Global hypoellipticity for a class of pseudo-differential operators on the torus
Fernando de ?vila Silva, Rafael Borro Gonzalez, Alexandre Kirilov, Cleber de Medeira
aa r X i v : . [ m a t h . A P ] M a y GLOBAL HYPOELLIPTICITY FOR A CLASS OFPSEUDO-DIFFERENTIAL OPERATORS ON THE TORUS
FERNANDO DE ´AVILA SILVA, RAFAEL BORRO GONZALEZ,ALEXANDRE KIRILOV, AND CLEBER DE MEDEIRA
In memory of Todor V. Gramchev
Abstract.
We show that an obstruction of number-theoretical nature appears as anecessary condition for the global hypoellipticity of the pseudo-differential operator L = D t + ( a + ib )( t ) P ( D x ) on T t × T Nx . This condition is also sufficient when thesymbol p ( ξ ) of P ( D x ) has at most logarithmic growth. If p ( ξ ) has super-logarithmicgrowth, we show that the global hypoellipticity of L depends on the change of sign ofcertain interactions of the coefficients with the symbol p ( ξ ) . Moreover, the interplaybetween the order of vanishing of coefficients with the order of growth of p ( ξ ) playsa crucial role in the global hypoellipticity of L . We also describe completely theglobal hypoellipticity of L in the case where P ( D x ) is homogeneous. Additionally,we explore the influence of irrational approximations of a real number in the globalhypoellipticity. Contents
1. Introduction 12. The constant coefficient operators 42.1. Global hypoellipticity and Liouville numbers 53. The variable coefficient operators 83.1. A necessary condition 93.2. Sufficient conditions 124. Logarithmic growth 134.1. Reduction to normal form 144.2. Change of sign 174.3. A particular class of operators 215. Super-logarithmic growth 225.1. Order of vanishing 256. Homogeneous operators 306.1. Sum of homogeneous operators 31References 351.
Introduction
We investigate the global hypoellipticity of pseudo-differential operators of the form(1.1) L = D t + ( a + ib )( t ) P ( D x ) , ( t, x ) ∈ T × T N , where a ( t ) and b ( t ) are real smooth functions on T , and P ( D x ) is a pseudo-differentialoperator of order m ∈ R defined on T N ≃ R N / (2 π Z N ). The operator P ( D x ) is given Date : November 8, 2018.2010
Mathematics Subject Classification.
Primary 35B10, 35H10, 35S05.During this work the second author was supported by PNPD/CAPES - Brazil in the GraduateProgram in Mathematics, PPGM-UFPR. by(1.2) P ( D x ) · u = X ξ ∈ Z N e ix · ξ p ( ξ ) b u ( ξ ) , where p = p ( ξ ) ∈ S m ( Z N ) is the toroidal symbol of P ( D x ) and b u ( ξ ) = 1(2 π ) N Z T N e − ix · ξ u ( x ) dx, ξ ∈ Z N , are the Fourier coefficients of u .The operator L is said to be globally hypoelliptic on T × T N if the conditions u ∈ D ′ ( T × T N ) and Lu ∈ C ∞ ( T × T N ) imply that u ∈ C ∞ ( T × T N ) . Even in the case of vector fields, the investigation of global hypoellipticity on thetorus is a challenging problem that still have open questions. Perhaps the questionwithout an answer that is most famous and seemingly far from a solution is the Green-field and Wallach conjecture. It states that: if a smoothly closed manifold M admitsa globally hypoelliptic vector field X , then M is diffeomorphic to a torus and X issmooth conjugated to a Diophantine vector field (see [22]).This conjecture has a geometric version stated in terms of cohomology-free dynamicalsystems, known as Katok conjecture, and was proved only in some few cases and indimensions 2 and 3. For more details we refer the works of G. Forni [17], J. Hounie[27], and A. Kocsard [28].With respect to the differential case of the operator we are interested, with P ( D x ) = D x and N = 1 , J. Hounie has proved in Theorem 2.2 of [26] that L = D t + ( a + ib )( t ) D x is globally hypoelliptic on T if and only if b ( t ) does not change sign and either b = 0or a is an irrational non-Liouville number, where a . = (2 π ) − Z π a ( t ) dt and b . = (2 π ) − Z π b ( t ) dt. We recall that S. Greenfield and N. Wallach have proved in [21] that the aboveconditions on a and b means that the constant coefficient operator D t + ( a + ib ) D x is globally hypoelliptic. Therefore, the global hypoellipticity of D t + ( a + ib ) D x is a necessary condition for the global hypoellipticity of the operator with variablecoefficients D t + ( a + ib )( t ) D x .We prove that this necessity remains valid for any pseudo-differential operator P ( D x )defined on the N -dimensional torus, that is, if the operator L defined in (1.1) is globallyhypoelliptic then the constant coefficient operator(1.3) L = D t + ( a + ib ) P ( D x ) , ( t, x ) ∈ T × T N , is also globally hypoelliptic (see Theorem 3.5).We also show that the global hypoellipticity of L and the control of the sign of theimaginary part of the functions t ∈ T
7→ M ( t, ξ ) . = ( a + ib )( t ) p ( ξ ) , ξ ∈ Z N , for sufficiently large | ξ | , are sufficient conditions to the global hypoellipticity of L (seeTheorem 3.6).Although the global hypoellipticity of L cannot be removed in the study of theglobal hypoellipticity of L , the converse of Theorem 3.6 in general does not hold; unlike LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 3 the differential case P ( D x ) = D x . In Sections 4 and 5 we exhibit examples of globallyhypoelliptic operators in which the imaginary part of the functions t ∈ T
7→ M ( t, ξ )changes sign for infinitely many indexes ξ ∈ Z N (see Examples 4.5, 4.6, 4.11, 4.13, andthe first example in Subsection 5.1).We point out that our results are not a consequence of the Hounie’s abstract resultsin [26], even when our operator fits in the conditions assumed in that work. Dependingon P ( D x ), the scales of Sobolev spaces used by Hounie are different from the usualSobolev spaces, which implies in a different notion of global hypoellipticity. We referthe reader to [1], Section 3.3, for more details.In Section 2 we study operators with constant coefficients giving a special attentionto the case when P ( D x ) is a homogeneous operator of rational degree m on T , seeTheorem 2.4. In this case, our main contribution is to shed light on the connectionsbetween hypoellipticity and certain approximations of real numbers, which are notconsidered in [21]. Indeed, in our approach the global hypoellipticity depends on thefollowing approximations (cid:12)(cid:12)(cid:12)(cid:12) τ | ξ | m + p ( ± (cid:12)(cid:12)(cid:12)(cid:12) , ( τ, ξ ) ∈ Z × Z ∗ where the numbers τ / | ξ | m can be irrational, depending on m. As a consequence of Theorem 2.4, if m = ℓ/q is irreducible, with ℓ, q ∈ N , thenthe operator D t + α ( D x ) m/ is globally hypoelliptic if and only if α q is an irrationalnon-Liouville number. Notice that the global hypoellipticity of this operator does notdepend on ℓ. For example, D t + √ D x ) ℓ/ q is globally hypoelliptic if and only if q isodd.Regarding the case of variable coefficients, one of the contributions of this work isto show that the global hypoellipticity of the operator L defined in (1.1) is related tothe growth of the real and imaginary parts of the symbol p ( ξ ) when | ξ | → ∞ .In Section 4, we give a complete characterization for the global hypoellipticity of L when either α ( ξ ) or β ( ξ ) has at most logarithmic growth, where p ( ξ ) = α ( ξ ) + iβ ( ξ ) , ξ ∈ Z N . When both α ( ξ ) and β ( ξ ) have at most logarithmic growth we show that the changeof sign of the functions t ∈ T
7→ ℑM ( t, ξ ) = a ( t ) β ( ξ ) + b ( t ) α ( ξ ) , ξ ∈ Z N , does not play any role in the global hypoellipticity of L . More precisely, we provethat L , defined in (1.1), is globally hypoelliptic if and only if L , defined in (1.3) isglobally hypoelliptic. This equivalence comes from the reduction to normal form, thatis a technique well explored in the works [1, 13, 14, 15, 16, 32].If β ( ξ ) has at most logarithmic growth, but α ( ξ ) has super-logarithmic growth, thenwe prove that L is globally hypoelliptic if and only if L is globally hypoelliptic and b ( t ) does not change sign. This result remains valid if we exchange β ( ξ ) by α ( ξ ) and b ( t ) by a ( t ), see Subsection 4.2.When α ( ξ ) and β ( ξ ) have super-logarithmic growth, the interactions between thefunctions a ( t ) β ( ξ ) and b ( t ) α ( ξ ) play a larger role. In this case, the operator L may benon-globally hypoelliptic even if L is globally hypoelliptic and both a ( t ) and b ( t ) donot change sign (see Examples 4.13, 5.5 and the second example in Subsection 5.1). F. DE ´AVILA, R. GONZALEZ, A. KIRILOV, AND C. DE MEDEIRA
On the other hand, L may be globally hypoelliptic even if both a ( t ) and b ( t ) changessign provided that α ( ξ ) and β ( ξ ) go to infinity with the same order of growth (seeExample 5.6).In the case where both the parts α ( ξ ) and β ( ξ ) have super-logarithmic growth and α ( ξ ) /β ( ξ ) → K, as | ξ | → ∞ , we show that L is not globally hypoelliptic if the function t ∈ T → a ( t ) + b ( t ) K changes sign (Corollary 5.2). In particular, if p ( ξ ) has super-logarithmic growth with α ( ξ ) = o ( β ( ξ )) , then L is not globally hypoelliptic if a ( t ) changes sign. Analogously, L is not globally hypoelliptic when β ( ξ ) = o ( α ( ξ )) and b ( t ) changes sign (see Corollary5.3).Another contribution we give is to present (in Subsection 5.1) a relation between theglobal hypoellipticity of the operator L and the order of vanishing of the coefficients a ( t ) and b ( t ). We emphasize that this phenomenon is more common in the study ofthe global solvability of vector fields on the torus (see [4, 5, 12, 18]).In Section 6, we describe completely the global hypoellipticity of L in the case where P ( D x ) is homogeneous (see Theorem 6.1 and Corollary 6.2). For these operators theconverse of Theorem 3.6 holds. Moreover, we analyze the case of sums of homogeneousoperators extending Theorem 1.3 of [6], see Corollary 6.6.For more results on the problem of global hypoellipticity and global solvability ofequations and systems of equations on the torus we refer the reader to the works[2, 3, 8, 9, 10, 11, 19, 20, 24] and the references therein.2. The constant coefficient operators
By following the approach introduced by Greenfield and Wallach in [21], we maycharacterize the global hypoellipticity of the operator(2.1) L = D t + P ( D x ) , ( t, x ) ∈ T × T N , by means of a control in its symbol L ( τ, ξ ) = τ + p ( ξ ) , ( τ, ξ ) ∈ Z × Z N . Theorem 2.1.
The operator L in (2.1) is globally hypoelliptic if and only if there existpositive constants C, M and R such that | τ + p ( ξ ) | > C ( | τ | + | ξ | ) M , for all | τ | + | ξ | > R. The proof of this result follows the same ideas of the differential case made in [21].Note that, if the imaginary part of p ( ξ ) does not approach to zero rapidly, then theestimate in Theorem 2.1 is verified. More precisely, if there exists M ≥ | ξ |→∞ | ξ | M |ℑ p ( ξ ) | > , then the operator L = D t + P ( D x ) is globally hypoelliptic.This type of condition appears in Theorem 5.3 of [16], where the authors studiedthe relation between global hypoellipticity and simultaneous inhomogeneous Siegelconditions. LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 5
On the other hand, without this control in the imaginary part, for instance when ℑ p ( ξ ) ≡
0, Diophantine phenomena appear. When the symbol is homogeneous ofrational positive degree, we present a new relation between global hypoellipticity andLiouville numbers in Theorem 2.4.We observe that each toroidal symbol p in the class S m ( Z N ) can be extended to anEuclidean symbol e p ∈ S m ( R N ) such that p = e p | Z N (see Theorem 4.5.3 of [33]). Definition 2.2.
