We will compare the foIlowing ideas: analytic hypoeIlipticity on open subsets of Euclidean space; global analytic hypoeIlipticity; analytic hypoeIlipticity in the sense of germs. We present a new operator which posseses Treves curves, yet is analytic hypoelliptic in the sense of germs. That is, the analog of the Treves conjecture, in the sense of germs, is false

Hanges, Nicholas

English

Cadernos Espinosanos (E-Journal)

Resenhas IME-USP 2002, Vol. 5, No. 3, 159 - 169. Remarks on Analytic Hypoellipticity 1 Nicholas Hanges Abstract: We will compare the foIlowing ideas: analytic hypoeIlipticity on open subsets of Euclidean space; global an-alytic hypoeIlipticity; analytic hypoeIlipticity in the sense of germs. We present a new operator which posseses Treves curves, yet is analytic hypoelliptic in the sense of germs. That is, the analog of the Treves conjecture, in the sense of germs, is false. Key words: Analytic hypoellipticity, sum of squares. 1 Introduction The problem of analyticity of solutions of partial differential equations is one of the oldest in analysis. lndeed , Hilbert [19) wrote the following , concerning his 19th problem: "One of the most remarkable facts in the elements of the theory of analytic functions appears to me to be this: That there exist partial differential equations whose integraIs are ali of necessity analytic functions of the independent variables, that is, in short, equations susceptible of none but analytic solutions. The best known partial differential equations of this kind are the potential equation and certain linear differential equations investigated by Picard; 11 In today's language we say that the Laplace operator is analytic hypoelliptic. This is a property it shares with all elliptic linear partial differential equations with analytic coefficients. When one leaves the class of elliptic equations, the question of analytic regularity becomes quite complicated . We begin by presenting a short survey comparing various notions of analytic hypoellipticity. This is the foeus of Sections 2 through 5. In Section 6 we present a new operator 1i on 1R3 . The operator 11. has an infinite number of Treves curves over the origin, yet is analytic hypoelliptie at the origin in the sense of germs. We state our results concerning 1i in Section 7. We give a short outline of the proofs in Section 8. Complete details will appear elsewhere. 1 This article is an expanded version of a lecture gi ven at the Workshop on Geometric Analysis of PDE and Several Complex Variables , Serra Negra SP, Brazil, August 2001. We thank the organizers for their generosity and hospitality. 159 160 Nicholas Hanges 2 Sums of Squares and Analytic Hypoellipticity Let M be a real analytic manifold and let Xl, ... X r be real analytic vector fields on M. We study an operator P of the form "sum of squares". That is P has the form P = Xl 2 + ... + X r 2. Definition 1 We say that P is analytic hypoelliptic (in the strong sense) on M if for every open O C M we have the following: Pu analytic on O implies that u is analytic on O. Here u is a distribution on O. We always assume that the X j satisfy a "finite type" condition. That is, at each point of M, the Lie algebra generated by the Xj (under the commutation bracket) has dimension equal to dimM. Under these conditions a classical result of Hormander [21] guarantees the hypoellipticity of P. However analytic hypoellipticity will not hold unless further assumptions are made. If we assume that E, the characteristic set of P, is a symplectic manif~ld and that the principal symbol of P vanishes precisely to second order on E, then P is analytic hypoelliptic. This follows from a result of Treves [27] and Tartakoff [26]. Further work in this direction around this time was done by Métivier and Sjostrand. See for example [23], [24]. When E is not symplectic, analytic hypoellipticity may fail. We have the following example due to Baouendi - Goulaouic [1]. Consider the operator on ~3 given by (1) If we use natural coordinates (x, y, t; ç, TJ, r) on the cotangent bundle T* (~3), we see that the characteristic set E for B is defined by the equations ç = TJ = x = o. E is not symplectic and B is not analytic hypoelliptic on any open set that inter-sects x = O. Note that E = {f = 9 = h = O} with df, dg, dh independent and the following