平方和算子的分析

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 12, December 1998, Pages 4993–5001 S 0002-9947(98)01987-4 GLOBAL ANALYTIC REGULARITY FOR SUMS OF SQUARES OF VECTOR FIELDS PAULO D. CORDARO AND A. ALEXANDROU HIMONAS Abstract. We consider a class of operators in the form of a sum of squares of vector �elds with real analytic coe�cients on the torus and we show that the zero order term may influence their global analytic hypoellipticity. Also we extend a result of Cordaro-Himonas. 1. Introduction and Results Let Ω be an open set in RN , or more generally a real analytic manifold, and A(Ω) be the set of real analytic functions in Ω. We shall consider operators of the form P = − ν∑ j=1 X2j + X0 + a,(1.1) where X0, . . . , Xν , are real vector �elds with coe�cients inA(Ω), and a is a complex valued function in A(Ω). We shall discuss the analytic regularity of the solutions to the equation Pu = f , for a given function f 2 A(Ω). To be more precise and to state our results we shall need the following de�nitions. We recall that the operator P is said to be analytic hypoelliptic (hypoelliptic) in Ω if for any U open subset of Ω the conditions u 2 D0(U) and Pu 2 A(U) (Pu 2 C1(U)) imply that u 2 A(U) (u 2 C1(U)). The operator P is said to be globally analytic hypoelliptic (hypoelliptic) in Ω if the conditions u 2 D0(Ω) and Pu 2 A(Ω) (Pu 2 C1(Ω)) imply that u 2 A(Ω) (u 2 C1(Ω)). Also, we recall that a point x0 2 Ω is of finite type if the Lie algebra generated by the vector �elds X0, � � � , Xν spans the tangent space of Ω at x0. By the celebrated sum of squares theorem of Ho¨rmander [Ho] the �nite type condition is su�cient for the hypoellipticity of P in the more general case where P has C1 coe�cients, while in the analytic category, which is our situation here, Derridj [D] proved that the �nite type condition is also necessary for hypoellip- ticity. Baouendi and Goulaouic [BG] discovered that the �nite type condition is not su�cient for the analytic hypoellipticity of P . They showed that if P is the operator in R3 de�ned by P = (∂x)2 + (x∂y)2 + (∂t)2, then the equation Pu = 0 has a non-analytic solution near x = 0. After, several authors including Hel�er Received by the editors January 23, 1996 and, in revised form, November 26, 1996. 1991 Mathematics Subject Classification. Primary 35H05, 35N15; Secondary 32F10, 58G15. Key words and phrases. Analytic hypoellipticity, global, torus, sum of squares of vector �elds, �nite type, subelliptic. The �rst author was partially supported by CNPq Grant 304825/89-1, and the second author by NSF Grant DMS 91-01161. c©1998 American Mathematical Society 4993 4994 PAULO D. CORDARO AND A. ALEXANDROU HIMONAS [H], Pham The Lai-Robert [PR], Metivier [M1], Hanges-Himonas [HH1], [HH2], and Christ [Ch1], [Ch2] found di�erent classes of operators satisfying the �nite type condition and failing to be analytic hypoelliptic. In [CH], most of these classes of operators were proved to be globally analytic hypoelliptic