We say that a toroidal symbol p ( ξ ) is homogeneous of degree m if ithas an Euclidean extension e p ( ξ ) such that p ( ξ ) = | ξ | m e p ( ξ/ | ξ | ) , ξ ∈ Z N ∗ . In order to not overload our notation, we will use the notation p ( ξ/ | ξ | ) in place of e p ( ξ/ | ξ | ).When the symbol p ( ξ ) is homogeneous of degree m , it follows from Theorem 2.1 thatthe operator L given by (2.1) is globally hypoelliptic if and only if there exist positiveconstants C, M, and R, such that (cid:12)(cid:12)(cid:12)(cid:12) τ | ξ | m + p (cid:18) ξ | ξ | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) > C ( | τ | + | ξ | ) M , (2.2)for all ( τ, ξ ) ∈ Z × Z N ∗ which satisfy | τ | + | ξ | > R. Global hypoellipticity and Liouville numbers.
Let P ( D x ) be an operatoron T with symbol p ( ξ ) homogeneous of degree m . In this case p ( ξ ) = | ξ | m p ( ± , for all ξ ∈ Z ∗ . Thus, when m <
0, by using condition (2.2) we see that the operator L is globallyhypoelliptic if and only if p ( ± = 0. Similarly, when m = 0, it follows that L isglobally hypoelliptic if and only if p ( ± / ∈ Z .The case in which m is a positive rational number and ℑ p ( ±
1) = 0 , is more inter-esting. We now move to describe it.By using the notations p (1) = α + iβ and p ( −
1) = e α + i e β, we have(2.3) (cid:12)(cid:12)(cid:12)(cid:12) τ | ξ | m + p (cid:18) ξ | ξ | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) τξ m + ( α + iβ ) (cid:12)(cid:12)(cid:12)(cid:12) , if ξ > , (cid:12)(cid:12)(cid:12)(cid:12) τ | ξ | m + ( e α + i e β ) (cid:12)(cid:12)(cid:12)(cid:12) , if ξ < . In this case, when β = 0 (respectively e β = 0) we must control the approximations ofthe real number α (respectively e α ) by numbers of the type τ / | ξ | m , for all ( τ, ξ ) ∈ Z × Z ∗ . Definition 2.3.
An irrational number λ is said to be a Liouville number if there existsa sequence ( j n , k n ) ∈ Z × N , such that k n → ∞ and (cid:12)(cid:12)(cid:12)(cid:12) λ − j n k n (cid:12)(cid:12)(cid:12)(cid:12) < ( k n ) − n , n ∈ N . Under the previous notation we have the following result:
F. DE ´AVILA, R. GONZALEZ, A. KIRILOV, AND C. DE MEDEIRA
Theorem 2.4. If p = p ( ξ ) is a homogeneous symbol of degree m = ℓ/q with ℓ, q ∈ N , and gcd( ℓ, q ) = 1 , then the operator L = D t + P ( D x ) , ( t, x ) ∈ T , is globally hypoelliptic if and only if α q is an irrational non-Liouville number whenever β = 0 , and e α q is an irrational non-Liouville number whenever e β = 0 .Proof. Inequality (2.2) is easily verified when β · e β = 0, consequently L is globallyhypoelliptic in this case. Therefore, in order to prove Theorem 2.4 it is enough toconsider either β = 0 or e β = 0.We start by considering the case β = 0 and e β = 0. In this situation, it followsfrom (2.2) and (2.3) that L is globally hypoelliptic if and only if there exist positiveconstants C, M and R such that(2.4) (cid:12)(cid:12)(cid:12)(cid:12) τξ m + α (cid:12)(cid:12)(cid:12)(cid:12) > C ( | τ | + ξ ) − M , for all ( τ, ξ ) ∈ Z × N such that | τ | + ξ > R. Since β = 0, if α = 0 then p ( ξ ) = 0 for all ξ > L is not globallyhypoelliptic. From now on, without loss of generality, we assume that α > . When α q is a rational number, we prove that L is not globally hypoelliptic byexhibiting infinitely many ( τ, ξ ) ∈ Z × N such that | τ /ξ m − α | = 0 . We then write α q = e p/ e q , with e p, e q ∈ N . By prime factorization we have e q = q γ · · · q γ r r and e p = p σ · · · p σ s s . Since gcd( ℓ, q ) = 1 , there exists ( x i , y i ) ∈ N and ( v j , w j ) ∈ N such that ℓx i − qy i = γ i , i = 1 , . . . , r and qv j − ℓw j = σ j , j = 1 , . . . , s. Define τ n = n ℓ q y · · · q y r r p v · · · p v s s and ξ n = n q q x · · · q x r r p w · · · p w s s . It follows that e qτ qn = e pξ ℓn , for all n ∈ N ; hence τ n ( ξ n ) m = τ n ( ξ n ) ℓ/q = (cid:18) e p e q (cid:19) /q = α, for all n ∈ N , and then L is not globally hypoelliptic.From now on, assume that α q is an irrational number.If L is not globally hypoelliptic, it follows from (2.4) that there exists a sequence( τ n , ξ n ) ∈ Z × N such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ n ξ ℓ/qn − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ( | τ n | + ξ n ) − n , | τ n | + ξ n > n. By taking j n = − τ qn and k n = ξ ℓn we obtain (cid:12)(cid:12)(cid:12)(cid:12) j n k n + α q (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − τ n ξ ℓ/qn ! q + α q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ n ξ ℓ/qn − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q X j =1 τ n ξ ℓ/qn ! q − j α j − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 7
Since q X j =1 ( τ n /ξ ℓ/qn ) q − j α j − goes to qα q − , as n goes to infinity, it follows that (cid:12)(cid:12)(cid:12)(cid:12) j n k n + α q (cid:12)(cid:12)(cid:12)(cid:12) C (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ n ξ ℓ/qn − α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < C ( | τ n | + ξ n ) − n Ck − n/ℓn , for all n, where the constant C > j n and k n .The estimate above implies that α q is a Liouville number.Assuming that α q is a Liouville number, let us show that L is not globally hypoel-liptic. Indeed, if α q is a Liouville number, then there is a sequence ( j n , k n ) ∈ N ,j n + k n > n , such that | j n − α q k n | < ( j n + k n ) − n . By multiplying this inequality by(2.5) j ( q − ℓ +( ℓ − e pℓn k e qqn , where e p and e q are positive integers such that e pℓ − e qq = 1, we obtain | ( j ℓ + e q ( ℓ − n k e qn ) q − α q ( k e pn j q − ℓ − e pn ) ℓ | < j ( q − ℓ +( ℓ − e pℓn k e qqn ( j n + k n ) − n . Suppose, by contradiction, that L is globally hypoelliptic. By (2.4), there existpositive constants C , M and R such that | τ − αξ ℓ/q | > C ( | τ | + ξ ) − M , for all ( τ, ξ ) ∈ N × N such that τ + ξ > R. Since | τ q − α q ξ ℓ | = | τ − αξ ℓ/q | · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q X κ =1 τ q − κ ( αξ ℓ/q ) κ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > C ( τ + ξ ) − M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q X κ =1 τ q − κ ( αξ ℓ/q ) κ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q X κ =1 (cid:0) j ℓ + e q ( ℓ − n k e qn (cid:1) q − κ α κ − (cid:0) k e pn j q − ℓ − e pn (cid:1) ℓ ( κ − q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > e C, ( j n > , k n > , for some e C >
0, it follows that C e C (cid:0) j ℓ + e q ( ℓ − n k e qn + k e pn j q − ℓ − e pn (cid:1) M j ( q − ℓ +( ℓ − e pℓn k e qqn ( j n + k n ) − n , for all n ∈ N . Now, by taking K = max { ℓ + e q ( ℓ − , e q, e p, q − ℓ − e p, ( q − ℓ + ( ℓ − e pℓ, e qq } we obtain0 < C e C ( j ℓ + e q ( ℓ − n k e qn + k e pn j q − ℓ − e pn ) M j ( q − ℓ +( ℓ − e pℓn k e qqn ( j n + k n ) − n ( j Kn k Kn + k Kn j Kn ) M j Kn k Kn ( j n + k n ) − n = 2 M ( j n + k n ) − n +2 K ( M +1) , F. DE ´AVILA, R. GONZALEZ, A. KIRILOV, AND C. DE MEDEIRA for all n ∈ N , which is a contradiction, since the right-hand side goes to zero as n goesto infinity.Finally, in the case in which β = 0 and e β = 0 , a slight modification in the previousarguments give us that L is globally hypoelliptic if and only if e α q is an irrational non-Liouville number. Analogously, if β = 0 and e β = 0 , then L is globally hypoelliptic ifand only if both α q and e α q are irrational non-Liouville numbers. (cid:3) As consequence of Theorem 2.4 we obtain the following examples.
Example 2.5.
Let m = ℓ/q be a positive rational number with gcd ( ℓ, q ) = 1, then L = D t + α ( D x ) m/ is globally hypoelliptic if and only if α q is an irrational nonLiouville number. In particular, for the non-Liouville number α = √ L = D t + α ( D x ) / is globally hypoelliptic while L = D t + α ( D x ) / is not. Example 2.6.
Let λ = P ∞ n =1 − n ! be the Liouville constant. For each integer q > λ q / λ q p / α = λ q p / L = D t + α ( D x ) / q is not globally hypoellipticfor each integer q > The variable coefficient operators
In this section we study the global hypoellipticity of the operator (1.1), which werecall L = D t + ( a + ib )( t ) P ( D x ) , ( t, x ) ∈ T × T N , where a ( t ) and b ( t ) are real valued smooth functions on T and P ( D x ) is a pseudo-differential on T N with symbol p = p ( ξ ) , ξ ∈ Z N . Without any assumption about the behavior of p ( ξ ) , as | ξ | → ∞ , we will present anecessary condition and, also, sufficient conditions for the global hypoellipticity of L. First, we show that the global hypoellipticity of L = D t + ( a + ib ) P ( D x ) , where a = (2 π ) − Z π a ( t ) dt and b = (2 π ) − Z π b ( t ) dt, is necessary for the global hypoellipticity of L (Theorem 3.5). After this, we will showthat this condition is also sufficient provided that the imaginary part of the function t ∈ T
7→ M ( t, ξ ) . = ( a + ib )( t ) p ( ξ ) , ξ ∈ Z N , does not change sign, for all | ξ | large enough (Theorem 3.6).By using partial Fourier series in the variable x, we can write a distribution u in D ′ ( T × T N ) as u = X ξ ∈ Z N b u ( t, ξ ) e ixξ , where b u ( t, ξ ) = (2 π ) − N h u ( t, · ) , e − ix · ξ i . Hence, the equation ( iL ) u = f lead us to con-sider the differential equations(3.1) ∂ t b u ( t, ξ ) + i M ( t, ξ ) b u ( t, ξ ) = b f ( t, ξ ) , t ∈ T , for all ξ ∈ Z N . LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 9
With the notations M ( ξ ) = (2 π ) − Z π M ( t, ξ ) dt = ( a + ib ) p ( ξ )and(3.2) Z M = { ξ ∈ Z N ; M ( ξ ) ∈ Z } , we have: Lemma 3.1. If u ∈ D ′ ( T × T N ) and iLu = f ∈ C ∞ ( T × T N ) , then equation (3.1) implies that b u ( · , ξ ) belongs to C ∞ ( T ) , for all ξ ∈ Z N . Moreover, for each ξ / ∈ Z M , equation (3.1) has a unique solution, which can be written in the following two ways: b u ( t, ξ ) = 11 − e − πi M ( ξ ) Z π exp (cid:18) − i Z tt − s M ( r, ξ ) dr (cid:19) b f ( t − s, ξ ) ds, (3.3) or b u ( t, ξ ) = 1 e πi M ( ξ ) − Z π exp (cid:18) i Z t + st M ( r, ξ ) dr (cid:19) b f ( t + s, ξ ) ds. (3.4)Furthermore, we have the following characterization for the global hypoellipticity of L . Proposition 3.2.
The following statements are equivalent:i) L is globally hypoelliptic;ii) There exist positive constants C, M, and R such that | τ + M ( ξ ) | > C ( | τ | + | ξ | ) − M , for all ( | τ | + | ξ | ) > R ;iii) There exist positive constants e C, f M , and e R such that | − e ± πi M ( ξ ) | > e C | ξ | − f M , for all | ξ | > e R. The equivalence i ) ⇔ ii ) follows from Theorem 2.1 and the equivalence ii ) ⇔ iii ) isa technical result that is a slight modification of the proof of Lemma 3.1 of [6].3.1. A necessary condition.
