic functions in U n defined in Ω and w A q M(Ω). In each case C is a constant that is independent of {u α }. a) For 0 < p <, M p (u α, x) C β α M p (u β, x) (3.) b) where C = C(p, q, M, n). u α BMO Ω,µ C u β BMO Ω,µ (3.2) β α where C = C(q, M, n). c) If Ω is a δ-john domain, 0 < p <, then u α p,ω,µ C β α u β p,ω,µ (3.3) where C = C(p, q, M, δ, n). d) If Ω is a Lip k -extension domain with constant N, 0 < k, then u α L k,ω C β α u β L k,ω (3.4) where C = C(N, n, k). Proof of Theorem 3.. We first prove c). Assertions a), b) and d) then follow. We use the following theorem which is a special case of Theorem 3. in [7] to supply a brief proof. The basic local results can be derived from the mean value property of harmonic functions and improvement of reverse Holder inequalities. The global result in John domains follows by patching together the weak local results and requires the special geometry of these domains ( see [6] and [4] ). Theorem 3.2 Suppose that 0 < p <, Ω is a δ-john domain, w A q M(Ω) and u and v are harmonic in Ω. If there is a constant A such that u 2,Q A v 2,2Q, (3.5) for all cubes Q with 2Q Ω, then there is a constant B, depending only on A,p,n,q,δ and M, so that inf u c c R p,ω,w B inf v c c R p,ω,w. (3.6) 6

7 Indeed if {u α } is a system of conjugate harmonic functions in Ω, then u = u α and v = u β are harmonic for each α. Moreover (.5) shows that (3.5) β α holds and so (3.3) follows from (3.6). Since cubes are John domains, a) and b) of Theorem 3. follow from c). Locally the Lipschitz norm u L k,ω, 0 < k, is equivalent to the norm sup Q Ω Q (k/n) u u Q,Q where the supremum is over all local cubes Q. See [5] for this result. Hence if Ω is a Lip k -extension domain then d) follows using Theorem 2.3. References [] J.E. Gilbert and M.A.M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge Univ. Press, Cambridge, 99. [2] F.W. Gehring and O. Martio, Lipschitz classes and quasiconformal mappings,ann. Acad. Sci. Fenn. Ser. AIMath.0 (985), [3] G.H. Hardy and J.E. Littlewood, Some properties of conjugate functions, J. Reine Agnew. Math.,67(932), [4] T. Iwaniec and C.A. Nolder, Hardy-Littlewood inequality for quasireguular mappings in certain domains in R n, Ann. Acad. Sci. Fenn. Series A.I. Math., 0(985), [5] N.G. Meyers, Mean oscillation over cubes and Holder continuity, Proc. Amer. Math. Soc., 5(964), [6] C.A. Nolder, A Privalov and a Hardy-Littlewood Theorem for Harmonic Functions and Quasiregular Mappings, Ph.D. dissertation, Univ. of Mich., 985. [7] C.A. Nolder, Hardy-Littlewood theorems for solutions of elliptic equations in divergence form, Indiana Univ. Math. Jour., 40(99), no., [8] C.A. Nolder, Hardy-Littlewood theorems for A-harmonic tensors,ill. Jour. Math.,43, no. 4,(Winter 999), [9] I.I. Privalov, Sur les fonctions conjugees,bull. Soc. Math. France,44 (96), [0] J. Ryan (editor), Clifford Algebras in Analysis and Related Topics, CRC Press, 996. [] J. Ryan and D. Struppa (editors), Dirac Operators in Analysis, Addison Wesley Longman, 998. [2] E.M. Stein and G. Weiss, On the theory of harmonic functions of several variables, I: The theory of H p -spaces, Acta Math, 03(960),