on the torus. The pur- pose of this article is to extend Theorem 1.1 in [CH] for the case where lower order terms are present, and to show that if the vector �eld X0 in (1.1) is complex, then the zero order term, a, may influence the global analytic hypoellipticity of P . We start with the extension of a result in [CH]. Theorem 1.1. Let P be an operator of the form (1.1) on the torus TN = Tm�Tn, with variables (x, t), x = (x1, . . . , xm), t = (t1, . . . , tn), and Xj = n∑ k=1 ajk(t) ∂ ∂tk + m∑ k=1 bjk(t) ∂ ∂xk , j = 0, . . . , ν, are real vector fields with coefficients in A(Tn), and a = a(x, t) 2 A(Tm+n) is complex-valued. If the following two conditions hold: (i) Every point of Tm+n is of finite type; (ii) The vector fields ∑n k=1 ajk(t) ∂ ∂tk , j = 1, . . . , ν, span Tt(Tn) for every t 2 Tn, then the operator P is globally analytic hypoelliptic in TN . Remark. A generalization of [CH] has been also obtained by Christ [Ch3] under the assumption of a certain symmetry condition, which does not hold here because of the dependence of a on x. A di�erent generalization has been proved by Tartako� [T3] under the restriction ν = n, but with P in a more general form and assumed to satisfy a maximal estimate. However his method could be used for Theorem 1.1 too. Also, we mention the related work of Chen [C], Komatsu [Ko], Derridj-Tartako� [DT], Metivier [M2], Sjo¨strand [S], Tartako� [T1], [T2], and Treves [T1]. Theorem 1.1 is only a partial result on the problem of global analytic hypoellipticity and there is no doubt that more general results are valid, although it is far from clear what is a necessary and su�cient condition for global analytic hypoellipticity. Next, in the 2-dimensional torus we consider the case where in (1.1) X0 is com- plex. While in the above theorem the zero order term did not play any role, we shall show that this is not the case in the following situation. In T2 let P be the operator de�ned by P = −�LL + a, a 2 C,(1.2) where L = ∂t + ib(t)∂x, with b 2 A(T1), and real-valued.(1.3) Then we have the following results. Theorem 1.2. Let t0 2 T1 be a zero of b of odd order. If a 2 C − f0g, then the operator P defined by (1.2) is analytic hypoelliptic near T1 � ft0g. Theorem 1.3. Let P be as in (1.2). If all zeros of b are of odd order and if a 2 C− f0g, then P is globally analytic hypoelliptic. Conversely, if b has a zero of odd order and if a = 0, then P is not globally analytic hypoelliptic Such phenomena have been studied in the past for an operator on the Heisenberg group related to the Lewy operator by Stein [St], and Kwon [Kw]. REGULARITY FOR SUMS OF SQUARES OF VECTOR FIELDS 4995 2. Proof of Theorem 1.1 We start with a lemma about a global subelliptic estimate. Lemma 2.1. Let Xj , j = 0, . . . , ν, be real C1 vector fields in TN and a 2 C1(TN ). If all points of TN are of finite type for X0, . . . , Xν , then there exist ε > 0 and C > 0 such that kukε � C (kPuk0 + kuk−1) , u 2 C1(TN ),(2.1) where P is of the form (1.1). Proof. Since the �nite type condition holds at every point, there exists a local subelliptic estimate near each point (see [Ho], [K], [OR], [RS]) and this implies that the following property holds true: u 2 H0(TN ), Pu 2 H0(TN ) =) u 2 H�(TN ). Then by the closed graph theorem the following global estimate holds: kukε � C1 (kPuk0 + kuk0) , u 2 C1(TN ),(2.2) for some ε > 0 and C1 > 0. By Lions’ Lemma for any δ > 0 there exists Cδ such that kuk0 � δkukε + Cδkuk−1, u 2 C1(TN ).(2.3) Applying (2.3) in (2.2) and selecting δ appropriately small give (2.1). The proof of Lemma 2.1 is complete. Now let u 2 D(TN ) such that Pu = f, with f 2 A(TN ).(2.4) By Ho¨rmander’s theorem u 2 C1(TN ). To show that P is globally analytic hy- poelliptic in TN it su�ces to show that u 2 A(TN ).(2.5) Since by our hypothesis P is elliptic in t, it su�ces to show that there exists B > 0 such that k∂αx uk0 � Bjαj+1α!, 8α 2 Nm0 .(2.6) Since a and f are in A(TN ), there exists A > 0 such that k∂αx ak1 � Ajαj+1α!, α 2 Nm0 ,(2.7) and k∂αx fk0 � Ajαj+1α!, α 2 Nm0 .(2.8) Since kuk0 � kukε, the basic inequality (2.1) implies the following weaker inequality: kuk0 � C(kPuk0 + kuk−1), u 2 C1(TN ),(2.9) which is what we need for proving (2.6). If we apply (2.9) with u replaced with ∂αx u, then we obtain k∂αx uk0 � C(k∂αx Puk0 + k[P, ∂αx ]uk0 + k∂αx uk−1).(2.10) We have k∂αx uk−1 � k∂α−ejx uk0,(2.11) 4996 PAULO D. CORDARO AND A. ALEXANDROU HIMONAS where ej is an element of the orthonormal basis of Rm such that the corresponding αj � 1. Also, by their form Xj , j = 0, . . . , ν, commute with ∂αx and we have [P, ∂αx ]u = a∂ α x u− ∂αx (au) = − ∑ β<α ( α β ) ∂α−βx a∂ β x u. Therefore k[P, ∂αx ]uk0 � ∑ β<α ( α β ) k∂α−βx ak1k∂βxuk0. Then by using (2.6) and (2.7) we obtain k[P, ∂αx ]uk0 � α! ∑ β<α Ajα−βj+1Bjβj+1.(2.12) By (2.8), (2.10), (2.11) and (2.12) we obtain k∂αx uk0 � C Ajαj+1α! + α! ∑ β<α Ajα−βjBjβj+1 + Bjαj(α− ej)!  .(2.13) We look for B of the form B = MA, for some M > 1,(2.14) such that (2.6) holds. By (2.13) it su�ces to choose M such that for all α 2 Nm0 we have C(Ajαj+1α! + α! ∑ β<α Ajα−βj+jβj+2M jβj+1 + AjαjM jαj(α− ej)!) � Ajαj+1M jαj+1α! By simplifying we obtain that the last inequality follows from C 1 + AM ∑ β<α M jβj + 1 A M jαj  � M jαj+1.(2.15) Since for M > 1 we have∑ β<α M jβj � [( M M − 1 )m − 1 ] M jαj,(2.16) by (2.16) we see that for (2.15) to hold it su�ces that C ( 1 M jαj+1 + A [( M M − 1 )m − 1 ] + 1 AM ) � 1.(2.17) Since the left-hand side of (2.17) goes to zero as M goes to in�nite, we conclude that there exist M > 1 such that (2.17) holds. And therefore (2.6) holds with B = MA. This completes the proof of Theorem 2.1. REGULARITY FOR SUMS OF SQUARES OF VECTOR FIELDS 4997 3. Proof of Theorems 1.2 & 1.3 We start with a being a function of t; i.e. in T2 we consider the operator P = −�LL + a, a = a(t) 2 A(T1),(3.1) where L = ∂t + ib(t)∂x, with b 2 A(T1), and real-valued. We shall work near a zero of b(t), which for simplicity we will assume to be t = 0. Then we may assume that b(t) = tkg(t), g(t) 6= 0, −δ � t � δ, some δ > 0.