7. SOME APPLICATIONS TO HYPERCOMPLEX FUNCTION THEORY

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1 7. SOME APPLICATIONS TO HYPERCOMPLEX FUNCTION THEORY Objekttyp: Chapter Zeitschrift: L'Enseignement Mathématique Band (Jahr): 41 (1995) Heft 34: L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am: Nutzungsbedingungen Die ETHBibliothek ist Anbieterin der digitalisierten Zeitschriften. Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften. Die Rechte liegen in der Regel bei den Herausgebern. Die auf der Plattform eperiodica veröffentlichten Dokumente stehen für nichtkommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden. Das Veröffentlichen von Bildern in Print und OnlinePublikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt. Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber. Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit. Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem OnlineAngebot oder durch das Fehlen von Informationen. Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind. Ein Dienst der ETHBibliothek ETH Zürich, Rämistrasse 101, 8092 Zürich, Schweiz,

2 7. SOME APPLICATIONS TO HYPERCOMPLEX FUNCTION THEORY The Clifford algebra associated with R" endowed with the Euclidean metric is the enlargement of R* to a unitary algebra sd n not generated (as an algebra) by any proper subspace of R" and such that 2 x2x = \x\ 2, for By polarization, this identity becomes any xe R n. for any x,yer n. In particular, if {ej}] =l is the standard basis of R n one should hâve Consequently, ej =?1 and eje k =?e k ej for any j= k.ln particular, any élément u e srf n can be uniquely represented in the form u = X^ =o S '/ = k u i e i> w *tn u i e wnere e i stands for the product e i} e??? i2... e ik if /= (ii, / 2,..., 4) (we make the convention that e 0 := 1). More detailed accounts on thèse matters can be found in [BDS], [Mi], The higher dimensional analogue of the form dz extensively used in the complex analysis of one variable is the j^valued (n? l) For a compact Lipschitz domain Q in R", we let do stand for the usual surface measure induced on ôq by the Euclidean metric onr", and let N dénote the outward unit normal to Q defined daalmost everywhere on ÔQ. As R" C ja4, the vector valued function TV can also be regarded as a ja4 valued function on ÔQ. In fact, if i dénotes the inclusion of ôq into R", then An j/,, valued function u defined on an open subset Q of R" is called integrally continuons, etc, provided the sd n valued (n l) wco has the corresponding property. Recall the generalized CauchyRiemann operator Let &(Q) be as defined at the beginning of 6. We also make the following définition.

3 DEFINITION 7.1. (1) If u= I,' I ui e I is an srf n valued function defined on QçR" whose components (w 7)j are differentiable functions at a point aeq, then we define the action of D on u at aeq by (2) If u and f are two locally integrable.sd n valued functions on Q, then we say that Du =f in the distribution sensé on Q provided for any realvalued, smooth functions \ / 5 compactly supported in Q. (3) A locally (n \) integrable, srf n valued function u is called Clifford differentiable at aeq if the limit exists in sd n. The solutions of the (gêneralized) CauchyRiemann équations Du = 0 are called monogenic functions. The theorems we are about to describe now are more or less immédiate corollaries of the results obtained so far and we shall omit the proofs. Theorem 7.2. Let u be a integrally continuous.sd n valued function on the open set Q of R" C sd n. The following are équivalent. (1) There exists fe LL l l0c (Q,._<) such that for any Qe (2) There exists a realvalued, positive function gell (Q, loc) such that for any Qe

4 (3) For any Qe &(Q) and any s>o, there exists ô>o such that for any subdivision (2/)/e/ of Q and any Jçl such that (4) Thefunction u is Clifford differentiable almost everywhere on Q, u' is locally integrable on Q and for any Qe M(Q). (5) Thefunction u is Clifford differentiable almost everywhere on Q, u' is locally integrable on Q and for any compact Lipschitz domain KC Q. (6) Du, taken in the distribution sensé, belongs to Lloc (Q, stf n ). If thèse équivalent conditions arefulfilled, thenalso u' = Du a.e. on Q. Theorem 7.3. Let u be a n valued, uniformly (n \)integrable function in R n, which is absolutely continuous in the spécial Lipschitz domain Q of R" C sd n. Also, suppose that supp u is compact. u is integrable on bq, and that Du is integrable on Q. Then The next application is a refined version of the Pompeiu intégral repré sentationformula for j^valued functions ([Mo], [Te]). To this effect, we shall call a locally (n 1) integrable function u meancontinuous at aeq if Also, let (ù n stand for the area of the unit sphère in R n.