condition on the Poisson brackets {J,g} = 1, {f, h} = O, {g, h} = O. E is foliated by the integral curves of the Hamilton field of h (bicharacteristics of h). We will say that such a E is of Baouendi - Goulaouic type. These bicharacteristics are examples of what are now called Treves curves. Indeed, let (0,1) C ~ denote the open unit interval. We have the following : Remarks on Analytic Hypoellipticity 161 Definition 2 Let E C T* M be an analytic submanifold and let , : (0,1) -+ E be a non-constant analytic curve. We call, a Treves curve for E if (2) for ali t E (0,1). Note that T""Y(t)~ is the tangent space to E at the point ,(t) and (T-y(t)E).l is the orthogonal space with respect to the symplectic formo In [27] Treves conjectured that when the characteristic set E is a manifold and contains such curves, the associated operator is not analytic hypoelliptic. Later [28], Treves extended his conjecture. Both conjectures are still open at this time. Recent progress has been made by Chanillo [4]. Next we have the example of Métivier [22]: M = âx 2 + (x 2 + l)ây 2. (3) M is not analytic hypoelliptic on any open set containing the origino (M is elliptic away from the origin.) Note that the characteristic set Eis given by E={Ç"=x=y=O}, and hence is of Baouendi-Goulaouic type. 3 Global Analytic Hypoellipticity Definition 3 We say that P is globally analytic hypoelliptic on M if for every distribution u on M we have the following: Pu analytic on M implies that u is analytic on M. We have the surprising fact that the Baouendi - Goulaouic operator is globally analytic hypoelliptic on the 3 dimensional torus 1f3. This result is due to Cordaro - Himonas [12]. Other interesting global results have been obtained by Himonas - Petronilho [20] and by Bergamasco - Cordaro - Malagutti [3] . Later Tartakoff [25] proved that the Baouendi-Goulaouic operator B=âx2+â/+x2ât2 is globally analytic hypoelliptic on Here §l is the circIe in y space and V is any open set in (x, t) space. Note that here the Treves curves are compacto 162 Nicholas Hanges 4 Several Complex Variables We turn to related results fram the theory of functions of several complex variables. We begin with the results of Chen [6] and [5]. Theorem 1 Let n 2: 2. Let 1) C C" be a bounded complete Reinhardt domain with real analytic pseudoconvex boundary. Then (1) The 8- Neumann problem is analytic hypoelliptic. (2) The Szegõ projector preserves analyticity. Similar results in this direction were obtained by Derridj and Tartakoff [13]. Note that Christ [9] has shown that there exist domains in CZ as above, such that the associated "sum of squares" is not analytic hypoelliptic (in the strong sense). Here Treves curves exist and are compacto Christ [11] has also shown the existence of bounded pseudoconvex domains with analytic boundary such that the Szego projector does not preserve analyticity. The analytic singularities of the Bergman and Szego kernels are determined by the Treves curves for tube domains in CZ, Francsics - Hanges [14], [15] . .Jt is not clear how these off-diagonal singularities affect the mapping properties of these kernels. 5 Analytic Hypoellipticity in the Sense of Germs Definition 4 We say that P is analytic hypoelliptic at x E M if we have the following: Pu analytic near x implies that u is analytic near X. Here u is a distribution defined near X. Consider the tube domain Here m 2: 3 is an odd integer. If we parametrize the boundary using Xl = lRzI , YI = S.SzI, X2 = lRz2 , we see that the natural CR structure is generated by the vector field 8 m-l 8 . 8 L= -8 +mYI -8 +l-8 . Xl x2 YI (4) L is analytic hypoelliptic at any point where YI = O. On the other hand, L is not analytic hypoelliptic at any point where YI i- O. Much more general results of this nature were obtained by Baouendi and Treves [2]. AIso observe that the operator LL is analytic hypoelliptic at any point where YI = O. Remarks on Analytic Hypoellipticity 163 On the other hand, the sum of squares is not analytic hypoelIiptic at any point where YI = O. This is due to Grigis -S}ostrand [16] when m = 3 and to Hanges - Himonas [18] for general odd m. A proof valid for alI m > 3 was later given by Christ [8] . Observe that T and LL differ by a first order operator. It would be interesting to study the influence of lower order terms on regularity questions. AIso note that the characteristic set here has codimension 2. However the set of characteristic points that are "weakly pseudoconvex" is a manifold of Baouendi-Goulaouic type. 6 A New Example Let (Xl, X2, t) be coordinates on ]R3 and let where and Ô 2 (ô 2 6 x = (-ô ) + -ô ) . Xl x2 Note that 1l is hypoelIiptic with 10ss of one derivative, see [21]. The charac-teristic set is given by Observe that ç =j:. O on I;, hence I; is of Baouendi- Goulaouic type. As a conse-quence I; is foliated by Treves curves. 