Our first result in this section is the following:
Proposition 3.3. If L is globally hypoelliptic, then the set Z M defined in (3.2) isfinite. Proof. If Z M is infinite, then there exists a sequence { ξ n } such that | ξ n | is increasingand M ( ξ n ) ∈ Z . Set c n = exp (cid:18) − Z t n ℑM ( r, ξ n ) dr (cid:19) , where t n ∈ [0 , π ] is such that Z t n ℑM ( r, ξ n ) dr = max t ∈ [0 , π ] Z t ℑM ( r, ξ n ) dr. For each ξ n the function b u ( t, ξ n ) = c n exp (cid:18) − i Z t M ( r, ξ n ) dr (cid:19) is smooth on T and satisfies the equation ∂ t b u ( t, ξ n ) + i M ( t, ξ n ) b u ( t, ξ n ) = 0 . Moreover, | b u ( t, ξ n ) | , for all t ∈ [0 , π ] , and | b u ( t n , ξ n ) | = 1 . Hence, u = ∞ X n =1 b u ( t, ξ n ) e ixξ n ∈ D ′ ( T × T N ) \ C ∞ ( T × T N ) , and satisfies Lu = 0 . Therefore, L is not globally hypoelliptic. (cid:3) Remark . In the case of constant coefficients the previous result implies that L isnot globally hypoelliptic if Z M is infinite. Therefore, every time we assume that L isglobally hypoelliptic it is understood that Z M is finite.Now we present our first general result on global hypoellipticity. Theorem 3.5. If L is globally hypoelliptic, then L is globally hypoelliptic.Proof. We assume that L is not globally hypoelliptic and prove that L is not globallyhypoelliptic.By Proposition 3.2, there is a sequence { ξ n } such that | ξ n | is strictly increasing, | ξ n | > n, and(3.5) | − e − πi M ( ξ n ) | < | ξ n | − n , for all n ∈ N . By Proposition 3.3, it is enough to consider the case where Z M is finite and ξ n Z M , for all n. For each n, we may choose t n ∈ [0 , π ] so that R tt n ℑM ( r, ξ n ) dr , for all t ∈ [0 , π ] . Indeed, for all t ∈ [0 , π ] we write Z tt n ℑM ( r, ξ n ) dr = Z t ℑM ( r, ξ n ) dr − Z t n ℑM ( r, ξ n ) dr, and it is enough to consider t n satisfying Z t n ℑM ( r, ξ n ) dr = max t ∈ [0 , π ] Z t ℑM ( r, ξ n ) dr. By passing to a subsequence, we may assume that there exists t ∈ [0 , π ] such that t n → t , as n → ∞ . Let I be a closed interval in (0 , π ) such that t I. Consider φ belonging to C ∞ c ( I, R ) , such that 0 φ ( t ) R π φ ( t ) dt > . For each n, we define b f ( · , ξ n ) as being the 2 π − periodic extension of(1 − e − πi M ( ξ n ) ) exp (cid:18) − Z tt n i M ( r, ξ n ) dr (cid:19) φ ( t ) . Since p ( ξ ) increases slowly, R tt n ℑM ( r, ξ n ) dr t ∈ [0 , π ] , and since (3.5)holds, it follows that b f ( · , ξ n ) decays rapidly. Hence, f ( t, x ) = ∞ X n =1 b f ( t, ξ n ) e ixξ n ∈ C ∞ ( T × T N ) . LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 11
In order to exhibit a distribution u ∈ D ′ ( T × T N ) \ C ∞ ( T × T N ) such that iLu = f, we consider b u ( t, ξ n ) = 11 − e − πi M ( ξ n ) Z π exp (cid:18) − Z tt − s i M ( r, ξ n ) dr (cid:19) b f ( t − s, ξ n ) ds. Note that 1 − e − πi M ( ξ n ) = 0 , since ξ n Z M , and b u ( · , ξ n ) ∈ C ∞ ( T ) (Lemma 3.1).Moreover, for t, s ∈ [0 , π ] such that t − s > , we have (cid:12)(cid:12)(cid:12)(cid:12) − e − πi M ( ξ n ) exp (cid:18) − Z tt − s i M ( r, ξ n ) dr (cid:19) b f ( t − s, ξ n ) (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18)Z tt − s ℑM ( r, ξ n ) dr + Z t − st n ℑM ( r, ξ n ) dr (cid:19) = exp (cid:18)Z tt n ℑM ( r, ξ n ) dr (cid:19) , while for t, s ∈ [0 , π ] such that t + s < , we have (cid:12)(cid:12)(cid:12)(cid:12) − e − πi M ( ξ n ) exp (cid:18) − Z tt − s i M ( r, ξ n ) dr (cid:19) b f ( t − s, ξ n ) (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18)Z tt − s ℑM ( r, ξ n ) dr + Z t − s +2 πt n ℑM ( r, ξ n ) dr (cid:19) =exp (cid:18)Z tt n ℑM ( r, ξ n ) dr + 2 π ℑM ( ξ n ) (cid:19) . Since ℑM ( ξ n ) → , by (3.5), the estimates above imply that | b u ( t, ξ n ) | π, for all t ∈ T and for n sufficiently large. Hence, b u ( · , ξ n ) increases slowly and u ( t, x ) = ∞ X n =1 b u ( t, ξ n ) e ixξ n ∈ D ′ ( T × T N ) . If t > sup I, then t n > sup I, for all n sufficiently large, and | b u ( t n , ξ n ) | = Z t n − sup It n − inf I φ ( t n − s ) ds = Z π φ ( t ) dt > , on the other hand, if t < sup I, then t n < inf I, for all n sufficiently large, and | b u ( t n , ξ n ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z t n − inf I +2 πt n − sup I +2 π exp (cid:18) − Z t n t n − s i M ( r, ξ n ) dr (cid:19) × exp (cid:18) − Z t n − s +2 πt n i M ( r, ξ n ) dr (cid:19) φ ( t n − s + 2 π ) ds (cid:12)(cid:12)(cid:12)(cid:12) = e π ℑM ( ξ n ) Z π φ ( s ) ds > (1 / Z π φ ( s ) ds > , which implies that b u ( · , ξ n ) does not decay rapidly.Hence u ∈ D ′ ( T × T N ) \ C ∞ ( T × T N ) , and since iLu = f (by Lemma 3.1), it followsthat L is not globally hypoelliptic. (cid:3) Sufficient conditions.
We now present sufficient conditions to the global hy-poellipticity of L . Theorem 3.6.
If the operator L given by (1.3) is globally hypoelliptic and the function ℑM ( t, ξ ) = a ( t ) β ( ξ ) + b ( t ) α ( ξ ) does not change sign, for sufficiently large | ξ | , then theoperator L given by (1.1) is globally hypoelliptic.Proof. Let u ∈ D ′ ( T × T N ) be a distribution such that iLu = f, with f ∈ C ∞ ( T × T N ) . We will show that u ∈ C ∞ ( T × T N ) . By using partial Fourier series in the variable x, it follows that iLu = f if and onlyif(3.6) ∂ t b u ( t, ξ ) + i M ( t, ξ ) b u ( t, ξ ) = b f ( t, ξ ) , for all t ∈ T and for all ξ ∈ Z N . Lemma 3.1 implies that b u ( · , ξ ) ∈ C ∞ ( T ) , for each ξ ∈ Z N . Moreover, since Z M isfinite (thanks to Remark 3.4), for | ξ | sufficiently large the equation (3.6) has a uniquesolution, which may be written in the form (3.3) or (3.4). Since t
7→ ℑM ( t, ξ ) doesnot change sign for | ξ | large enough, we conveniently write b u ( t, ξ ) = 11 − e − πi M ( ξ ) Z π exp (cid:18) − i Z tt − s M ( r, ξ ) dr (cid:19) b f ( t − s, ξ ) ds, if ξ is such that ℑM ( t, ξ ) , for all t ∈ T , and b u ( t, ξ ) = 1 e πi M ( ξ ) − Z π exp (cid:18) i Z t + st M ( r, ξ ) dr (cid:19) b f ( t + s, ξ ) ds, if ξ is such that ℑM ( t, ξ ) > , for all t ∈ T . Since L is globally hypoelliptic, then there exist positive constants C, M, and R, so that(3.7) | − e ± πi M ( ξ ) | > C | ξ | − M , for all | ξ | > R (Proposition 3.2).Hence, for | ξ | sufficiently large, the solution b u ( · , ξ ) of (3.6) satisfies | b u ( t, ξ ) | πC | ξ | M k b f ( · , ξ ) k ∞ . Similar estimates holds true for the derivatives ∂ nt b u ( t, ξ ).Thus, the rapid decaying of the sequence b f ( · , ξ ) implies that b u ( · , ξ ) decays rapidly.Therefore, u ∈ C ∞ ( T × T N ); consequently, L is globally hypoelliptic. (cid:3) In the next sections we will see situations where L is globally hypoelliptic, butthere exist infinitely many indexes ξ such that ℑM ( t, ξ )) changes sign. That is,the assumption that ℑM ( t, ξ ) does not change sign is not necessary for the globalhypoellipticity of the operator L. LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 13 Logarithmic growth
From now on, the speed in which the symbol p ( ξ ) goes to infinity will play a crucialpoint in the study of the global hypoellipticity of L = D t + ( a + ib )( t ) P ( D x ) , ( t, x ) ∈ T × T N . We recall that p ( ξ ) = α ( ξ )+ iβ ( ξ ) , where both α ( ξ ) and β ( ξ ) are real-valued functionsin S m ( Z N ) . In particular(4.1) | α ( ξ ) | C | ξ | m and | β ( ξ ) | C | ξ | m , as | ξ | → ∞ . In this section our goal is to deal with the case where either α ( ξ ) or β ( ξ ) has at mostlogarithmic growth. Definition 4.1.
A function r : Z N → C has at most logarithmic growth if r ( ξ ) = O (log( | ξ | )) , as | ξ | → ∞ , that is, there are positive constants κ and n such that(4.2) | r ( ξ ) | κ log( | ξ | ) , for all | ξ | > n . When this condition fails, we will say that r ( ξ ) has super-logarithmic growth .When either α ( ξ ) or β ( ξ ) has at most logarithmic growth, the global hypoellipticityof L is completely characterized by the following: Theorem 4.2.
Let p ( ξ ) = α ( ξ ) + iβ ( ξ ) ∈ S m ( Z N ) be a symbol. i) If α ( ξ ) = O (log( | ξ | )) and β ( ξ ) = O (log( | ξ | )) , then L is globally hypoelliptic ifand only if L is globally hypoelliptic. ii) If α ( ξ ) = O (log( | ξ | )) and β ( ξ ) has super-logarithmic growth, then L is globallyhypoelliptic if and only if L is globally hypoelliptic and a ( t ) does not changesign. iii) If α ( ξ ) has super-logarithmic growth and β ( ξ ) = O (log( | ξ | )) , then L is globallyhypoelliptic if and only if L is globally hypoelliptic and b ( t ) does not changesign. In the particular case where β ≡ Corollary 4.3.
If the symbol p ( ξ ) is real-valued, then the operator L is globallyhypoelliptic if and only if L is globally hypoelliptic and either i ) p ( ξ ) = O (log( | ξ | )); or ii ) p ( ξ ) has super-logarithmic growth and b ( t ) does not change sign. Remark . When p ( ξ ) is a real-valued symbol having at most logarithmic growth,item i) shows that the behaviour of the function b ( t ) plays no role in the global hy-poellipticity of pseudo-differential operators of type (1.1), what means that the famouscondition ( P ) of Nirenberg-Treves, see [30] and [31], is neither necessary nor sufficientto guarantee global hypoellipticity.On the other hand, item ii) is according to the known result for vector fields L = D t + ( a + ib )( t ) D x on T studied by Hounie in [26]. We recall that in this case, thecondition L globally hypoelliptic means that either b = 0 or a is an irrational non-Liouville number. We split the proof of Theorem 4.2 in two subsections. In Subsection 4.1 we proveitem i ) by using an argument of reduction to normal forms. The proof of items ii ) and iii ) are treated in Subsection 4.2, where the change of sign of the coefficients play animportant role.In Subsection 4.3 we show that the techniques developed in previous subsections canbe applied to study a particular case where the symbol has super logarithmic growth.Before proceeding with the proofs, we present two examples which illustrate thatthe condition ℑM ( t, ξ ) does not change sign in Theorem 3.6 is not necessary for theglobal hypoellipticity of L. Example 4.5. If P ( D x ) = ( − ∆ x ) m/ on T N , with m < , then by item i ) of theTheorem 4.2 the operator L = D t + [1 + i sin( t )]( − ∆ x ) m/ is globally hypoelliptic since L = D t + ( − ∆ x ) m/ , is globally hypoelliptic by Theorem 2.1. Notice that ℑM ( t, ξ ) = sin( t ) | ξ | m changessign for all | ξ | > . Example 4.6.
