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: Lj=∂/∂tj+(∂ϕ/∂tj)(t,A)A, j=1,2,…,n, where A:D(A)⊂H→H is a self-adjoint linear operator, positive with 0∈ρ(A), in a Hilbert space H, and ϕ=ϕ(t,A) is a series of nonnegative powers of A-1 with coefficients in C∞(Ω), Ω being an open set of Rn, for any n∈N, different from what happens in the work of Hounie (1979) who studies the problem only in the case n=1. 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 t′(s)=-∇Reϕ0(t(s)), s≥0, t(0)=t0∈Ω,ϕ0:Ω→C being the first coefficient of ϕ(t,A). Besides, to get over the problem out of the elliptic region, that is, in the points t∗∈Ω such that ∇Reϕ0(t∗) = 0, we will use the techniques developed by Bergamasco et al. (1993) for the particular operator A=1-Δ:H2(RN)⊂L2(RN)→L2(RN).
1. 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: (1)Lj=∂∂tj+∂ϕ∂tjt,AA,j=1,2,…,n,where A:D(A)⊂H→H is a self-adjoint linear operator, positive with 0∈ρ(A), in a Hilbert space H, and ϕ(t,A) is a series of nonnegative powers of A-1 with coefficients in C∞(Ω), Ω being an open set of Rn.
The map ϕ=ϕ(t,A) is given by (2)ϕt,A=∑k=0∞ϕktA-k,with convergence in L(H), uniform in compacts of Ω, and ϕk∈C∞(Ω)=C∞(Ω;C) for every k∈N∪{0}.
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 (3)Lj,0≔∂∂tj+∂Reϕ0∂tjtA,j=1,2,…,n.
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 C∞ 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 ϕ0.
To be more specific, we are going to explore the properties of the gradient system generated by the C∞ function Reϕ0, that is, the system (4)t′s=-∇Reϕ0ts,s≥0,t0=t0∈Ω,to get that for every point t0∈Ω∖E, where E≔{t∗∈Ω:∇Reϕ0(t∗)=0}, there exists an open set U⊂Ω with t0∈U and U∩E=⌀, such that for each u∈C∞(U;H-∞) which fulfill (5)∑j=1nLj,0udtj=f in U,with f∈Λ1C∞(U;H∞), then u is actually in C∞(U;H∞).
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 A:D(A)⊂H→H be a self-adjoint linear operator, positive with 0∈ρ(A), in a Hilbert space H with inner product (·,·)H and norm ·H. Therefore, A is a sectorial operator with Reσ(A)>0 (see [6], for a definition) and, for each real s, let Hs be its fractional power space associated, that is, for s≥0, Hs≔{A-sf:f∈H} with inner product (u,v)s≔(Asu,Asv)H, for u,v∈Hs, where the operator A-s is given by (6)A-s≔1Γs∫0∞θs-1e-Aθdθ, the one which is injective whose inverse is denoted by As:Hs→H, {e-Aθ:θ≥0} being the analytic semigroup generated by -A, and, for s<0, Hs is the topological dual space of H-s; that is, Hs≔H-s∗.
That way, as the spaces Hs are Hilbert spaces, we obtain that, for each real s, H-s is the topological dual of Hs.
Now, we put H∞≔⋂s∈RHs; with the topology projective limit, we mean the topology generated by the family of norms ·ss>0, and H-∞≔⋃s∈RHs, with the topology weak star, namely, the one such that “a net (xλ)λ∈Λ in H-∞ converges to x∈H-∞ if, and only if, the net (〈xλ-x,u〉)λ∈Λ converges to zero, in C, when λ runs in directed set Λ, for every u∈H∞.” That is, H-∞ is the topological dual space of H∞.
When we have A=1-Δ:H2(RN)⊂L2(RN)→L2(RN), A fulfill the properties above, the fractional power spaces are the usual Sobolev spaces in RN, and, as we well know, in this case holds (7)H∞⊂C∞RN∩L2RN,H-∞⊂DF′RN∩S′RN,where D(F)′(RN) stands for the finite order distribution on RN and S′(RN) for the tempered distribution on RN (go to [7, 8] for a proof).
On the other hand, let (8)ϕt,A=∑k=0∞ϕktA-k,with convergence in L(H), uniform in compacts of Ω, where Ω is an open set of Rn, and ϕk∈C∞(Ω) for every k∈N∪{0}.
We define, for j=1,2,…,n, the differential operators Lj:C∞(Ω;H∞)→C∞(Ω;H∞), by(9)Lju≔∂u∂tj+∂ϕ∂tjt,AAu.
Taking the leading coefficient of ϕ(t,A), that is, ϕ0∈C∞(Ω), we also define, for each j=1,2,…,n, the differential operator Lj,0:C∞(Ω;H∞)→C∞(Ω;H∞), by (10)Lj,0u≔∂u∂tj+∂Reϕ0∂tjtAu.
It is easy to see that, for each j=1,2,…,n, the operator given by (11)Lj,0∗u≔-∂u∂tj+∂Reϕ0∂tjtAu is the adjoint of Lj,0.
Indeed, if φ,ψ∈Cc∞(Ω;H∞), by the fact that A is self-adjoint, integrating by parts, we see (12)Lj,0φ,ψ=∫Ω∂φt∂tj+∂Reϕ0∂tjtAφt,ψtHdt=-∫Ωφt,∂ψt∂tjHdt+∫Ωφt,∂Reϕ0∂tjtAψtHdt=φ,Lj,0∗ψ.
Observe that supp(Lj,0u)⊂supp(u), for every u∈C∞(Ω).