7 Treves Curves for 1-l and Statements of Results The Treves curves (bicharacteristics) are contained in I; and are integral curves of the non-degenerate vector field which is the Hamilton field of the function xl6 - x26. 164 Nicholas Hanges Characteristic points have the form where OcO OCO ° X1<'2 - X2<'1 = . The Treves curve through such a point is given by the equations t = 0, r = ° and where ° < s < 211". Xl (s) = X~ cos(s) - xg sin(s) X2( s) = xg cos(s) + x~ sin(s) 6(s) =efcos(s) -egsin(s) 6(s) = eg cos(s) + ef sin(s) Note that these curves are always compacto When X O :/= 0, each curve, is the intersection of ~ with the torus When x O = 0, each curve is a circle in espace, centered at 0, with radius equal to leol. In particular, the origin is the projection of an infinite number of compact Treves curves. Given p, q > ° we define We have the following results concerning the operator 1i. Theorem 2 Let n C IR 3 be open. lf n intersects the hyperplane {t = O}, then 1i is not analytic hypoelliptic (in the strong sense) on n . Theorem 3 The operator 1i is globally analytic hypoelliptic on np ,q for every p,q> o. Theorem 4 The operator 1i is analytic hypoelliptic at the origino Note that Theorem 2 is consistent with the Treves conjectures. Also observe that Theorem 4 is an immediate consequence of Theorem 3. Even though the origin is the projection of an infinite number ofTreves curves, it is impossible to find a solution that is singular there. That is, the analog of Treves conjecture in the sense of germs, is false. Remarks on Analytic Hypoellipticity 165 8 Construction of the Inverse of 1l If u E S, the Schwartz space of rapidly decreasing functions on lR 3 , we define the partial Fourier transform by u(ç, t) = [ e-i<x,ç>u(x, t)dx. 1J'i.2 We will use polar coordinates (p, <p) in ç space, that is 6 = pcos(<p),6 = psin(<p). If we have u, I E S such that 1lu = I, then this is equivalent to 82 '11 ( ) 82 '11 ( ) 2 2 , ( ) - 1' ( ) 8<p2 p, <p, t + 8t 2 \p, <p, t - t P u p, <p, t - p, <p, t . Next we expand '11, j in Fourier series 00 u(p, <p, t) = L Uk (p, t)eik'P, k=-oo 00 j(p, <p, t) = L A (p, t)eik'P. k=-oo It follows that where 82 2 2 2 H p2,k2 = 8t2 - P t - k . Note that Hp2,k2 has an inverse on S(lR) for all p :I O and k E lR. Indeed, an explicit integral formula exists for the inverse, see for example [27]. Hence we have where Kk,p(t, t') is the distribution kernel of the inverse of Hp2,k2. It follows that the solution u is given by u(x, t) = 100 121T ei(x,pCOS(<P)+X2psin(<p)) :(!-. eik<p 100 K (t t') i (p t')dt,pdpdt.p ~ _ k,p, Jk, (211')2 . o o k=-oo 00 We may write u = f-/, where we define (f-/)(x,t) = [ E(x,t;x',t')/(x',t')dx'dt' Jff.3 166 Nicholas Ranges where the kernel Eis determined from above. There are several ways to write E. One way to write E is in terms of Bessel functions of integral order. Recall that satisfies Bessel's equation Hence it follows that E(x, t; x', ti) = 100 k'f;;oo eik (8-8') Jk (rp)Jk (ri p)Kk,p(t, ti) ~;. Note that we have used the notation Xl = rcos(B),X2 = rsin(B) with similar notation for x'. The following formula is useful in obtaining estimates that are uniform in k and p: Kk,p(t, ti) = (1/ v7iJ 100 e-$k 2 / p K 1 (s; tvp, ti v'PJds. Here K 1 is the inverse for the Hermite operator It follows from our construction and classical estimates, see for example [27], that we have Proposition 1 1{ is an isomorphism 01 the Schwartz space S(]R3) with inverse [. An important preliminary observation is the following Proposition 2 Let (x, t; x', ti) E ]R3 X ]R3. Assume that either t =1= ti or Ixl =1= Ix'l. Then E is analytic near (x, t; x', ti). The results of Section 7 follow from these Propositions. Complete details will appear elsewhere. Remarks on Analytic Hypoellipticity 167 References [1] M. S. Baouendi and