Assume that P ( D x ) = ( − ∆ x ) m/ + i ( − ∆ x ) n/ on T N , where m n > . Theorem 4.2 item iii ) implies that the operator L = D t + [1 + cos( t ) − i ][( − ∆ x ) m/ + i ( − ∆ x ) n/ ]is globally hypoelliptic, since a ( t ) = 1 + cos( t ) > L = D t + (1 − i )[( − ∆ x ) m/ + i ( − ∆ x ) n/ ]is globally hypoelliptic. Indeed, the assumptions m n > τ, ξ ) ∈ Z × Z N ∗ such that | ξ | > /n , we have | τ + (1 − i )( | ξ | m + i | ξ | n ) | > || ξ | n − | ξ | m | > . Hence, L is globally hypoelliptic by Section 2.Notice that ℑM ( t, ξ ) = (1 + cos( t )) | ξ | n − | ξ | m changes sign for infinitely manyindexes, since m n > . Reduction to normal form.
In this subsection we show that, under the assump-tion of growth at most logarithm of the symbol, the study of the global hypoellipticityof L and L are equivalent.In this situation we have α ( ξ ) = O (log( | ξ | )) and β ( ξ ) = O (log( | ξ | )) , as | ξ | → ∞ , and the proof of item i ) of Theorem 4.2 follows from Corollary 4.8 bellow.We introduce the following (formal) operators: for each distribution u ∈ D ′ ( T × T N ) , we set Ψ a ( u ) = X ξ ∈ Z N e − i ( A ( t ) − a t ) p ( ξ ) b u ( t, ξ ) e ixξ , and Ψ b ( u ) = X ξ ∈ Z N e ( B ( t ) − b t ) p ( ξ ) b u ( t, ξ ) e ixξ , LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 15 where A ( t ) = Z t a ( s ) ds and B ( t ) = Z t b ( s ) ds. Proposition 4.7. If β ( ξ ) = O (log( | ξ | )) , then Ψ a is an isomorphism which satisfies(4.3) Ψ − a ◦ L ◦ Ψ a = L a on both the spaces D ′ ( T × T N ) and C ∞ ( T × T N ) , where L a . = D t + ( a + ib ( t )) P ( D x ) . Analogously, if α ( ξ ) = O (log( | ξ | )) , then Ψ b is an isomorphism which satisfies(4.4) Ψ − b ◦ L ◦ Ψ b = L b on both the spaces D ′ ( T × T N ) and C ∞ ( T × T N ) , where L b . = D t + ( a ( t ) + ib ) P ( D x ) . The proof of this proposition consists in to show that Ψ a and Ψ b are well definedoperators, in this case they are evidently linear operators with inverseΨ − a ( v ) = X ξ ∈ Z N e i ( A ( t ) − a t ) p ( ξ ) b v ( t, ξ ) e ixξ , and Ψ − b ( v ) = X ξ ∈ Z N e − ( B ( t ) − b t ) p ( ξ ) b v ( t, ξ ) e ixξ , respectively, on both the spaces D ′ ( T × T N ) and C ∞ ( T × T N ) . Moreover, identities(4.3) and (4.4) are easily verified.Before starting this proof, let us state the reduction to the normal form:
Corollary 4.8. If β ( ξ ) = O (log( | ξ | )) (respectively α ( ξ ) = O (log( | ξ | ))), then L isglobally hypoelliptic if and only if L a (respectively L b ) is globally hypoelliptic. Proof.
The validity of the identity L a = Ψ − a ◦ L ◦ Ψ a on both D ′ ( T × T N ) and C ∞ ( T × T N ) imply that L is globally hypoelliptic if and only if L a is globally hypoelliptic.In fact, assume that L is globally hypoelliptic and let u ∈ D ′ ( T × T N ) such that L a u = f ∈ C ∞ ( T × T N ). Since v = Ψ a ( u ) ∈ D ′ ( T × T N ) satisfy Lv = Ψ a ( f ) ∈ C ∞ ( T × T N ) , it follows that v ∈ C ∞ ( T × T N ) , since L is globally hypoelliptic.Hence, u = Ψ − a ( v ) ∈ C ∞ ( T × T N ) , which implies that L a is globally hypoelliptic.The converse is similar.Analogously, the validity of the identity L b = Ψ − b ◦ L ◦ Ψ b on both D ′ ( T × T N ) and C ∞ ( T × T N ) will imply that L is globally hypoelliptic if and only if L b is globallyhypoelliptic. (cid:3) The following estimates will be useful in the proof of Proposition 4.7.
Lemma 4.9.
Consider p ∈ S m ( Z N ) . Given k ∈ N , there are positive constants C and n such that | ∂ kt ( e − i ( A ( t ) − a t ) p ( ξ ) ) | C | ξ | km e β ( ξ )( − A ( t )+ a t ) , and | ∂ kt ( e ( B ( t ) − b t ) p ( ξ ) ) | C | ξ | km e α ( ξ )( B ( t ) − b t ) , for each | ξ | > n .Proof. For k = 0 these estimates are evident. If the first estimate holds for ℓ ∈{ , , . . . , k } , then we have: | ∂ k +1 t ( e − i ( A ( t ) − a t ) p ( ξ ) ) | | p ( ξ ) | k X ℓ =0 (cid:18) kℓ (cid:19) | ∂ ℓt e − i ( A ( t ) − a t ) p ( ξ ) |× sup t ∈ T | ∂ k − ℓt ( a ( t ) − a t ) | C | p ( ξ ) | k X ℓ =0 (cid:18) kℓ (cid:19) | ∂ ℓt e − i ( A ( t ) − a t ) p ( ξ ) | C | p ( ξ ) | e β ( ξ )( − A ( t )+ a t ) k X ℓ =0 (cid:18) kℓ (cid:19) | ξ | ℓm C | ξ | ( k +1) m e β ( ξ )( − A ( t )+ a t ) , where we are using | p ( ξ ) | C | ξ | m , as | ξ | → ∞ .The second estimate can be obtained by using similar arguments. (cid:3) Proof of Proposition 4.7.
We have to verify only that Ψ a and Ψ b are well defined linearoperators on both D ′ ( T × T N ) and C ∞ ( T × T N ) whenever β ( ξ ) = O (log( | ξ | )) and α ( ξ ) = O (log( | ξ | )) , respectively.Fixed u ∈ D ′ ( T × T N ) , we must study the behavior of the Fourier coefficients ψ a ( t, ξ ) = e − i ( A ( t ) − a t ) p ( ξ ) b u ( t, ξ ) , for all ξ ∈ Z N , and ψ b ( t, ξ ) = e ( B ( t ) − b t ) p ( ξ ) b u ( t, ξ ) , for all ξ ∈ Z N . Given u ∈ D ′ ( T × T N ) , it follows by Lemma 4.9 the existence of positive constants C and M such that(4.5) |h ψ a ( t, ξ ) , φ i| C | ξ | M k φ k M sup t ∈ T | e β ( ξ )( − A ( t )+ a t ) | , and(4.6) |h ψ b ( t, ξ ) , φ i| C | ξ | M k φ k M sup t ∈ T | e α ( ξ )( B ( t ) − b t ) | , for | ξ | large enough, where k φ k M . = max {| ∂ α φ ( t ) | ; α M, t ∈ T } . If β ( ξ ) = O (log( | ξ | )) , by (4.2) there exist κ > n ∈ N such that(4.7) | β ( ξ ) | log( | ξ | κ ) , for all | ξ | > n . LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 17
Now, take δ < δ > δ − A ( t ) + a t δ , for all t ∈ T . The inequalities (4.7) and (4.8) imply that, for all | ξ | > n we have: β ( ξ )( − A ( t ) + a t )) ( log( | ξ | κδ ) , if β ( ξ ) > , log( | ξ | − κδ ) , if β ( ξ ) < . Hence,(4.9) e β ( ξ )( − A ( t )+ a t ) | ξ | δ , as | ξ | → ∞ , where δ = max { κδ , − κδ } .With similar ideas, by using the fact that α ( ξ ) = O (log( | ξ | )) , we obtain δ > e α ( ξ )( B ( t ) − b t ) | ξ | δ , as | ξ | → ∞ . Then, by (4.5), (4.6) and the last two inequalities |h ψ a ( t, ξ ) , φ i| C | ξ | M + δ k φ k M and |h ψ b ( t, ξ ) , φ i| C | ξ | M + δ k φ k M , for all | ξ | sufficiently large and for all φ ∈ C ∞ ( T ); thus Ψ a · u ∈ D ′ ( T × T N ) andΨ b · u ∈ D ′ ( T × T N ) . Finally, if u ∈ C ∞ ( T × T N ) , then Lemma 4.9 and the rapid decaying of b u ( · , ξ ) implythat for each k ∈ N we obtain C k > M k ∈ R such that | ∂ kt ψ a ( t, ξ ) | C k | ξ | M k e β ( ξ )( − A ( t )+ a t ) k X j =0 | ∂ kt b u ( t, ξ ) | and | ∂ kt ψ b ( t, ξ ) | C k | ξ | M k e α ( ξ )( B ( t ) − b t ) k X j =0 | ∂ kt b u ( t, ξ ) | , for | ξ | large enough.By using again (4.9) and (4.10), and from the rapid decaying of b u ( · , ξ ) , it followsthat Ψ a ( u ) and Ψ b ( u ) are in C ∞ ( T × T N ), what finishes the proof of Proposition 4.7. (cid:3) (cid:3) Change of sign.
Our focus now is to prove item ii ) of Theorem 4.2, in which α ( ξ )has at most logarithmic growth and β ( ξ ) has super-logarithmic growth. Notice that, inthis case, the global hypoellipticity of L cannot be reduced to the global hypoellipticityof a constant coefficient operator.The proof of item iii ) of Theorem 4.2 consists in slight modifications of the techniquesused in the proof of item ii ) . Since the argument is quite similar, it will be omitted.
