Cauchy-Riemann distributions and boundary values of analytic functions
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1 Research Collection Doctoral Thesis Cauchy-Riemann distributions and boundary values of analytic functions Author(s): Straube Emil J. Publication Date: 1983 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library
2 Diss. ETH No CAUCHY-RIEMANN DISTRIBUTIONS AND BOUNDARY VALUES OF ANALYTIC FUNCTIONS A dissertation submitted to the SWISS FEDERAL INSTITUTE OF TECHNOLOGY ZURICH for the degree of Doctor of Mathematics presented by Emil J. Straube dipl. math. ETH born August citizen of Alt St. Johann SG accepted on the recommendation of Prof. Dr. K. Osterwalder examiner Prof. Dr. J. Moser co-examiner 1983
3 Abstract Let M be a generating C Cauchy-Riemann (= CR) manifold imbedded in C «?(M) the usual space of testfunctions and V' (M) its dual. We give a coordinate free defini tion of CR-distributions which includes the ordinary CRfunctions as well as the so-called "weak solutions" of the tangential CR-equations. In a neighborhood U(P) of P c M let M be given by Ys = ys(zx' zm'xm+l'"*'xn* ' s = m + l''''n ' wnere m is the complex dimension of T (M) (z...z ) e Q (x...x ) e 0.We prove the following regularity re sults: a) any CR-distribution on fi *fi is in fact in C (Q J'(Q )) b) any sequence of CR-distributions con verging in P'(K *fl ) converges in C (!) ^(il)). Con cerning the structure of CR-distributions we have: locally any CR-distribution is the derivative of some say C CR-function where only "restrictions of the complex deriva- 9 3 tions tt..- * to the manifold" are involved. 3z 3z m+1 n Let now M be a hypersurface f analytic in a one sided neighborhood of P M. We say that f admits a distribution boundary value in Q '(ft xft.) if lim J" f(z+en(z))mz) <3F (z) n+ M exists for all < > e *){Q x 0.) ; here n {z) denotes the normal to M at z & M df the volume element on M = (z+cnlz) z e M). These boundary values are easily seen to be CR-distributions. As a corollary of the above mentioned Structure Theorem we get: if M is such that any suitably regular CR-function can be continued analytically to one side of M then the same is true of CR-distributions. The analytic functions f which admit a boundary value are
4 characterized by either one of the following two properties: a) polynomial boundedness in l/dist(z^m) near M b) local existence of an analytic primitive of f which is continuous up to M. Finally we define distribution boundary values on manifolds of higher codimension and apply our techniques to extend the "Edge-of-the-Wedge" Theorem to the case of gener ating edges (but otherwise of arbitrary dimension).
5 Zusammenfassung Sei M eine erzeugende ("generating") C -glatte Cauchy- Riemann (= CR) -Mannigfaltigkeit die in C eingebettet 1st S(M) der Raum der C Funktionen mit kompaktem Trager 8)' (M) der dazugehsrige Dualraum. Wir geben eine koordinatenfreie Definition fur CR-Distributionen welche sowohl die klassischen CR-Funktionen als auch die sogenannten "schwachen LSsungen" der tangentialen CR-Gleichungen umfasst. durch In einer Umgebung U(P) von P e. M sei M gegeben ys = ys(zl VXm+l V s = m + 1 n. Hier bezeichnet m die komplexe Dimension von C ferner Tp(M) sei (z...z ) en (x...x ) e«.. Wir zeigen: 1 mi m+1 n 2 a) jede CR-Distribution auf n «!1 gehort zu C (SI 3>' (n.)) b) jede Folge von CR-Distributionen die in id' (Q «8J konvergiert konvergiert in C (n i?'(n2)). Was die lokale Struktur von CR-Distributionen anbelangt gilt: Lokal 1st jede CR-Distribution Ableitung einer C (zum Beispiel) CR-Funktion. Dabei kommen nur Ableitungen vor die "Restriktionen auf M " der komplexen Ableitungen sind. o Z. 0 Z. m+1 n Nun sei M eine Hyperflache f analytisch in einer einseitlgen Umgebung von P M. Wir sagen f habe in?'(n xn ) einen Distributionenrandwert falls lim / f(z+cn(z))*(z) df (z) n+ M existiert fiir alle <(i t 0* (n x8j? dabei bezeichnen n(zo die Normale zu M in z^ e. M df das Flachenelement auf M. = {z+cn(z) z_ e M). Man sieht leicht dass diese Randwerte CR-Distributionen sind. Der oben erwahnte Struktursatz liefert sofort: falls M die Eigenschaft hat dass jede geniigend glatte CR-Funktion analytisch nach einer Seite von M fortgesetzt werden kann dann ksnnen auch CR-Distributionen fort-
6 gesetzt werden. Analytische Funktionen f welche einen Distributionenrandwert zulassen werden durch jede der folgen- den zwei Eigenschaften charakterisiert: a) polynomiale Beschranktheit in l/dist(z^m) in der Nahe von M b) lokale Existenz einer analytischen primitiven Funktion F von f welche stetig nach M fortgesetzt werden kann. Schliesslich definieren wir Randwerte auch fiir Mannigfaltiqkeiten hsherer Codimension und benutzen unsere Methoden um das "Edge-of-the- Wedge" Theorem auszudehnen auf den Fall wo die "Kante" erzeugend ("generating") sonst aber von beliebiger Dimension 1st.
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