Local Hypoellipticity by Lyapunov Function
We treat the local hypoellipticity, in the first degree, for a class of abstract differential operators complexes; the ones are given by the following differential operators: , , where is a self-adjoint linear operator, positive with , in a Hilbert space , and is a series of nonnegative powers of with coefficients in , being an open set of , for any , different from what happens in the work of Hounie (1979) who studies the problem only in the case . We provide sufficient condition to get the local hypoellipticity for that complex in the elliptic region, using a Lyapunov function and the dynamics properties of solutions of the Cauchy problem ′, , being the first coefficient of . Besides, to get over the problem out of the elliptic region, that is, in the points ∗ such that ∗ = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator .
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 , in a Hilbert space , and is a series of nonnegative powers of with coefficients in , being an open set of .
The map is given by with convergence in , uniform in compacts of , and for every
We will observe, using a method from [1–3], that the local hypoellipticity of the differential complex generated by the operators above is equivalent to the local hypoellipticity of a simpler complex, namely, the one generated by the differential operators
The local solvability of the transpose of this complex in top degree was firstly studied in . There, the authors consider a method, a result from  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  to study the problem in the nonelliptic one, the only case we suppose the analyticity of .
To be more specific, we are going to explore the properties of the gradient system generated by the function , that is, the system to get that for every point , where , there exists an open set with and , such that for each which fulfill with , then is actually in .
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.
2. The Complex in Study
Let be a self-adjoint linear operator, positive with , in a Hilbert space with inner product and norm . Therefore, is a sectorial operator with (see , for a definition) and, for each real , let be its fractional power space associated, that is, for , with inner product , for , where the operator is given by the one which is injective whose inverse is denoted by , being the analytic semigroup generated by , and, for , is the topological dual space of ; that is, .
That way, as the spaces are Hilbert spaces, we obtain that, for each real , is the topological dual of .
Now, we put ; with the topology projective limit, we mean the topology generated by the family of norms , and , with the topology weak star, namely, the one such that “a net in converges to if, and only if, the net converges to zero, in , when runs in directed set , for every .” That is, is the topological dual space of .
When we have , fulfill the properties above, the fractional power spaces are the usual Sobolev spaces in , and, as we well know, in this case holds where stands for the finite order distribution on and for the tempered distribution on (go to [7, 8] for a proof).
On the other hand, let with convergence in , uniform in compacts of , where is an open set of , and for every .
We define, for , the differential operators , by
Taking the leading coefficient of , that is, , we also define, for each , the differential operator , by
It is easy to see that, for each , the operator given by is the adjoint of .
Indeed, if , by the fact that is self-adjoint, integrating by parts, we see
Observe that , for every .
In the same way we can see that, for each , the operator is the adjoint of , where is the series , whose coefficients are the complex conjugated of the ones from .
That observation allows us to define and on distributions, and , putting just recalling that is the topological dual space of , where the last one is equipped with the inductive limit.
The operators and , defined above, can be used to define complexes of differential operator, , and also with , by where , , are the basic elements from the canonical basis of the -module .
Thus, we get the global form of these complexeswith where where, for every nonnegative integer , stands for the exterior derivative in the variable in , being and .
Consequently, and , condition which defines the concept of a differential complex.
Of course, just by restriction, we see that and define complexes on currents with coefficients in (see ); that is, we can look at
In these conditions, we can introduce the kind of hypoellipticity that we are going to work with.
Definition 1. Let be an open set of . Given , an open set of , one says that an operator is hypoelliptic in , in the first degree, when, for every distribution such that , one actually has .
When is hypoelliptic in , where , one says that is globally hypoelliptic (in ) and when is hypoelliptic in , for every open set , one says that is locally hypoelliptic in .
We should say that, in this work, our concern is the regularity of the distributions 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 is locally hypoelliptic in the whole . What we actually are going to do is to show that is locally hypoelliptic in , where , set we will call the elliptic region of and , and after that, using the techniques we have learned from , we will consider and get the local hypoellipticity for associated.
In other words, in the general case, we do not have the total information about which allows us to obtain its local hypoellipticity in , but our knowledge of the dynamics properties of the solution of the Cauchy problemwill give us the local hypoellipticity in and the nature, or noble structure, of the operator will be used to solve the problem out of , that is, in some neighborhood of .
The analysis we will do below in will be strongly inspired by the study made in , 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 let us point out that as was done in [1–3] we can isolate the “principal part” of and conclude that to study its hypoellipticity is equivalent to study the hypoellipticity of the simpler operator .
Lemma 2. For each and each open set , is hypoelliptic in if and only if is hypoelliptic in .
Proof. We just have to define, for each , 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 ) and a bound.
Therefore, we can define the operator , .
Thus, this one can be used to generate an automorphism of and , for each , putting It is not hard to see that defines an automorphism, because is invertible for every , which extends to another , just by taking its adjoint.
From the definition of it is just a calculation to get, for , the equality If we define, for , equality (28) tells us that As the same equality above it is true for ; our claim holds.
3. 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.
Lemma 3. Let , consider the Cauchy problemand let be the set of all equilibrium points of it.
If, for each , indicates the maximal time of existence of the solution , , of this problem, then, for each and with , there exist an open set with and , such that (i) for every ,(ii) whenever (when , the symbol stands for the union of all open balls with radius and center in some point of ),(iii) when ,(iv)
As we have seen in Lemma 2, we just need to study the complex generated by . That fact will be implicit in the results we establish below.