Proof of item ii ) of Theorem 4.2. We recall that the hypothesis in this case are β ( ξ )has super-logarithmic growth and α ( ξ ) = O (log( | ξ | )), hence, in view of Corollary 4.8we may assume that b ( t ) is constant, b ≡ b . Sufficiency:
Assume that a ( t ) does not change sign and that L is globally hypoelliptic.Let u ∈ D ′ ( T × T N ) be such that iLu = f ∈ C ∞ ( T × T N ) . We will show that u ∈ C ∞ ( T × T N ) . By the Fourier series in the variable x, we are led to the equations b f ( t, ξ ) = ∂ t b u ( t, ξ ) + i M ( t, ξ ) b u ( t, ξ )(4.11) = ∂ t b u ( t, ξ ) + (cid:2) − ( b α ( ξ ) + a ( t ) β ( ξ )) + i ( a ( t ) α ( ξ ) − b β ( ξ )) (cid:3)b u ( t, ξ ) , for all t ∈ T and for all ξ ∈ Z N . By applying Lemma 3.1 to equation (4.11), it follows that t ∈ T b u ( t, ξ ) is smooth,for each ξ ∈ Z N . Since Z M is finite (Remark 3.4), for | ξ | sufficiently large, equation(4.11) has a unique solution. This solution can be written by b u ( t, ξ ) = 11 − e − πi M ( ξ ) Z π exp (cid:18) − Z tt − s i M ( r, ξ ) dr (cid:19) b f ( t − s, ξ ) ds, if ξ is such that a ( t ) β ( ξ ) , for all t ∈ T , and b u ( t, ξ ) = 1 e πi M ( ξ ) − Z π exp (cid:18)Z t + st i M ( r, ξ ) dr (cid:19) b f ( t + s, ξ ) ds, if ξ is such that a ( t ) β ( ξ ) > , for all t ∈ T . Since α ( ξ ) = O (log( | ξ | )) , there exists K > e α ( ξ ) sb | ξ | K | b | , for | ξ | sufficiently large and s ∈ [0 , π ] . Thus, for | ξ | large enough and such that a ( t ) β ( ξ ) , for all t ∈ T , we have (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − Z tt − s i M ( r, ξ ) dr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = exp (cid:18) α ( ξ ) sb + Z tt − s a ( r ) β ( ξ ) dr (cid:19) e α ( ξ ) sb | ξ | K | b | . Similarly, for | ξ | large enough and such that a ( t ) β ( ξ ) > , for all t ∈ T , we have (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18)Z t + st i M ( r, ξ ) dr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = exp (cid:18) − α ( ξ ) sb − Z t + st a ( r ) β ( ξ ) dr (cid:19) e − α ( ξ ) sb | ξ | K | b | . Finally, as in the proof of Theorem 3.6, the global hypoellipticity of L give us acontrol as in (3.7), and the rapid decaying of b f ( · , ξ ) imply that b u ( · , ξ ) decays rapidly.Hence, u belongs to C ∞ ( T × T N ) and L is globally hypoelliptic. Necessity:
By Theorem 3.5, it is enough to prove that the changing of sign of a ( t ) implies that L is not globally hypoelliptic.We will exhibit a smooth function f ( t, x ) = ∞ X n =1 b f ( t, ξ n ) e ixξ n , LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 19 for which iLu = f has a solution in D ′ ( T × T N ) \ C ∞ ( T × T N ) . Our assumptions on β ( ξ ) imply that we may choose a sequence { ξ n } , such that | ξ n | is strictly increasing, | ξ n | > n, and | β ( ξ n ) | > log( | ξ n | n ) , for all n ∈ N . By passing to a subsequence we may assume that either β ( ξ n ) > , for all n, or β ( ξ n ) < , for all n. Without loss of generality, we also may assume that b α ( ξ n ) + a β ( ξ n ) , for all n ∈ N . Indeed, in the other case, it is enough to consider − L and to change thevariable t by − t. Suppose we are in the case β ( ξ n ) > n (the other case is similar).Set M a = max t,s π Z tt − s a ( r ) dr = Z t t − s a ( r ) dr. Since a ( t ) changes sign, M a > s ∈ (0 , π ); moreover, without loss of generality(by performing a translation in the variable t ) we may assume that t and σ . = t − s belong to the open interval (0 , π ) . Let φ ∈ C ∞ c (( σ − ǫ, σ + ǫ )) be a function such that 0 φ ( t ) , and φ ( t ) = 1 ina neighborhood of [ σ − ǫ/ , σ + ǫ/ . We then define b f ( · , ξ n ) by the 2 π − periodic extension of the function(1 − e − πi M ( ξ n ) ) φ ( t ) exp (cid:18) i Z t t ℜ M ( r, ξ n ) dr (cid:19) e − β ( ξ n ) M a e α ( ξ n )( t − t ) b . Since b α ( ξ n )+ a β ( ξ n )
0, we have that 1 − e − πi M ( ξ n ) is bounded and for t ∈ [0 , π ]we have e α ( ξ n )( t − t ) b e | α ( ξ n ) | π | b | , which increases slowly, since α ( ξ ) = O (log( | ξ | )) . Moreover, by using estimate (4.1), the term e − β ( ξ n ) M a will imply that b f ( · , ξ n ) decaysrapidly, since β ( ξ n ) > log( | ξ n | n ) . By Proposition 3.3 we may assume that Z M is finite and, by passing to a subse-quence, that 1 − e − πi M ( ξ n ) = 0 , then we define b u ( t, ξ n ) = 11 − e − πi M ( ξ n ) Z π exp (cid:18) − Z tt − s i M ( r, ξ n ) dr (cid:19) b f ( t − s, ξ n ) ds. For all s, t ∈ [0 , π ] , we have (cid:12)(cid:12)(cid:12)(cid:12) − e − πi M ( ξ n ) b f ( t − s, ξ n ) (cid:12)(cid:12)(cid:12)(cid:12) e | α ( ξ n ) | π | b | e − β ( ξ n ) M a . Thus, | b u ( t, ξ n ) | Z π exp (cid:18) − β ( ξ n ) (cid:16) M a − Z tt − s a ( r ) dr (cid:17)(cid:19) e α ( ξ n ) sb e | α ( ξ n ) | π | b | ds πe | α ( ξ n ) | π | b | . This estimate imply that the sequence b u ( · , ξ n ) increases slowly, since α ( ξ ) = O (log( | ξ | )) . Hence u = ∞ X n =1 b u ( t, ξ n ) e ixξ n ∈ D ′ ( T × T N ) . Note that | b u ( t , ξ n ) | > Z s + δ/ s − δ/ exp (cid:18) − β ( ξ n ) (cid:16) M a − Z t t − s a ( r ) dr (cid:17)(cid:19) ds. Since M a − R t t − s a ( r ) dr > s is a zero of order even, the Laplace Method forIntegrals implies that | b u ( t , ξ n ) | > C ( β ( ξ n )) − / > C (1 + | ξ n | ) − m/ > C − m/ | ξ n | − m/ , where C and m are positive constants and do not depend on n. This estimate impliesthat b u ( · , ξ n ) does not decay rapidly.Hence, u ∈ D ′ ( T × T N ) \ C ∞ ( T × T N ) . Therefore, L is not globally hypoelliptic,since iLu = f by Lemma 3.1.When β ( ξ n ) < n, we repeat the constructions above, where now we use M a = min t,s π Z tt − s a ( r ) dr. The proof of Theorem 4.2 - item ii ) is complete. (cid:3) (cid:3) Remark . In the proof of sufficiency in Theorem 4.2 item ii ) , was not necessarysuppose that β ( ξ ) has super-logarithmic growth. Moreover, we observe that this proofis not consequence of Theorem 3.6, since t
7→ ℑM ( t, ξ ) = a ( t ) β ( ξ ) + b ( t ) α ( ξ ) maychange sign, even when a ( t ) does not change sign.We finish this subsection with an additional example which also exhibit a globallyhypoelliptic operator in a situation in which t ∈ T
7→ ℑM ( t, ξ ) changes sign, forinfinitely many indexes ξ. Example 4.11.
Let b ( t ) be a 2 π − periodic extension of a real smooth nonzero functiondefined on (0 , π ) with integral equals to zero. Let a ( t ) be the 2 π − periodic extensionof the function 1 − ψ, where ψ ∈ C ∞ c ((0 , π ) , R ) , ψ ( t ) ψ ≡ b ( t ) . If P ( D x ) has symbol p ( ξ ) = 1 + i ( | ξ | log(1 + | ξ | )), then L = D t + ( a ( t ) + ib ( t )) P ( D x ) , ( t, x ) ∈ T × T N , is globally hypoelliptic by Theorem 4.2 - item ii ). Note that, t
7→ ℑM ( t, ξ ) = a ( t ) | ξ | log(1 + | ξ | ) + b ( t ) , changes sign for all indexes ξ ∈ Z N . LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 21
A particular class of operators.
The aim of this subsection is to notice thatthere is a particular class of operators, which includes cases in which both α ( ξ ) and β ( ξ )have super-logarithmic growth, where the study of the global hypoellipticity followsfrom adaptations of the techniques used in the proof of Theorem 4.2.For example, if p ( ξ ) = α ( ξ ) + i (1 + α ( ξ )) and α ( ξ ) has super-logarithmic growth, wecannot apply Theorem 4.2 to study the global hypoellipticity of the operator D t + (cos ( t ) + i sin( t )) P ( D x ) , but notice that ℑM ( t, ξ ) splits in the form[sin( t ) + cos ( t )] α ( ξ ) + cos ( t ) . Hence, ℑM ( t, ξ ) satisfies the assumptions concerning the speed of growth which wasassumed in Theorem 4.2. We claim that the operator above is not globally hypoelliptic,since [sin( t ) + cos ( t )] changes sign.More generally, with similar arguments of those used in the proof of Theorem 4.2, wemay give a complete answer about the global hypoellipticity of the operator L, givenby (1.1), in the case where ℑM ( t, ξ ) splits in the following way:(4.12) ℑM ( t, ξ ) = ˜ a ( t ) γ ( ξ ) + ˜ b ( t ) η ( ξ ) , where ˜ a ( t ) and ˜ b ( t ) are real smooth functions on T , and γ ( ξ ) and η ( ξ ) are real valuedtoroidal symbols, such that either γ ( ξ ) = O (log( | ξ | )) or η ( ξ ) = O (log( | ξ | )) . Theorem 4.12.
Let L be the operator defined in (1.1) and assume that the decom-position (4.12) is true. Then L is globally hypoelliptic if and only if L is globallyhypoelliptic and ˜ a ( t ) (respectively ˜ b ( t ) ) does not change sign whenever γ ( ξ ) (respec-tively η ( ξ ) ) has super-logarithmic growth. Observe that, under the assumptions in Theorem 4.12 and assuming that γ ( ξ ) hassuper-logarithmic growth, the converse of Theorem 3.6 holds true provided that thefunction ℑM ( t, ξ ) = ˜ a ( t ) γ ( ξ ) + ˜ b ( t ) α ( ξ ) changes sign if and only if ˜ a changes sign.However, as we saw in Example 4.11, this property does not hold in general.Bellow we present other interesting examples in this direction. Example 4.13. If a ( t ) and b ( t ) do not vanish identically and are R − linearly dependentfunctions, we may write ℑM ( t, ξ ) = b ( t )( α ( ξ ) + λβ ( ξ )) , with λ ∈ R \ { } . In this case, Theorem 4.12 gives a complete answer about the globalhypoellipticity of L. When α ( ξ ) + λβ ( ξ ) = O (log( | ξ | )) , L is globally hypoelliptic if and only if L isglobally hypoelliptic.For instance, if a ( t ) = − b ( t ) and β ( ξ ) = 1 + α ( ξ ) , then ℑM ( t, ξ ) = − a ( t ) . Hence, L is globally hypoelliptic even if b ( t ) changes sign.When α ( ξ ) + λβ ( ξ ) has super-logarithmic growth, L is globally hypoelliptic if andonly if L is globally hypoelliptic and b ( t ) does not change sign. Example 4.14.
