Local Hypoellipticity by Lyapunov Function

and Applied Analysis 3 just recalling that D(Ω;H) is the topological dual space of C c (Ω;H ∞ ), where the last one is equipped with the inductive limit. The operators Lj and Lj,0, defined above, can be used to define complexes of differential operator, L : Λ p C ∞ (Ω;H ∞ ) 󳨀→ Λ p+1 C ∞ (Ω;H ∞ ) , L : Λ p D 󸀠 (Ω;H −∞ ) 󳨀→ Λ p+1 D 󸀠 (Ω;H −∞ ) , (15) 0 ≤ p ≤ n, and L0 : Λ p C ∞ (Ω;H ∞ ) 󳨀→ Λ p+1 C ∞ (Ω;H ∞ ) , L0 : Λ p D 󸀠 (Ω;H −∞ ) 󳨀→ Λ p+1 D 󸀠 (Ω;H −∞ ) , (16) also with 0 ≤ p ≤ n, by Lu fl ∑ |J|=p n ∑ j=1 LjuJdtj ∧ dtJ, for u = ∑ |J|=p uJdtJ, L0u fl ∑ |J|=p n ∑ j=1 Lj,0uJdtj ∧ dtJ, for u = ∑ |J|=p uJdtJ, (17) where dtJ = dtj 1 ∧ ⋅ ⋅ ⋅ ∧ dtj p , J = {j1 < ⋅ ⋅ ⋅ < jp} ⊂ In = {1, 2, . . . , n}, are the basic elements from the canonical basis of the C(Ω)-module ΛC(Ω). Thus, we get the global form of these complexes Lu fl dtu + ω (t, A) ∧ Au, L0u fl dtu + Reω0 (t) ∧ Au, (18)


Introduction
In this work, we want to lay down sufficient condition for the local hypoellipticity, in the first degree, of the differential complex given by the following operators: where  : () ⊂  →  is a self-adjoint linear operator, positive with 0 ∈ (), in a Hilbert space , and (, ) is a series of nonnegative powers of  −1 with coefficients in  ∞ (Ω), Ω being an open set of R  .
The local solvability of the transpose of this complex in top degree was firstly studied in [3].There, the authors consider a method, a result from [4] we might add, to get the local solvability and they assume that the leading coefficient is analytic.Here, we will just assume that the leading coefficient is  ∞ and use dynamic property to obtain the local hypoellipticity in the elliptic region and, after that, use some of the techniques developed in [5] to study the problem in the nonelliptic one, the only case we suppose the analyticity of  0 .
In order to do that, we need to clarify every concept in the set above which we will work with in this paper.
We begin the work introducing, in a precise way, the complex of differential operators which we want to study and talking about its local hypoellipticity in the "elliptic region" and after that its hypoellipticity out of it.