Theorem 4. In the conditions above, given , there exists an open set , with , such that is hypoelliptic in .
Proof. Indeed, given and with , let and be the ones given by the lemma above.
Also, let be the analytic semigroup generated by the minus sectorial operator . As we well know, for every whenever (see ).
Now, for (or ) and for , inspired in work , we define the linear operatorwhere the integration path is , .
In the same way, we can define in each open subset of .
We have to say that the value is well defined because the function is a Lyapunov function for Cauchy problem (31), so for every and ; hence we may apply the semigroup in and, for the case when , , endowed with the weak star topology, is complete.
Besides, it is not hard to see that maps into and into , for every open subset
On the other hand, let , consider , and define .
From this, for every we have, by Lemma 3, that ; hence , so, integrating by parts and using the fact that is the solution of (31), we see that for In resumeThus, if has , for each we may choose , with in some neighborhood of of . Then, and we have So, by (34), we have Since , we have . So if we show that is in , then the theorem follows, once was arbitrary.
Indeed, on one hand, since is constant in we have as long as .
On the other, for each there exist a neighborhood in , for it, and such that whenever and .
So, for where and stands for the components, , of the vector .
Observe that , because, for fixed, we only have to a finite number of in . Otherwise, there exists a sequence in with , so ; that is, , but it cannot be true, because when .
Finally, it is not hard to see that if is fixed, for every we have that 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 , yet. However, the next result shows us that there might exist points in , where we can not obtain the hypoellipticity.
Proposition 5. If is a local minimal point for , then has a neighborhood in , where is not hypoelliptic.
Proof. Indeed, let be an open set of , where for all .
Take and define by It follows that is well defined and .
Now, it is pretty easy to see that in , so . However, since , we do not have , and the claim is true.
Remark 6. It is easy to see that when is an isolated saddle point, then is an open map in the same neighborhood of .
We finish this section restricting us to the case where the operator and the Hilbert space are , , and , 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 situation allows us to use the Fourier transform to get the regularity of the solutions of the equation by studying its Fourier transform decay rate in infinity, the same way the authors do to lay down work .
Just for completeness of this paper, we write below the technical lemma shown in  which we are also going to need here, with a little alteration, which does not change its proof.
Lemma 7 (see Lemma in ). Suppose that is an analytic function.
Let and let be an open ball contained in such that is connected by piecewise smooth paths and take . Then there exist (a)an open neighborhood of ;(b)a constant and ;(c)a family of piecewise smooth paths , such that one has the following:(I), for every ;(II) , for all and all ;(III) the length of is such that for all ;(IV) if , then one of the following properties holds:(IV)1,(IV)2.
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 “ is connected by piecewise smooth paths” instead of the one stating that “ is connected,” only, as the authors consider there. We made this because our data need not be constantly equal to zero on , as they have there, but the fact that “ is connected by piecewise smooth paths” allows us to get that is constant on , an alteration which does not change the proof that we have in .
Another thing, the hypothesis that “ is connected by piecewise smooth paths” is always satisfied when is discrete, just taking with radius as small as it needs to be a singleton.
Finally, the proof of Lemma 7 lies on the Łojasiewicz-Simon inequality, which can be obtained without the hypothesis of analyticity of if we suppose, for example, that the second derivative of in is an isomorphism, as we can see in .
We are now in position to prove our final theorem.
Theorem 8. Suppose that is an analytic function.
Let , with , , and suppose that one of the following properties holds: (i) is an open map at ; that is, transforms neighborhoods of in neighborhoods of .(ii)There is such that , where is taken from Lemma 7.Then, for some neighborhood of .
Proof. Well, applying the Fourier transform in variable to the equality we getwhere the “hat” stands for the Fourier transform in the variable , is the symbol of the operator , and is the one obtained in the last lemma.
Multiplying equality (42) by and using the product rule we may write Also by Lemma 7, considering the family of paths and integrating the equality above along , for and , we get so, for all and holds and henceAt this point, we divide the proof into two cases.
Case 1. The conclusion IV of Lemma 7 holds.
In this case, we use Theorem 8 hypothesis (ii); therefore for every we have thatThanks to the fact that , for every we also have for all (in particular, for ), and the map is , for all , where we have written .
Thus, using these facts and conclusion (III) from Lemma 7 in inequality (46) we obtain, for each real , all and Now, observe that, by the Minköwski inequality for integrals, we have This and (47) give us that for all real .
Case 2. The conclusion of Lemma 7 holds.
In this situation, by Lemma 7, we are actually using Theorem 8 hypothesis (i) so estimate (46) gives us, for each real ,From that we see that, to take care of , we may use the same method we have used in Case and since for the same real , the exponential decay of gives us that, for every real , and hence for all and , completing the proof of this case.
From the cases we have studied above, we conclude that for all .
Finally, differentiating with respect to the equation we get so we can repeat the procedure we have made above to conclude that for all ; thus the induction will show us , and the proof is done.
4. Final Comments
We must make some comments to ensure the reader that the question we have treated here was not done in  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  the author considers a different class of operators in the specific space , topological dual space of the space , the one which is an inductive limit of Hilbert spaces.
There, the author does a systematic study of the problem , for , where , , is a pseudodifferential operator which has no need to be in the same class as our operator , .
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 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 , the spaces admit the characterizationand are equipped with the norm