G. Goulaouic, Non-analytic hypoellipticity for some de-generate operators, Bull. Amer. Mat. Soco 78 (1972) , pp.483- 486. [2] M. S. Baouendi and F. Treves, A microlocal version of Bochner 's tube theo-rem, Indiana Univ. Math. J. 31 (1982), pp.885-895. [3] A. P. Bergamasco, P. D. Cordaro, P. A. Malagutti , Globally hypoelliptic sys-tems of vector fields, J . of Func. Anal. 114, pp.267- 285. [4] S. Chanillo, Kirillov theory, Treves strata, Schrodinger equations and analytic hypoellipticity of sums of squares, preprint. [5] S.C. Chen, Real analytic regularity of the Szegõ projection on circular do-mains, Pacific J. Math. 148 (1991) , pp.225- 235. [6] S.C. Chen, Global analytic hypoellipticity of the [j-Neumann problem on cir-cular domains, Invent. Math. 92(1988) , pp .173-185. [7] M. Christ, A necessary condition for analytic hypoellipticity, Mathematical Research Letters, 1 , pp.241-248, (1994). [8] M. Christ, Analytic hypoellipticity, representations of nilpotent groups, and a nonlinear eigenvalue problem, Duke. Math. J. (1993) , no. 3, pp.595-639 . [9] M. Christ, Analytic hypoellipticity breaks down for weakly pselldoconvex Rein-hardt domains, Internat. Math . Res. Notices 1991, no. 3, pp.31-40. [10] M. Christ, Remarks on the breakdown of analyticity for 8b and Szego kernels, ICM - 90 Satellite Conference Proceedings, Harmonic Analysis, Springer -Verlag, 1991. [11] M. Christ , The Szegõ projection need not preserve global analyticity, AnnaIs of Math. 143 (1990) , pp .301- 330. [12] P. D. Cordaro and A. A. Himonas, Global analytic hypoellipticity for a class of degenerate elliptic operators on the torus, MRL 1 (1994), pp.501-510. [13] M. Derridj and D. Tartakoff, Microlocal analyticity for the canonical solution to 8b on some rigid weakly pseudoconvex hypersurfaces in C2, Comm. PDE 20 (1995), pp.1647-1667. 168 Nicholas Hanges [14] G. Franesies and N. Hanges, Analytic singularities of the Bergman kemel for tubes, Duke Mathematieal Journal, 8 (3) (2001) , pp.539- 580. [15] G. Franesies and N. Hanges, Treves curves and the Szegõ kemel, Indiana University Mathematies Journal , 47 (1998), pp.995-1009. [16] A. Grigis and J. Sjostrand, Front d'onde analytique et sommes de carres de champs de vecteurs, Duke Math. J. 52 (1985), pp.35- 51. [17] N. Hanges and A. A. Himonas, Analytic hypoellipticity for generalized Baouendi - Goulaouic operators, Journal of Functional Analysis, 125 (1)(1994) , pp.309-325. [18] N. Hanges and A. A. Himonas, Singular solutions for sums of squares of vector fields , Comm. PDE 16 (8/9) (1991) pp.1503-1511 . [19] D. Hilbert, Mathematical problems, Bulletin of the AMS, 37(4)(2000) , pp.407- 436. [20] A. A. Himonas and G. Petronilho, On global hypoellipticity of degenerate elliptic operators, Math . Z. 230 (1999) , pp. 241-257. [21] L. Hormander, Hypoelliptic second order differential equations, Aeta. Math. 119 (1967) , pp.147-171. [22] G. Métivier , Non-hypoellipticité analytique pour D; + (x2 + y2)D~ , C. R. Aead. Sei. Paris Sér. I Math. 292 (1981) , no. 7, pp.401- 404. [23] G. Métivier , Une classe di opérateurs non hypoelliptiques analytiques, Indiana Univ. Math. J. 29 (1980) , pp.823 - 860. [24] J. Sjostrand, Analytic wavefront sets and operators with multiple character-istics, Hokkaido Mathematieal Journal , 12 (1983) pp.392- 433. [25] D. Tartakoff, Global (and local) analyticity for second order operators con-structed from rigid vector fields on products of to ri, Trans. AMS 348 (1996), 2577- 2583. [26] D. Tartakoff, On the Local Real Analyticity of Solutions to Ob and the ã Neumann Problem, Aeta. Math . 145(1980) pp.117- 204. Remarks on Analytic Hypoellipticity 169 [27) F. Treves, Analytic hypoellipticity of a class of psel.ldodijJerential operators with dOl.lble characteristics and applications to the 8 - Nel.lmann problem, Communications in PDE 3 (1978) , pp.475- 642. [28] F . Treves, Symplectic geometry and analytic hypo-ellipticity, Proceedings of Symposia in Pure Mathematics, 65 (1999) , pp.201-219. Nicholas Hanges Department of Mathematics Lehman College - CUNY Bronx, New York 10468 nhanges@alpha.lehman.cuny.edu USA 

Remarks on Analytic Hypoellipticity

https://www.revistas.usp.br/resenhasimeusp/article/download/75049/78606

Remarks on Analytic Hypoellipticity

Abstract

Similar works

Full text

Available Versions

Cadernos Espinosanos (E-Journal)