The Complex in Study
Let  : () ⊂  →  be a self-adjoint linear operator, positive with 0 ∈ (), in a Hilbert space  with inner product (⋅, ⋅)  and norm ‖ ⋅ ‖  .Therefore,  is a sectorial operator with Re() > 0 (see [6], for a definition) and, for each real , let   be its fractional power space associated, that is, for  ≥ 0,   fl { −  :  ∈ } with inner product (, V)  fl (  ,   V)  , for , V ∈   , where the operator  − is given by the one which is injective whose inverse is denoted by   :   → , { − :  ≥ 0} being the analytic semigroup generated by −, and, for  < 0,   is the topological dual space of  − ; that is,   fl ( − ) * .That way, as the spaces   are Hilbert spaces, we obtain that, for each real ,  − is the topological dual of   .Now, we put  ∞ fl ⋂ ∈R   ; with the topology projective limit, we mean the topology generated by the family of norms (‖ ⋅ ‖  ) >0 , and  −∞ fl ⋃ ∈R   , with the topology weak star, namely, the one such that "a net (  ) ∈Λ in  −∞ converges to  ∈  −∞ if, and only if, the net (⟨  − , ⟩) ∈Λ converges to zero, in C, when  runs in directed set Λ, for every  ∈  ∞ ."That is,  −∞ is the topological dual space of  ∞ .
When we have  = 1 − Δ :  2 (R  ) ⊂  2 (R  ) →  2 (R  ),  fulfill the properties above, the fractional power spaces are the usual Sobolev spaces in R  , and, as we well know, in this case holds where D  () (R  ) stands for the finite order distribution on R  and S  (R  ) for the tempered distribution on R  (go to [7,8] for a proof).
In the same way we can see that, for each  = 1, 2, . . ., , the operator just recalling that D  (Ω;  −∞ ) is the topological dual space of  ∞  (Ω;  ∞ ), where the last one is equipped with the inductive limit.
The operators   and  ,0 , defined above, can be used to define complexes of differential operator, 0 ≤  ≤ , and also with 0 ≤  ≤ , by where Thus, we get the global form of these complexes with where where, for every nonnegative integer ,   stands for the exterior derivative in the  variable in Ω, being Consequently, L ∘ L = 0 and L 0 ∘ L 0 = 0, condition which defines the concept of a differential complex.
Of course, just by restriction, we see that L and L 0 define complexes on currents with coefficients in  ∞ (Ω;  −∞ ) (see [2]); that is, we can look at In these conditions, we can introduce the kind of hypoellipticity that we are going to work with.
is hypoelliptic in , in the first degree, when, for every distribution We should say that, in this work, our concern is the regularity of the distributions  ∈  ∞ (Ω;  −∞ ) in the " variable," by which we mean the regularity relatively to the scale of spaces   , where the distributions have their image.
To be more precise, in this work, we are not able, yet, to show in the more general framework that L : What we actually are going to do is to show that L is locally hypoelliptic in Ω 0 fl Ω \ E, where E fl { * ∈ Ω : ∇Re 0 ( * ) = 0}, set we will call the elliptic region of L and L 0 , and after that, using the techniques we have learned from [5], we will consider  fl 1 − Δ and get the local hypoellipticity for L associated.
In other words, in the general case, we do not have the total information about L which allows us to obtain its local hypoellipticity in Ω, but our knowledge of the dynamics properties of the solution of the Cauchy problem will give us the local hypoellipticity in Ω 0 and the nature, or noble structure, of the operator 1 − Δ will be used to solve the problem out of Ω 0 , that is, in some neighborhood of E.
The analysis we will do below in Ω 0 will be strongly inspired by the study made in [9], where the author considers the same kind of problem as us, but only in one dimension, getting complete characterization of the global hypoellipticity, in the abstract framework, by the conditions () and ().Such conditions, however, we will not assume, explicitly, here.
Before we start to study the hypoellipticity of the operator L let us point out that as was done in [1][2][3] we can isolate the "principal part" of L and conclude that to study its hypoellipticity is equivalent to study the hypoellipticity of the simpler operator L 0 .

Lemma 2. For each 0 ≤ 𝑝 ≤ 𝑛 and each open set 𝑈 ⊂ Ω,
is hypoelliptic in  if and only if is hypoelliptic in .
Proof.We just have to define, for each  ∈ Ω, the operator and to observe that the composition (, ) is the sum of an operator of type Schrödinger (hence, infinitesimal generator of a group of linear operators; see [10]) and a bound.
From the definition of U it is just a calculation to get, for  = 1, 2, . . ., , the equality As the same equality above it is true for  ∈  ∞ (;  −∞ ); our claim holds.