(3.2) If we expand P , we obtain P = −∂2t − b2(t)∂2x − ib0(t)∂x + a(t). If u 2 C1(Tt,D0(Tx)), then by taking Fourier transform with respect to x we obtain P̂ u(ξ, t) = −u^tt(ξ, t) + [ξ2b2(t) + ξb0(t) + a(t)]u^(ξ, t). If we multiply by Z u and integrate in t 2 (−δ, δ), then we obtain∫ δ −δ P̂ u(ξ, t) Z u(ξ, t)dt = − ∫ δ −δ u^tt(ξ, t) Z u(ξ, t)dt + ∫ δ −δ [ξ2b2(t) + ξb0(t) + a(t)]ju^(ξ, t)j2dt. Then we integrate by parts and use the Cauchy-Schwarz inequality to obtain:∫ δ −δ ju^t(ξ, t)j2dt + ∫ δ −δ [ξ2b2(t) + ξb0(t)]ju^(ξ, t)j2dt � ∫ δ −δ [ 1 2 − a(t)]ju^(ξ, t)j2dt + 1 2 ∫ δ −δ jP̂ u(ξ, t)j2dt + jZu(ξ, t)u^t(ξ, t) ∣∣δ t=−δj. (3.3) Now let us assume that we have started with some r > 0. And δ above has been chosen to be in the interval (0, r). If we assume that Pu 2 A(T1 � (−r, r)),(3.4) and we use the fact that the operator P is elliptic near (T1 � f−δg) [ (T1 � fδg), then by (3.3) and (3.4) we obtain∫ δ −δ ju^t(ξ, t)j2dt + ∫ δ −δ [ξ2b2(t) + ξb0(t)]ju^(ξ, t)j2dt(3.5) � ∫ δ −δ [ 1 2 + kak1]ju^(ξ, t)j2dt + O(e−εjξj), for some ε > 0. Next we shall absorb the term ( 1 2 + kak1) ∫ δ −δ ju^(ξ, t)j2dt in the left-hand side of (3.5) by using the following (Poincar�e inequality) argument. We write u^(ξ, t) = u^(ξ,−δ) + ∫ t −δ u^t(ξ, s)ds. 4998 PAULO D. CORDARO AND A. ALEXANDROU HIMONAS Then we obtain ju^(ξ, t)j2 � 2c2e−2εjξj + 4δ ∫ δ −δ jut(ξ, t)j2dt, which implies that∫ δ −δ ju^(ξ, t)j2dt � 4c2δe−2εjξj + 8δ2 ∫ δ −δ jut(ξ, t)j2dt.(3.6) If we choose δ such that (3.2) is true and furthermore 8δ2(12 + kak1) < 12 , then by using (3.6), relation (3.5) gives∫ δ −δ jut(ξ, t)j2dt + ∫ δ −δ [ξ2b2(t) + ξb0(t)]ju^(ξ, t)j2dt . e−εjξj.(3.7) Very similar to the above arguments applied to the operator Q = −L�L+ a give the inequality ∫ δ −δ jvt(ξ, t)j2dt + ∫ δ −δ [ξ2b2(t)− ξb0(t)]jv^(ξ, t)j2dt . e−εjξj,(3.8) for any v 2 C1(Tt,D0(Tx)) with Qv 2 A(T1 � (−r, r)). To summarize, we have the following lemma. Lemma 3.1. Let P be given by (3.1) with b as in (3.2), and r > 0 be a given number. If δ 2 (0, r) is such that 8δ2( 1 2 + kak1) < 12 , then the following hold: 1. Any u 2 C1(Tt,D0(Tx)) with Pu 2 A(T1 � (−r, r)) satisfies inequality (3.7) for some � > 0. 2. Let Q = −L�L + a. Then any v 2 C1(Tt,D0(Tx)) with Qv 2 A(T1 � (−r, r)) satisfies inequality (3.8) for some � > 0. Now we assume that b(t) has a zero of odd order at t = 0; without loss of generality we can assume b0(t) � 0, −δ � t � δ,(3.9) and we have the following proposition: Proposition 3.2. Let P be as in (3.1), b be as in (3.2) and (3.9), and r > 0 be a given number. If δ 2 (0, r) is such that 8δ2(12 + kak1) < 12 , then the following hold: 1. For any solution u 2 C1(Tt,D0(Tx)) to Pu = f, f 2 A(T1 � (−r, r)) there exist constants c > 0 and ε > 0, which may depend on u, such that ju^(ξ, t)j � ce−εjξj, ξ > 0, jtj � δ.(3.10) 2. For any solution v 2 C1(Tt,D0(Tx)) to Qv = f, f 2 A(T1� (−r, r)) there exist constants c > 0 and ε > 0, which may depend on v, such that jv^(ξ, t)j � ce−εjξj, ξ < 0, jtj � δ.