ON ANALYTIC MICROHYPOELLIPTICITY OF LINEAR PARTIAL DIFFERENTIAL OPERATORS OF PRINCIPAL TYPE (FOURIER, TRANSFORM, PHASE, SYMBOL)

ALEXANDROS TAXIARHIS HIMONAS, Purdue University

Abstract

We consider a linear partial differential operator P = P(x,D) with real-analytic, complex-valued coefficients defined near x(,0) (ELEM) (//R)('n). We denote by p(,m)(x,(xi)) = A(x,(xi)) + iB(x,(xi)) its principal symbol and let (xi)('0) (ELEM) (//R)('n) - 0 be such that p(,m)(x(,0),(xi)('0)) = 0 and grad(,(xi))p(,m)(x(,0),(xi)('0)) (NOT=) 0. In fact we can assume grad(,(xi))A(x(,0),(xi)('0)) (NOT=) 0. In chapter (I) we prove that if H(,A)B(x(,0),(xi)('0)) = 0 and H(,A)H(,A)B(x(,0),(xi)('0)) (NOT=) 0 or H(,B)H(,A)B(x(,0),(xi)('0)) (NOT=) 0, then P is analytic hypoelliptic at (x(,0),(xi)('0)). In Chapter (II) we prove that, if there is an even integer k (GREATERTHEQ) 2 satisfying: (1) H(,A)('j)B(x(,0),(xi)('0)) = 0 for 0 (LESSTHEQ) j < k and H(,A)('k)B(x(,0),(xi)('0)) (NOT=) 0; (2) H(,H(,A))j(,B)(x(,0),(xi)('0)) is collinear to H(,A)(x(,0),(xi)('0)) for 0 (LESSTHEQ) j < k/2. Then P is analytic hypoelliptic at (x(,0),(xi)('0)). In the Appendix we use the same method to give a proof to a well known result, namely: If H(,A)B(x(,0),(xi)('0)) > 0 than P is again analytic hypoelliptic at (x(,0),(xi)('0)). The hypotheses of the above results expressed in terms of Hormander's numbers k = k (x(,0),(xi)('0)),A,B and s = s (x(,0),(xi)('0)),A,B , correspond to the cases k = 2, k even (GREATERTHEQ) 2, s (GREATERTHEQ) k/2 and k = 1 respectively.

Degree

Ph.D.

Subject Area

Mathematics

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