When a ( t ) and b ( t ) are R − linearly independent functions, L may benot globally hypoelliptic even if both a ( t ) and b ( t ) do not change sign. Indeed, we mayfind non-zero integers p and q so that a ( t ) p + b ( t ) q changes sign (see Lemma 3.1 of [7]). If, for instance, α ( ξ ) = qγ ( ξ ) and β ( ξ ) = pγ ( ξ ) , in which γ ( ξ ) has super-logarithmicgrowth, then Theorem 4.12 implies that L is not globally hypoelliptic.5. Super-logarithmic growth
The purpose of this section is to present additional results about the global hypoel-lipticity of the operator L given by (1.1), which we recall L = D t + ( a + ib )( t ) P ( D x ) , ( t, x ) ∈ T × T N , where a ( t ) and b ( t ) are real smooth functions on T , and P ( D x ) is a pseudo-differentialoperator on T N , with symbol p ( ξ ) = α ( ξ ) + iβ ( ξ ) , ξ ∈ Z N . We consider a more general situation where either α ( ξ ) or β ( ξ ) has super-logarithmicgrowth and we present a necessary condition for the global hypoellipticity of L, whichis given by a control in the sign of certain functions.Precisely, assume that β ( ξ ) has super-logarithmic growth and let E α,β . = (cid:8) K ∈ R ; there exists { ξ n } ⊂ Z N satisfying ( ∗ ) (cid:9) , ( ∗ ) | ξ n | → ∞ ; α ( ξ n ) /β ( ξ n ) → K, as n → ∞ ; | β ( ξ n ) | > n log( | ξ n | ) , for all n ∈ N . In this case, we prove that L is not globally hypoelliptic if there exists K ∈ E α,β such that the function t ∈ T a ( t ) + b ( t ) K changes sign.An analogous result holds when α ( ξ ) has super-logarithmic growth. In this case, L is not globally hypoelliptic if there exists C ∈ E β,α such that the function t ∈ T b ( t ) + a ( t ) C changes sign.In particular, we obtain a necessary condition for the global hypoellipticity of L when either α ( ξ ) or β ( ξ ) has super-logarithmic growth and the limit lim | ξ |→∞ α ( ξ ) /β ( ξ )exists. When the order of growth of α ( ξ ) is faster (respectively slower) than the orderof growth of β ( ξ ) , the operator L is not globally hypoelliptic if b ( t ) (respectively a ( t ))changes sign (see Corollary 5.2). Theorem 5.1. If β ( ξ ) has super-logarithmic growth, K ∈ E α,β and the function t ∈ T a ( t ) + b ( t ) K changes sign, then L given by (1.1) is not globally hypoelliptic.Similarly, if α ( ξ ) has super-logarithmic growth, C ∈ E β,α , and t ∈ T b ( t ) + a ( t ) C changes sign, then L is not globally hypoelliptic.Proof. We consider the situation in which β ( ξ ) has super-logarithmic growth and K ∈ E α,β . The other situation is analogous.We assume that t ∈ T a ( t ) + b ( t ) K changes sign and prove that L is not globallyhypoelliptic.The assumptions on β and K imply that there exists a sequence { ξ n } such that | ξ n | is strictly increasing, | ξ n | > n, | β ( ξ n ) | > n log( | ξ n | ) , α ( ξ n ) /β ( ξ n ) → K, and ξ n Z M , for all n. Note that we are assuming that Z M is finite, otherwise, by Proposition 3.3,there is nothing to prove. LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 23
Without loss of generality, suppose that b α ( ξ n ) + a β ( ξ n ) , for all n. Indeed, ifnecessary we can consider − L and perform the change of variable t by − t. By using a subsequence, we may assume that either β ( ξ n ) < , for all n, or β ( ξ n ) > , for all n. Suppose first that β ( ξ n ) > , for all n. For each n, set M n = max t,s π (cid:26)Z tt − s a ( r ) + b ( r ) α ( ξ n ) β ( ξ n ) dr (cid:27) = Z t n t n − s n a ( r ) + b ( r ) α ( ξ n ) β ( ξ n ) dr. Again, by passing to a subsequence, there exist t and s such that t n → t and s n → s , as n → ∞ . Since α ( ξ n ) /β ( ξ n ) → K, as n → ∞ , it follows that Z t t − s a ( r ) + b ( r ) Kdr = max t,s π Z tt − s a ( r ) + b ( r ) Kdr . = M ab . Since a ( t ) + b ( t ) K changes sign, we have M ab > s ∈ (0 , π ) . Performing atranslation in the variable t, we may assume that t , s and σ . = t − s belong to(0 , π ) . Choose ǫ > < σ − ǫ and σ + ǫ < t . Consider φ belonging to C ∞ c (( σ − ǫ, σ + ǫ ) , R ) such that 0 φ ( t ) φ ( t ) = 1 for all t ∈ [ σ − ǫ/ , σ + ǫ/ . Finally, we define b f ( · , ξ n ) as being the 2 π − periodic extension of(1 − e − πi M ( ξ n ) ) φ ( t ) exp (cid:18) i Z t n t ℜM ( r, ξ n ) dr (cid:19) e − β ( ξ n ) M n . Note that 1 − e − πi M ( ξ n ) is bounded, since b α ( ξ n ) + a β ( ξ n ) . Thus, by estimate(4.1), the behaviour of the term e − β ( ξ n ) M n when | ξ n | → ∞ imply that b f ( · , ξ n ) decaysrapidly, since M n → M ab > β ( ξ n ) > log( | ξ n | n ) . It follows that f ( t, x ) = ∞ X n =1 b f ( t, ξ n ) e ixξ n ∈ C ∞ ( T × T N ) . Since ξ n Z M , we may define b u ( t, ξ n ) = 11 − e − πi M ( ξ n ) Z π exp (cid:18) − i Z tt − s M ( r, ξ n ) dr (cid:19) b f ( t − s, ξ n ) ds, which belongs to C ∞ ( T ) . For n large enough, the estimate | (1 − e − πi M ( ξ n ) ) − b f ( t − s, ξ n ) | e − β ( ξ n ) M n | b u ( t, ξ n ) | Z π exp (cid:18) − β ( ξ n ) (cid:16) M n − Z tt − s a ( r ) + b ( r ) α ( ξ n ) β ( ξ n ) dr (cid:17)(cid:19) ds π. Hence, b u ( · , ξ n ) increases slowly. Then u = ∞ X n =1 b u ( t, ξ n ) e ixξ n ∈ D ′ ( T × T N ) . We will show that u C ∞ ( T × T N ) . In fact, for n sufficiently large we have σ + ǫ < t n , from which we can infer that | b u ( t n , ξ n ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z t n − σ + ǫt n − σ − ǫ φ ( t n − s ) × exp (cid:18) − β ( ξ n ) (cid:18) M n − Z t n t n − s a ( r ) + b ( r ) α ( ξ n ) β ( ξ n ) dr (cid:19)(cid:19) ds (cid:12)(cid:12)(cid:12)(cid:12) . Since t n − s n → σ , we have t n − s n − σ − ǫ/ < − ǫ/ t n − s n − σ + ǫ/ > ǫ/ , for n large enough. Hence, for n large enough, we have( s n − ǫ/ , s n + ǫ/ ⊂ ( t n − σ − ǫ, t n − σ + ǫ )and φ ( t n − s ) = 1 , for s ∈ ( s n − ǫ/ , s n + ǫ/ . It follows that | b u ( t n , ξ n ) | > Z s n + ǫ/ s n − ǫ/ exp (cid:18) − β ( ξ n ) (cid:16) M n − Z t n t n − s a ( r ) + b ( r ) α ( ξ n ) β ( ξ n ) dr (cid:17)(cid:19) ds. For each n, the function[ s n − ǫ/ , s n + ǫ/ ∋ s φ n ( s ) . = M n − Z t n t n − s a ( r ) + b ( r ) α ( ξ n ) β ( ξ n ) dr vanishes at s n and φ n ( s ) > , for all s. Furthermore, since α ( ξ n ) /β ( ξ n ) → K and t n → t , there exists C > , which does not depend on n, such that | b u ( t n , ξ n ) | > Z s n + ǫ/ s n − ǫ/ e − β ( ξ n ) C ( s − s n ) ds. The Laplace Method for Integrals implies that | b u ( t n , ξ n ) | > e Cβ ( ξ n ) − / , where e C > n. As in the proof of necessity of item ii ) in Theorem 4.2, the previous estimate impliesthat b u ( · , ξ n ) does not decay rapidly. Hence, u belongs to D ′ ( T × T N ) \ C ∞ ( T × T N )and L is not globally hypoelliptic.Finally, in the case β ( ξ n ) < , for all n, we repeat the technique above, but now weuse M n = min t,s π (cid:26)Z tt − s a ( r ) + b ( r ) α ( ξ n ) β ( ξ n ) dr (cid:27) = Z t n t n − s n a ( r ) + b ( r ) α ( ξ n ) β ( ξ n ) dr and M ab = min t,s π Z tt − s a ( r ) + b ( r ) Kdr = Z t t − s a ( r ) + b ( r ) Kdr < . The proof of Theorem 5.1 is complete. (cid:3)
Corollary 5.2. If β ( ξ ) has super-logarithmic growth and α ( ξ ) /β ( ξ ) → K, as | ξ | → ∞ , then L is not globally hypoelliptic if a ( t ) + b ( t ) K changes sign. Similarly, if α ( ξ ) hassuper-logarithmic growth and β ( ξ ) /α ( ξ ) → C, as | ξ | → ∞ , then L is not globallyhypoelliptic if b ( t ) + a ( t ) C changes sign. LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 25
We say that β ( ξ ) goes to infinity faster than α ( ξ ), and use the notation α ( ξ ) = o ( β ( ξ )), if for all positive constant κ there exists a positive constant n such that | α ( ξ ) | κ | β ( ξ ) | , for all | ξ | > n . Note that, in this case, α ( ξ ) /β ( ξ ) → , as | ξ | → ∞ . Corollary 5.3. If β ( ξ ) has super-logarithmic growth and α ( ξ ) = o ( β ( ξ )) , then L isnot globally hypoelliptic if a ( t ) changes sign. If α ( ξ ) has super-logarithmic growth and β ( ξ ) = o ( α ( ξ )) , then L is not globally hypoelliptic if b ( t ) changes sign. Remark . The main contribution of Theorem 5.1 and its corollaries is in the casewhere both α ( ξ ) and β ( ξ ) have super-logarithmic growth. We invite the reader tocompare this result with items ii ) and iii ) in Theorem 4.2. Example 5.5. If a ( t ) = cos ( t ) , b ( t ) = − sin ( t ) , α ( ξ ) = p | ξ | and β ( ξ ) = p | ξ | + 1 , then Theorem 5.1 implies that L is not globally hypoelliptic. Note that α ( ξ ) /β ( ξ ) → , as | ξ | → ∞ , and a ( t ) + b ( t ) = cos ( t ) − sin ( t ) changes sign.Under the conditions in Corollary 5.2, in the case in which β ( ξ ) has super-logarithmicgrowth with lim inf | ξ |→∞ | ξ | M | β ( ξ ) | > , for some M > K . = lim | ξ |→∞ α ( ξ ) /β ( ξ ) , the operator L is globally hypoellipticprovided that a ( t ) + b ( t ) K never vanishes. In fact, for | ξ | sufficiently large, the function t
7→ ℑM ( t, ξ ) = β ( ξ )[ a ( t ) + b ( t ) α ( ξ ) /β ( ξ )]does not change sign. Moreover, L is globally hypoelliptic, since | a + b K | > | τ + M ( ξ ) | > |ℑM ( ξ ) | > | β ( ξ ) || a + b α ( ξ ) /β ( ξ ) | > | β ( ξ ) || a + b K | / , for | ξ | large enough. Hence, Theorem 3.6 implies that L is globally hypoelliptic. Example 5.6.
Assume that a ( t ) = 1 + sin( t ) and b ( t ) = 1 − sin( t ) . If α ( ξ ) = p | ξ | + ξ and β ( ξ ) = ξ, then α ( ξ ) /β ( ξ ) → , as | ξ | → ∞ , and a ( t ) + b ( t ) = 2 never vanishes.Hence, L is globally hypoelliptic.On the other hand, if a ( t ) + b ( t ) K does not change sign, but a ( t ) + b ( t ) K vanishes,then L may be non-globally hypoelliptic. In Subsection 5.1 we explore this phenomenonwhen K = 0 , where we present a non-globally hypoelliptic operator in the case in which a ( t ) does not change sign, a ( t ) vanishes (of finite order) at a singular point, both α ( ξ )and β ( ξ ) have super-logarithmic growth, and α ( ξ ) = o ( β ( ξ )) . Order of vanishing.
The idea here is to show that certain relations between theorder of vanishing of a ( t ) and the speed in which α ( ξ ) and β ( ξ ) go to infinity, play arole in the global hypoellipticity of the operators studied in this article.We start with an example which illustrates that the operator may be globally hy-poelliptic if, for ξ large, the functions ℑM ( t, ξ ) vanishes only of finite order, and theorder of vanishing at each zero is appropriated to absorb the growth of p ( ξ ) . This sit-uation is generalized in Theorem 5.7 and, in the sequence, we show that the converseof this result does not hold.
First example:
Let b ≡ a ∈ C ∞ ( T , R ) be a function such that a ( t ) = − ( t − π ) on a fixed interval ( π − ǫ, π + ǫ ) , a ( t ) is increasing on [0 , π − ǫ ) , and is decreasing on( π + ǫ, π ] . Note that, a − (0) ∩ [0 , π ] = { π } and a ( t ) < t ∈ [0 , π ) ∪ ( π, π ] . Setting p ( ξ ) = p | ξ | + i | ξ | p | ξ | , ξ ∈ Z , we have ℑM ( t, ξ ) = p | ξ | ( | ξ | a ( t ) + 1) . Note that ℑM ( · , ξ ) changes sign for all but a finite number of indexes ξ. We will prove that L = D t + ( a + i ) P ( D x ) , ( t, x ) ∈ T , is globally hypoelliptic, for this, given u ∈ D ′ ( T ) such that iLu = f, with f ∈ C ∞ ( T ) , we must show that u ∈ C ∞ ( T ) . Lemma 3.1 implies that b u ( · , ξ ) belongs to C ∞ ( T ) for all ξ, and for | ξ | > − a − , wemay write b u ( t, ξ ) = 11 − e − πi M ( ξ ) Z π exp (cid:18) − Z tt − s i M ( r, ξ ) dr (cid:19) b f ( t − s, ξ ) ds. For | ξ | > − a − , the term (1 − e − πi M ( ξ ) ) − is bounded; indeed, since a < π ℑM ( ξ ) = 2 π p | ξ | ( | ξ | a + 1) → −∞ , as | ξ | → ∞ . Moreover, for t, s ∈ [0 , π ] , we have −ℜ Z tt − s i M ( r, ξ ) dr = Z tt − s ℑM ( r, ξ ) dr = Z tt − s | ξ | p | ξ | a ( r ) + p | ξ | dr Z π +1 / √ | ξ | π − / √ | ξ | | ξ | p | ξ | a ( r ) + p | ξ | dr = − Z π +1 / √ | ξ | π − / √ | ξ | | ξ | p | ξ | ( r − π ) dr + 2 = 4 / , for | ξ | sufficiently large.Hence, the rapid decaying of b f ( · , ξ ) and estimates (4.1) will imply that b u ( · , ξ ) decaysrapidly. Hence, u ∈ C ∞ ( T ) and L is globally hypoelliptic.The following result generalizes the situation presented in the previous example. Theorem 5.7.