5 Theorem 7.4. Let Q be a compact d R" C sdn \)locally integrable function Lipschitz domain in (n be a n valued, uniformly continuons on Q and meancontinuous is absolutely which on R", we hâve almost everywhere on Q. Then, at almost every point aeq, and let u This extends the results in [Te], [Mo], [Bo], [BDS], [HL]. Moreover, a similar resuit is valid for the MartinelliBochner intégral représentation formula (cf. [HL]). Assume that Theorem 7.5. Q is an open subset of R" C sfn. Let u be a locally integrable, sdn valued function which is also locally X)integrable on Q. Let (C v ) v be an at most countable collection of {n closed subsets of Q such that each pair (u, C v ) satisfies one of the be a conditions (a) (y) stated in 6. Set A:= u v C v and also let f locally integrable srfn valued function on Q. Assume that at least one of the following conditions holds: A is closed, u is integrally continuons on Q\v4 and Du the distribution sensé on Q\^4; (1) (2) u'(x) = Then u is Clifford differentiable at f(x) for Du =f any point of each Q\A =f in and xe Q\A. in the distribution sensé on Q. / Note that, for = 0, Theorem 7.3 gives sufficient conditions for u to be monogenic. Thèse are substantially weaker than the ones presented in the literature (cf. e.g. [BDS]). In our final application we briefly explain how the above theorem extends to more gênerai linear first order differential operators. In doing so, it is convenient to slightly alter the définition of uniform locally and replace (4.1) by {n l) l)l) With this modification, the uniform locally {n condition becomes invariant under multiplication with locally bounded functions. function will be called locally integrally Also, a locally (n bounded in Q, if for any K e comp(q) there exist 0, k > 0 such that for

6 any Lipschitz (n C of R", C c X, with \x n\n \ l)submanifold (C) < 0 we hâve Consider now a linear, first order, differential operator where the ax,...,a.^valued functions x,...,a n are locally Lipschitz continuous on Q, and a 0 is a locally essentially bounded function on Q. Let P* stand for the formai transpose of P, i.e. Also, for any op(^)o ÇeR", the symbol of Pis defined by P (^) := Z" =1 Çye/tf/. Recall that for two.^valued, locally integrable functions u and f on Q we hâve that Pu = f in the distribution sensé, if for any realvalued test function \j/ on Q. Let übea locally (w? 1) integrable function on Q. We shall say that u is Pdifferentiable at x e Q provided that the limit exists in $/?. Proceeding as in Theorem 6.2, one can readily see that if u is actually differentiable at x e Q, and if ao(x)u(x)a then wis Pdifferentiable at x and Pu(x) = o (x)u(x) + Y," =laj(x)dju(x).1 aj(x)dju(x). The foliowing resuit is an extension of Theorem in [Hô]. Theorem 7.6. With the above définitions, consider u, two f locally integrable.<tf n valued functions on Q, and let (C v ) v be an at most countable collection of closed subsets of Q. Assume that u is also locally

7 integrally bounded. Suppose that at least one holds: for (1) (2) ao each =O conditions (a) Finally, set point of Pu =f v, and (y). the for of the following conditions pair (v, C v ) satisfies the condition (a); each v, the pair {v, C v ) satisfies one of the A:= u and assume that u is Pdifferentiable at each v Cv and that Pu{x) = f(x) for any xe Q\A. Then?l\A in the distribution sensé on Q. Let us finally note that, due to the noncommutativity of the Clifford 3, the results presented in this section are not in the most algebra s/n for n gênerai form. For instance, one could consider the Clifford differentiation operator defined for or dered pairs of j/^valued functions (u,v) by for which ail our techniques apply as well (cf. also [Hel,2]). However, leave the détails of this matter to the interested reader. we REFERENCES [BM] Bochner, Bochner, [BDS] Brackx, F., R. Delanghe and F. Sommen. Clifford Analysis. Pitman Adv. [Bo] GreenGoursat theorem. Math. Z. 63 (1955), and W.T. Martin. Several Complex Variables. Princeton Mathematical Séries, Vol. 10, Princeton Univ. Press, Princeton, N.J., S. S [Cr] Publ. Program, Craven, B.D. A note on Green's theorem. /. Austral. Math. Soc 4 (1964) [Fel] Fédérer, H. The GaussGreen theorem. Trans. Amer. Math. Soc. 58 (1965) [Fe2] [Fe3] [G] A note on the GaussGreen theorem. Proc. Amer. Math Soc 9 (1958) Géométrie Measure Theory. SpringerVerlag, Heidelberg, [Ha] Gheorghiev, G. L'évolution de la dérivée aréolaire en analyse hyper complexe.stud. Math. Bulgarica 11 (1991), Harrison, J. Stokes' theorem for nonsmooth chains. Bull Amer Math [HL] Henkin, G. M. and [H] [Hel] [He2] Soc. 29 (1993), J. Leiterer. Theory of Functions on Complex Manifolds. Monographs in Mathematics, Vol. 79, Birkhâuser Verlag, BaselBostonStuttgart, Henstock, R. A Riemann type intégral of Lebesgue power Canad J Math. 20 (1968), Hestenes, D. Multivector Calculus. J. Math. Anal. Appl. 24 (1968) Multivector Functions. /. Math. Anal. Appl. 24 (1968),