The Main Theorems
We begin our contribution introducing a very simple result, from the ordinary differential equations theory, whose proof will be left to the reader.As we have seen in Lemma 2, we just need to study the complex generated by L 0 .That fact will be implicit in the results we establish below.Theorem 4. In the conditions above, given  0 ∈ Ω \ E, there exists an open set  ⊂ Ω \ E, with  0 ∈ , such that L is hypoelliptic in .
In the same way, we can define  in each open subset  of .
Besides, it is not hard to see that for every open subset   ⊂ .
Finally, it is not hard to see that if  ∈ R is fixed, for every ℎ ∈  ∞ ([ 1 , ];  −∞ ) we have that  − ℎ ∈  ∞ ([ 1 , ];   ) and, by that, Putting all these results together we get that for every  ∈   holds so the second term in the sum above defines also an element of  ∞ (  ;  ∞ ); therefore  ∈  ∞ (  ;  ∞ ).But  =  in   and the proof is complete.
As we saw in the theorem above, we did not give the answer to our problem for points in the set E, yet.However, the next result shows us that there might exist points in E, where we can not obtain the hypoellipticity.Proposition 5.If  * ∈ E is a local minimal point for Re 0 , then  * has a neighborhood  in Ω, where L is not hypoelliptic.
It follows that  is well defined and  ∈  ∞ (;  −∞ ).Now, it is pretty easy to see that L 0  = 0 in , so L 0  ∈ Λ 1  ∞ (;  ∞ ).However, since ( * ) =  0 ∉  ∞ , we do not have  ∈  ∞ (;  ∞ ), and the claim is true.Remark 6.It is easy to see that when  * ∈ E is an isolated saddle point, then Re 0 is an open map in the same neighborhood of  * .
We finish this section restricting us to the case where the operator  : D() ⊂  →  and the Hilbert space  are  = 1 − Δ, D() =  2 (R  ), and  =  2 (R  ), the ones which have the properties we consider in the abstract framework above.
The reason that leads us to do this hypothesis is the fact that the nature of this operator in the  2 situation allows us to use the Fourier transform to get the regularity of the solutions of the equation L =  by studying its Fourier transform decay rate in infinity, the same way the authors do to lay down work [5].
Just for completeness of this paper, we write below the technical lemma shown in [5] which we are also going to need here, with a little alteration, which does not change its proof.Lemma 7 (see Lemma 4.4 in [5]).Suppose that Re 0 is an analytic function.
Let  * ∈ E and let  be an open ball contained in Ω such that  ∩ E is connected by piecewise smooth paths and take  0 ∈  ∩ E. Then there exist (c) a family (  ) ∈ * of piecewise smooth paths   : [0, 1] → , such that one has the following: (I)   (0) = , for every  ∈  * ; (II) Re 0 (  ()) ≤ Re 0 (), for all  ∈ [0, 1] and all  ∈  * ; (III) the length (  ) of   is such that (  ) ≤  for all  ∈  * ; (IV) if  ∈  * , then one of the following properties holds: The reader must observe that we have made a little alteration in the statement of Lemma 7; more precisely, we have made the hypothesis that "∩E is connected by piecewise smooth paths" instead of the one stating that " ∩ E is connected," only, as the authors consider there.We made this because our data Re 0 need not be constantly equal to zero on E, as they have there, but the fact that " ∩ E is connected by piecewise smooth paths" allows us to get that Re 0 is constant on  ∩ E, an alteration which does not change the proof that we have in [5].
Another thing, the hypothesis that " ∩ E is connected by piecewise smooth paths" is always satisfied when E is discrete, just taking  with radius as small as it needs to be  ∩ E a singleton.
Finally, the proof of Lemma 7 lies on the Łojasiewicz-Simon inequality, which can be obtained without the hypothesis of analyticity of Re 0 if we suppose, for example, that the second derivative of Re 0 in  * ∈ E is an isomorphism, as we can see in [11].
We are now in position to prove our final theorem.
Proof.Well, applying the Fourier transform in variable  ∈ R  to the equality L 0  =  we get where the "hat" stands for the Fourier transform in the variable , () = 1 + 4 2 || 2 is the symbol of the operator 1 − Δ, and  * is the one obtained in the last lemma.
At this point, we divide the proof into two cases.
In this case, we use Theorem 8 hypothesis (ii); therefore for every  ∈ R we have that Thanks to the fact that  ∈ Λ 1  ∞ (Ω;  ∞ ), for every  ∈ R we also have for all  ∈  * ; thus the induction will show us  ∈  ∞ ( * × R  ), and the proof is done.

Final Comments
We must make some comments to ensure the reader that the question we have treated here was not done in [2] because even though the kind of problem treated there is similar to that we study here, the structure of the operator we consider is different from that seen there.For example, our operator  is an abstract one in the Hilbert space framework, abstract as well, whereas in [2] the author considers a different class of operators in the specific space F 2 loc (R  ), topological dual space of the space F 2  (R  ), the one which is an inductive limit of Hilbert spaces.
Another situation we must point out is that if the operator  : () ⊂  →  fulfills all the properties we have made above to prove Theorem 4 and, besides these,  is separable and  −1 is compact, as we well know, in this case, the operator  admits the spectral resolution where   's are the eigenvalues of  and   :  →   are the sequence of projections into the eigenspaces   corresponding and the semigroup analytic is written like this: In this situation, for  ≥ 0, the spaces   admit the characterization
(a) an open neighborhood  * ⊂  of  * ; (b) a constant  > 0 and  > 0; and suppose that one of the following properties holds:(i) Re 0 is an open map at  * ; that is, Re 0 transforms neighborhoods of  * in neighborhoods of Re 0 ( * ).