Suppose that β ( ξ ) has super-logarithmic growth with lim inf | ξ |→∞ | ξ | M | β ( ξ ) | > , for some M > and α ( ξ ) = o ( β ( ξ )) . Assume that a ( t ) does not change sign andvanishes of finite order only. Write a − (0) = { t < · · · < t n } and let m j be the orderof vanishing of a ( t ) at t j , j = 1 , . . . , n. If for each j we have | α ( ξ ) /β ( ξ ) | /m j | α ( ξ ) | = O (log( | ξ | )) , then the operator L given by (1.1) is globally hypoelliptic. LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 27
Proof.
Given u ∈ D ′ ( T × T N ) such that iLu = f, with f ∈ C ∞ ( T × T N ) , we mustshow that u ∈ C ∞ ( T × T N ) . Without loss of generality, assume that a ( t ) is non-negative.By Lemma 3.1, the coefficients b u ( · , ξ ) are smooth on T , for all ξ ∈ Z N . Moreover,since ℑM ( ξ ) = β ( ξ ) a + b α ( ξ ) , a > α ( ξ ) = o ( β ( ξ )) , for | ξ | large enough wehave ℑM ( ξ ) = 0 , and then M ( ξ ) Z M . Hence, for | ξ | sufficiently large, we may write b u ( t, ξ ) = 11 − e − πi M ( ξ ) Z π exp (cid:18) − Z tt − s i M ( r, ξ ) dr (cid:19) b f ( t − s, ξ ) ds, if β ( ξ ) < , and b u ( t, ξ ) = 1 e πi M ( ξ ) − Z π exp (cid:18)Z t + st i M ( r, ξ ) dr (cid:19) b f ( t + s, ξ ) ds, if β ( ξ ) > . We must show that the sequence b u ( · , ξ ) decays rapidly. Notice that, | τ + M ( ξ ) | > |ℑM ( ξ ) | = | β ( ξ ) || ( a + b α ( ξ ) /β ( ξ )) | > C | ξ | − M , when | ξ | → ∞ and it follows by Proposition 3.2 that | − e − πi M ( ξ ) | − and | e πi M ( ξ ) − | − have at most polynomial growth.Now, let I = ∪ nj =1 I j be a neighborhood of a − (0) such that a ( t ) = ( t − t j ) m j a j ( t ) , t ∈ I j , where a j ( t ) > C j > , and m j is an even number, so a ( t ) does not change sign.For the indexes ξ such that β ( ξ ) < | ξ | is sufficiently large, we have β ( ξ )( a ( r ) + b ( r ) α ( ξ ) /β ( ξ )) < T \ I. Moreover, if β ( ξ )( a ( r ) + b ( r ) α ( ξ ) /β ( ξ )) = 0for a certain r in I j , then ( r − t j ) m j a j ( r ) = − b ( r ) α ( ξ ) /β ( ξ ) . In particular, | r − t j | = (cid:12)(cid:12)(cid:12)(cid:12) b ( r ) α ( ξ ) a j ( r ) β ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) /m j C ′ j (cid:12)(cid:12)(cid:12)(cid:12) α ( ξ ) β ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) /m j , where C ′ j = ( k b k ∞ /C j ) /m j .It follows that β ( ξ )( a ( r ) + b ( r ) α ( ξ ) /β ( ξ )) < T \ n [ j =1 " t j − C ′ j (cid:12)(cid:12)(cid:12)(cid:12) α ( ξ ) β ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) /m j , t j + C ′ j (cid:12)(cid:12)(cid:12)(cid:12) α ( ξ ) β ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) /m j . Hence, for the indexes ξ such that β ( ξ ) < | ξ | is sufficiently large, we obtain Z tt − s β ( ξ )( a ( r ) + b ( r ) α ( ξ ) /β ( ξ )) dr n X j =1 Z t j + C ′ j | α ( ξ ) β ( ξ ) | /mj t j − C ′ j | α ( ξ ) β ( ξ ) | /mj ( r − t j ) m j a j ( r ) β ( ξ ) + b ( r ) α ( ξ ) dr. Since Z t j + C ′ j | α ( ξ ) β ( ξ ) | /mj t j − C ′ j | α ( ξ ) β ( ξ ) | /mj ( r − t j ) m j a j ( r ) β ( ξ ) + b ( r ) α ( ξ ) dr K j (cid:12)(cid:12)(cid:12)(cid:12) α ( ξ ) β ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12) /m j | α ( ξ ) | , for some positive constant K j , and | α ( ξ ) /β ( ξ ) | /m j | α ( ξ ) | = O (log( | ξ | )) (by hypothesis),it follows that there exists a positive constant f M such that (cid:12)(cid:12)(cid:12)(cid:12) exp (cid:18) − Z tt − s i M ( r, ξ ) dr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = exp (cid:18)Z tt − s β ( ξ ) (cid:16) a ( r ) + b ( r ) α ( ξ ) β ( ξ ) (cid:17) dr (cid:19) | ξ | f M , for all the indexes ξ such that β ( ξ ) < | ξ | is sufficiently large.This same procedure may be used to verify that a similar estimate holds true forthe indexes ξ such that β ( ξ ) > | ξ | is sufficiently large.Finally, by using estimates above and (4.1), we may verify that the rapid decayingof b f ( · , ξ ) will imply that b u ( · , ξ ) decays rapidly. Therefore u ∈ C ∞ ( T × T N ) and L isglobally hypoelliptic. (cid:3) Remark . We have a similar result when α ( ξ ) has super-logarithmic growth,lim inf | ξ |→∞ | ξ | M | α ( ξ ) | > ,β ( ξ ) = o ( α ( ξ )) , and b ( t ) does not change sign and vanishes only of finite order.In the next example we show that the converse of Theorem 5.7 does not hold true. Second example:
Consider a ∈ C ∞ ( T , R ) as in the first example in this subsection.We will see that L = D t + ( a ( t ) + i ) P ( D x ) , where ( t, x ) ∈ T and p ( ξ ) = ξ + iξ , is notglobally hypoelliptic. Notice that a ( t ) does not change sign, but ℑM ( t, ξ ) = ξ a ( t ) + ξ changes sign for infinitely many indexes ξ ∈ Z . For ξ > ξ a ( t ) + ξ < , π − / p ξ ) ∪ ( π + 1 / p ξ, π ]and ξ a ( t ) + ξ = − ξ ( t − π ) + ξ > π − / √ ξ, π + 1 / √ ξ ) , so that M ξ . = max t,s ∈ [0 , π ] Z tt − s ℑM ( r, ξ ) dr = Z π +1 / √ ξπ − / √ ξ − ξ ( r − π ) + ξdr = 4 p ξ/ . LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 29
Let b f ( · , ξ ) be the 2 π − periodic extension of(1 − e − πi M ( ξ ) ) exp i Z π +1 / √ ξt ℜM ( r, ξ ) dr ! e − M ξ φ ξ ( t ) , in which φ ξ ∈ C ∞ c (( π − / √ ξ, π ) , R ) is given by φ ξ ( t ) . = ψ ( √ ξ ( t − π + 1 / √ ξ )) , with ψ ∈ C ∞ c (( − , , R ) , ψ , and ψ ≡ − / , / . Notice that 1 − e − πi M ( ξ ) is bounded, since a < ℑM ( ξ ) = ξ a + ξ < , for ξ large enough.With these definitions, by using (4.1) we may see that b f ( · , ξ ) decays rapidly. Hence f ( t, x ) . = X ξ> − a − b f ( t, ξ ) e ixξ ∈ C ∞ ( T ) . In order to exhibit u ∈ D ′ ( T ) \ C ∞ ( T ) such that iLu = f, consider b u ( t, ξ ) = 11 − e − πi M ( ξ ) Z π exp (cid:18) − Z tt − s i M ( r, ξ ) dr (cid:19) b f ( t − s, ξ ) ds, for ξ > − a − .Note that 1 − e − πi M ( ξ ) = 0; hence b u ( · , ξ ) is well defined and belongs to C ∞ ( T ).For s, t ∈ [0 , π ] , we have (cid:12)(cid:12)(cid:12)(cid:12) − e − πi M ( ξ ) b f ( t − s, ξ ) exp (cid:18) − Z tt − s i M ( r, ξ ) dr (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) k ψ k ∞ exp (cid:18) − (cid:16) M ξ − Z tt − s ℑM ( r, ξ ) dr (cid:17)(cid:19) . Thus | b u ( t, ξ ) | π, which implies that b u ( · , ξ ) increases slowly. It follows that u = X ξ> − a − b u ( t, ξ ) e ixξ ∈ D ′ ( T ) , and Lemma 3.1 implies that iLu = f. Finally, | b u ( π + 1 / p ξ, ξ ) | = Z / √ ξ / √ ξ φ ξ ( π + 1 / p ξ − s ) × exp − (cid:16) M ξ − Z π +1 / √ ξπ +1 / √ ξ − s ℑM ( r, ξ ) dr (cid:17)! ds. Since 2 / √ ξ is a zero of order at least two of θ ξ ( s ) . = M ξ − Z π +1 / √ ξπ +1 / √ ξ − s ℑM ( r, ξ ) dr > , it follows that θ ξ ( s ) ( s − / p ξ ) k θ ′′ ξ k ∞ ≤ ( s − / p ξ ) ξ ( k a k ∞ + 1) . Hence | b u ( π + 1 / p ξ, ξ ) | > Z / √ ξ / √ ξ ψ (2 − s p ξ ) e − ξ ( k a k ∞ +1)( s − / √ ξ ) ds > Z / (2 √ ξ )3 / (2 √ ξ ) e − ξ ( k a k ∞ +1)( s − / √ ξ ) ds = Z / (2 √ ξ ) − / (2 √ ξ ) e − ξ ( k a k ∞ +1) s ds. As previously mentioned, the Laplace Method for Integrals implies that | b u ( π + 1 / p ξ, ξ ) | > K/ξ, where K is a positive constant which does not depend on ξ. In particular, b u ( · , ξ ) doesnot decay rapidly and L is not globally hypoelliptic.6. Homogeneous operators
In the previous section we saw that, (in general) the converse of Theorem 3.6 doesnot hold, since there exist globally hypoelliptic operators of type (1.1) for which thefunction t ∈ T → ℑM ( t, ξ ) changes sign, for infinitely many indexes ξ. We present here a class of symbols where the converse holds. For instance, if p ( ξ )is homogeneous of order one, then this converse holds, since, in this case, condition( P ) of Nirenberg-Treves is necessary for the global hypoellipticity (see [25], Corollary26.4.8).We will see that the converse of Theorem 3.6 holds true in the case in which p ( ξ ) ishomogeneous of any positive degree.In the sequel, we present a class of operators composed of a sum of homogeneouspseudo-differential operators, for which the study of the global hypoellipticity followsfrom the techniques used in this article. Theorem 6.1.
Assume that the symbol of P ( D x ) is homogeneous of degree m.i ) If m then L is globally hypoelliptic if and only if L is globally hypoelliptic; ii ) If m > then L is globally hypoelliptic if and only if L is globally hypoelliptic,and the function t
7→ ℑM ( t, ξ ) does not change sign, for all ξ ∈ Z N \ { } . Proof. If m , the result follows from item i ) of Theorem 4.2. For the case in which m > , the presented conditions are sufficient thanks to Theorem 3.6. On the otherhand, if there exists ξ ∈ Z N \ { } such that t
7→ ℑM ( t, ξ ) changes sign, then t ( n | ξ | ) m ℑM ( t, ξ / | ξ | )changes sign for all n ∈ N . Now in order to show that L is not globally hypoelliptic, we may repeat the tech-niques in the proof of the necessity in item ii ) of Theorem 4.2. (cid:3) The following result is a consequence of Theorem 2.4 and Theorem 6.1.
LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 31
Corollary 6.2.
Let p = p ( ξ ) be a homogeneous symbol of degree m = ℓ/q, with ℓ, q ∈ N , and gcd( ℓ, q ) = 1. Write p (1) = α + iβ and p ( −
1) = e α + i e β. The operator L = D t + ( a + ib )( t ) P ( D x ) , ( t, x ) ∈ T , is globally hypoelliptic if and only if the following statements occur:i) the functions t ∈ T a ( t ) β + b ( t ) α and t ∈ T a ( t ) e β + b ( t ) e α do not changesign.ii) ( a α − b β ) q is an irrational non-Liouville number whenever a β + b α = 0, and( a e α − b e β ) q is an irrational non-Liouville number whenever a e β + b e α = 0 . Sum of homogeneous operators.
The techniques used in this article allow usto study the global hypoellipticity of operators of the type(6.1) L = D t + N X j =1 ( a j + ib j )( t ) P j ( D x j ) , ( t, x ) ∈ T × T N , where each P j ( D x j ) is homogeneous of degree m j (see Definition 2.2), so that its symbol p j ( ξ j ) satisfies p j ( ξ j ) = ( ξ m j j p j (1) , if ξ j > , | ξ j | m j p j ( − , if ξ j < . The results presented in this subsection generalize Theorem 1.3 of [6], see Corollary6.6 below.The constant coefficient operator L associated to the operator L given by (6.1) is L = D t + N X j =1 ( a j + ib j ) P j ( D x j ) . We also set, for j = 1 , . . . , N, M j ( t, ξ j ) . = ( a j + ib j )( t ) p j ( ξ j ) , M j ( ξ j ) . = ( a j + ib j ) p j ( ξ j ) M ( t, ξ ) . = N X j =1 M j ( t, ξ j ) , and M ( ξ ) . = N X j =1 M j ( ξ j ) . Theorem 6.3.
The operator L given by (6.1) is globally hypoelliptic if the followingsituations occur: i) L is globally hypoelliptic. ii) for each pair j, k ∈ { , . . . , N } ( j = k ) such that m j > and m k > , the setsof real-valued functions Υ r,s . = {ℑM j ( · , ( − r ) , ℑM k ( · , ( − s ) } , r, s ∈ { , } , are R − linearly dependent. iii) for each ξ j ∈ Z \ { } , the function t ∈ T → ℑM j ( t, ξ j ) does not change signwhenever m j > , j = 1 , . . . , N. On the other hand, if L is globally hypoelliptic, then conditions i ) and iii ) hold. We notice that L may be non-globally hypoelliptic if conditions i ) and iii ) hold, butcondition ii ) fails. For instance, consider the operator D t + i cos ( t ) D x + i √ ( t ) D x , ( t, x , x ) ∈ T . This operator satisfies i ) and iii ) , but ii ) fails, since cos ( t ) and √ ( t ) are R − linearlyindependent functions. Theorem 1.3 of [7] implies that this operator is not globallyhypoelliptic.Before presenting the proof of Theorem 6.3, we give an example which shows thatcondition ii ) , in general, is not necessary for the global hypoellipticity of L. Example 6.4.
Consider L = D t + i cos ( t ) D x + i sin ( t ) D x , ( t, x , x ) ∈ T . Note that ℑM ( t,
1) = cos ( t ) and ℑM ( t,
1) = sin ( t ) are R − linearly independentfunctions. Moreover, condition iii ) is satisfied and we have | τ + i ℑM ( ξ ) + i ℑM ( ξ ) | = | τ + i ( ξ + ξ ) / | > / , for all ( ξ , ξ ) ∈ Z \ { (0 , } . Hence, condition i ) is also satisfied.By using partial Fourier series in the variables ( x , x ) and proceeding as in the proofof Theorem 3.6, we see that L is globally hypoelliptic. Sketch of the proof of Theorem 6.3. Sufficient conditions:
Given a distribution u ∈ D ′ ( T t × T Nx ) such that iLu = f, with f ∈ C ∞ ( T × T N ) , we must show that u ∈ C ∞ ( T × T N ) . By using partial Fourier series in the variable x = ( x , . . . , x N ) , we are led to theequations ∂ t b u ( t, ξ ) + i b u ( t, ξ ) N X j =1 M j ( t, ξ j ) = b f ( t, ξ ) , t ∈ T , ξ = ( ξ , . . . , ξ N ) ∈ T N . Since L is globally hypoelliptic, proceeding as in the proof of Proposition 3.3 wesee that Z M = ( ξ ∈ Z N ; N X j =1 M j ( ξ j ) ∈ Z ) is finite. Hence, Lemma 3.1 implies that for all but a finite number of indexes ξ, b u ( t, ξ )is written as either b u ( t, ξ ) = 11 − e − πi M ( ξ ) Z π exp (cid:18) − i Z tt − s M ( r, ξ ) dr (cid:19) b f ( t − s, ξ ) ds, (6.2)or b u ( t, ξ ) = 1 e πi M ( ξ ) − Z π exp (cid:18) i Z t + st M ( r, ξ ) dr (cid:19) b f ( t + s, ξ ) ds, (6.3)where now M ( t, ξ ) = N X j =1 M j ( t, ξ j ) . Assume that m j > , for j = 1 , . . . , r, and m j , for j = r + 1 , . . . , N. LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 33
From formulas (6.2) and (6.3) we see that, in order to show that b u ( · , ξ ) decaysrapidly, it is enough to control the imaginary part of the functions t ∈ T → r X j =1 M j ( t, ξ j ) . Recall that the global hypoellipticity of L implies that the sequences (1 − e ± πi M ( ξ ) ) − increases slowly (Proposition 3.2).For the indexes ξ ∈ Z N such that ξ = · · · = ξ r = 0 , we have r X j =1 ℑM j ( t, ξ j ) = r X j =1 ℑM j ( t, , which does not depend on ξ. Suppose now that ξ ∈ Z N is such that ξ = 0 . Since ℑM j ( t, ξ j ) = | ξ j | m j ℑM j ( t, ± , ξ j = 0 , under assumption ii ) it follows that r X j =1 ℑM j ( t, ξ j ) = ℑM ( t, ± r X j =1 λ ± j | ξ j | m j + r X j =1 γ j ℑM j ( t, , where λ ± j and γ j are real numbers, j = 1 , . . . , r. Moreover, γ j = 0 when ξ j = 0 , and λ ± j = 0 when ξ j = 0 . An analogous formula holds if at least one ξ j is non-zero, for j = 1 , . . . , r. We note that we have a finite number of such formulas which we may use to represent r X j =1 ℑM j ( t, ξ j ) , for all indexes ξ such that at least one ξ j = 0 , j = 1 , . . . , r. By using these formulas and condition iii ) we may see that the rapid decaying of b f ( · , ξ ) implies that b u ( · , ξ ) decays rapidly (similar to which was done in the proof ofitem ii ) of Theorem 4.2).Therefore, conditions i ) − iii ) imply that L is globally hypoelliptic. Necessary conditions:
Proceeding as in the proof of Theorem 3.5, where now M ( t, ξ ) = N X j =1 M j ( t, ξ j ) , we see that condition i ) is necessary.The necessity of condition iii ) follows from Theorem 6.1. Indeed, if L j . = D t + ( a j + ib j )( t ) P j ( D x j ) , ( t, x j ) ∈ T , is not globally hypoelliptic, there exists ν ∈ D ′ ( T t,x j ) ) \ C ∞ ( T ) such that L j ν ∈ C ∞ ( T ) . Setting x ′ = ( x , . . . , x j − , x j +1 , . . . , x N ) , it follows that µ . = ν ⊗ x ′ ∈ D ′ ( T × T N − ) \ C ∞ ( T × T N − )and µ satisfies Lµ = L j ν ∈ C ∞ ( T × T N ) . Hence, L is not globally hypoelliptic ifcondition iii ) fails. (cid:3) (cid:3) It follows from Theorem 1.3 of [6] that condition ii ) of Theorem 6.3 is necessary if P j ( D x j ) = D x j , j = 1 , . . . , N. The next result gives a larger class of operators for whichthis necessity still holds true.
Theorem 6.5.
Assume that the operator L defined in (6.1) is globally hypoelliptic.Then, for all j, k ∈ { , . . . , N } such that j = k, m j , m k ∈ Z ∗ + and gcd( m j , m k ) = 1 ,the sets Υ r,s = {ℑM j ( · , ( − r ) , ℑM k ( · , ( − s ) } , r, s ∈ { , } , are R − linearly dependent.Proof. Let m and m be positive integers such that gcd ( m , m ) = 1 and assume that ℑM ( · ,
1) and ℑM ( · ,
1) are R − linearly independent functions in C ∞ ( T , R ) (theother possibilities are analogous).By Lemma 3.1 of [7] there exist non-zero integers p = q such that t
7→ ℑM ( t, p + ℑM ( t, q changes sign and has non-zero mean.Inspired by (2.5), we multiply this function by p ( m − m +( m − ℓ m q ℓ m , where ℓ and ℓ are non-negative integers such that ℓ m − ℓ m = 1 . Hence, thefunction t [ p ℓ ( m − m q ℓ ] m ℑM ( t,
1) + [ p m − m − ℓ q ℓ ] m ℑM ( t, , changes sign.Setting ˜ p = p ℓ ( m − m q ℓ and ˜ q = p m − m − ℓ q ℓ , it follows that ˜ p and ˜ q areintegers and n m m [˜ p m ℑM ( t,
1) + ˜ q m ℑM ( t, ℑM ( t, ˜ pn m ) + ℑM ( t, ˜ qn m )] . Notice that, changing the variable t by − t and considering − L (if necessary), wemay assume that ℑM (˜ p ) + ℑM (˜ q ) < . We then proceed as in the proof of necessity in item ii ) of Theorem 4.2 in order toshow that L . = D t + ( a + ib ) P ( D x ) + ( a + ib )( t ) P ( D x ) , ( t, x , x ) ∈ T , is not globally hypoelliptic. As before, this implies that L is not globally hypoelliptic.To be more precise, the technique to show that L is not globally hypoellipticconsists of using the change of sign of n m m [ ℑM ( t, ˜ p ) + ℑM ( t, ˜ q )] to construct asmooth function ˆ f ( t, x , x ) = ∞ X n =1 b f ( t, ˜ pn m , ˜ qn m ) e i (˜ pn m x +˜ qn m x ) LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 35 such that iLu = f has a solution in D ′ ( T ) \ C ∞ ( T ) . The Fourier coefficients b f ( · , ˜ pn m , ˜ qn m )are the 2 π − periodic extension ofΘ , ( n ) φ ( t ) exp (cid:18) i Z t t ℜ M ( r, ˜ pn m ) + ℜ M ( r, ˜ qn m ) dr (cid:19) e − n m m M , where Θ , ( n ) . = 1 − e − πi [ M (˜ pn m )+ M (˜ qn m )] and M = max t,s π Z tt − s ℑM ( r, ˜ p ) + ℑM ( r, ˜ q ) dr, which is supposed to be assumed in t = t and s = s , and φ is a smooth cutoff functionidentically one in a small neighborhood of t − s . (cid:3) Corollary 6.6.
Suppose that each symbol p j ( ξ j ) is real-valued and homogeneous,whose degree is a positive integer m j . Assume also thatgcd( m j , m k ) = 1 , for j = k, j, k ∈ { , . . . , N } . Under these assumptions, L given by (6.1) is globally hypoelliptic if and only if thefollowing occurs:i) L is globally hypoelliptic.ii) dim span { b j ∈ C ∞ ( T , R ); j = 1 , . . . , N } b j ( t ) does not change sign, for j = 1 , . . . , N. References [1] Fernando de ´Avila Silva, Todor V. Gramchev, and Alexandre Kirilov,
Global Hypoellipticity forFirst-Order Operators on Closed Smooth Manifolds,
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LOBAL HYPOELLIPTICITY FOR PSEUDO-DIFFERENTIAL OPERATORS 37
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