(3.11) Remark. f may be assumed to satisfy the correct estimate only for ξ > 0 in (1), and ξ < 0 in (2). REGULARITY FOR SUMS OF SQUARES OF VECTOR FIELDS 4999 Proof. Let ξ > 0. Then by (3.9) we obtain ξ2b2(t) + ξb0(t) � 0, and we can apply Lemma 4.1 in Cordaro-Himonas [CH] to show that ju^(ξ, t)j2 . ∫ δ −δ ju^t(ξ, t)j2dt + ∫ δ −δ [ξ2b2(t) + ξb0(t)]ju^(ξ, t)j2dt.(3.12) Therefore by (3.7) and (3.12) we obtain (3.10). If ξ < 0, then by (3.9) we obtain ξ2b2(t)−ξb0(t) � 0, and again we apply Lemma 4.1 in [CH] to obtain ju^(ξ, t)j2 . ∫ δ −δ jut(ξ, t)j2dt + ∫ δ −δ [ξ2b2(t)− ξb0(t)]ju^(ξ, t)j2dt.(3.13) By (3.8) and (3.13) we obtain (3.11). End of Proof of Theorem 1.2. Since by our hypothesis Pu is analytic, by Proposi- tion 3.2 u satis�es the estimate (3.10). To complete the proof it su�ces to show that u satis�es estimate (3.11) too. We have L(−�LLu + au) = Lf . Since a is a constant, it commutes with L and we obtain L(−�LL+a) = (−L�L+a)L. Therefore we have that Lu satis�es the equation (−L�L + a)(Lu) = Lf . Now by applying the second part of Proposition 3.2 for v = Lu we obtain that Lu satis�es estimate (3.11). If we solve the equation −�LLu + au = f for au, we obtain au = �L(Lu)+ f . Since a 6= 0, we obtain u = 1 a (�L(Lu) + f). Since both Lu and f satisfy estimate (3.11), the last relation implies that u satis�es the estimate (3.11) too. Since u satis�es both estimates (3.10) and (3.11), we obtain the inequality ju^(ξ, t)j . e−εjξj, ξ 2 R.(3.14) Relation (3.14) together with standard arguments (see for example [CH]) implies that u is analytic near T1 � ft0g. This completes the proof of Theorem 1.2. To prove Theorem 1.3 we shall need the following result in Bergamasco [B]. Lemma 3.3. Let L be as in (1.3) with b 6� 0. Then L is globally analytic hypoel- liptic in T2 if and only if the function b(t) does not change sign in T1. Proof. If b(t) does not change sign in T1, then condition (P) holds and by the work of Treves [Tr2] L is locally and therefore globally analytic hypoelliptic. If b(t) does change sign, then by using the stationary phase method one can construct a non-analytic solution in T2 to Lu = 0, [B]. Proof of Theorem 1.3. If a 6= 0, then by Theorem 1.2 P is analytic hypoelliptic near T1 � ft0g, for each zero, t0, of b. Therefore P is globally analytic hypoelliptic in T2. If a = 0, then P = −�LL. Since b(t) changes sign, by Lemma 3.3 there exists a global non-analytic solution u to the equation Lu = 0 in T2. This implies Pu = 0, and therefore P is not globally analytic hypoelliptic in T2. This completes the proof of Theorem 1.3. 5000 PAULO D. CORDARO AND A. ALEXANDROU HIMONAS 4. Final remarks 1. 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Treves, Analytic hypoelliptic partial differential operators of principal type, Comm. Pure Appl. Math. 24 (1971), 537-570. MR 45:5569 IME - USP, Caixa Postal 66281, CEP 05389-970, Sa˜o Paulo, SP, Brazil E-mail address: cordaro@ime.usp.br Dep