In the same way we can see that, for each j=1,2,…,n, the operator (13)Lj∗=-∂∂tj+∂ϕ¯∂tjt,AAis the adjoint of Lj, where ϕ¯(t,A) is the series ∑k=0∞ϕk¯(t)A-k, whose coefficients are the complex conjugated of the ones from ϕ(t,A).
That observation allows us to define Lj and Lj,0 on distributions, Lj:D′(Ω;H-∞)→D′(Ω;H-∞) and Lj,0:D′(Ω;H-∞)→D′(Ω;H-∞), putting (14)Lju,φ≔u,Lj∗φ,for u∈D′Ω;H-∞,φ∈Cc∞Ω;H∞,Lj,0u,φ≔u,Lj,0∗φ,for u∈D′Ω;H-∞,φ∈Cc∞Ω;H∞,just recalling that D′(Ω;H-∞) is the topological dual space of Cc∞(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, (15)L:ΛpC∞Ω;H∞⟶Λp+1C∞Ω;H∞,L:ΛpD′Ω;H-∞⟶Λp+1D′Ω;H-∞,0≤p≤n, and (16)L0:ΛpC∞Ω;H∞⟶Λp+1C∞Ω;H∞,L0:ΛpD′Ω;H-∞⟶Λp+1D′Ω;H-∞,also with 0≤p≤n, by (17)Lu≔∑J=p∑j=1nLjuJdtj∧dtJ,for u=∑J=puJdtJ,L0u≔∑J=p∑j=1nLj,0uJdtj∧dtJ,for u=∑J=puJdtJ, where dtJ=dtj1∧⋯∧dtjp, J={j1<⋯<jp}⊂In={1,2,…,n}, are the basic elements from the canonical basis of the C∞(Ω)-module ΛpC∞(Ω).
Thus, we get the global form of these complexes(18)Lu≔dtu+ωt,A∧Au,L0u≔dtu+Reω0t∧Au,with (19)ωt,A≔∑k=0∞ωktA-k∈Λ1C∞Ω;LH, where (20)ωkt≔∑j=1n∂ϕk∂tjtdtj, where, for every nonnegative integer k, dt stands for the exterior derivative in the t variable in Ω, being u∈ΛpC∞Ω;H∞oru∈ΛpD′(Ω;H-∞) and Au≔∑|J|=pAuJdtJ.
Consequently, L∘L=0 and L0∘L0=0, condition which defines the concept of a differential complex.
Of course, just by restriction, we see that L and L0 define complexes on currents with coefficients in C∞(Ω;H-∞) (see [2]); that is, we can look at (21)L:ΛpC∞Ω;H-∞⟶Λp+1C∞Ω;H-∞,for 0≤p≤n,L0:ΛpC∞Ω;H-∞⟶Λp+1C∞Ω;H-∞,for 0≤p≤n.
In these conditions, we can introduce the kind of hypoellipticity that we are going to work with.
Definition 1.
Let Ω be an open set of Rn. Given U, an open set of Ω, one says that an operator (22)M:C∞Ω;H-∞⟶Λ1C∞Ω;H-∞ is hypoelliptic in U, in the first degree, when, for every distribution u∈C∞(U;H-∞) such that Mu∈Λ1C∞(U;H∞), one actually has u∈C∞(U;H∞).
When M is hypoelliptic in U, where U=Ω, one says that M:C∞(Ω;H-∞)→Λ1C∞(Ω;H-∞) is globally hypoelliptic (in Ω) and when M:C∞(Ω;H-∞)→Λ1C∞(Ω;H-∞) is hypoelliptic in U, for every open set U⊂Ω, one says that M is locally hypoelliptic in Ω.
We should say that, in this work, our concern is the regularity of the distributions u∈C∞(Ω;H-∞) in the “x variable,” by which we mean the regularity relatively to the scale of spaces Hs, 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:C∞(Ω;H-∞)→Λ1C∞(Ω;H-∞) is locally hypoelliptic in the whole Ω. What we actually are going to do is to show that L is locally hypoelliptic in Ω0≔Ω∖E, where E≔{t∗∈Ω:∇Reϕ0(t∗)=0}, set we will call the elliptic region of L and L0, and after that, using the techniques we have learned from [5], we will consider A≔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(23)t′=-∇Reϕ0t,s≥0,t0=t0∈Ω,will give us the local hypoellipticity in Ω0 and the nature, or noble structure, of the operator 1-Δ will be used to solve the problem out of Ω0, that is, in some neighborhood of E.
The analysis we will do below in Ω0 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 (ψ) and (τ). 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 [1–3] 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 L0.
Lemma 2.
For each 0≤p≤n and each open set U⊂Ω, (24)L:ΛpC∞U;H-∞⟶Λp+1C∞U;H-∞is hypoelliptic in U if and only if (25)L0:ΛpC∞U;H-∞⟶Λp+1C∞U;H-∞is hypoelliptic in U.
Proof.
We just have to define, for each t∈Ω, the operator (26)αt,A≔Reϕ0t-ϕt,A=ϕ0t-ϕt,A-iImϕ0t and to observe that the composition α(t,A)A is the sum of an operator of type Schrödinger (hence, infinitesimal generator of a group of linear operators; see [10]) and a bound.
Therefore, we can define the operator U(t)≔eα(t,A)A, t∈Ω.
Thus, this one can be used to generate an automorphism of ΛpC∞(U;H∞) and ΛpC∞(U;H-∞), for each 0≤p≤n, putting (27)Uut≔Utut=eαt,AAut,for u∈C∞U;H∞,t∈U. It is not hard to see that U:C∞(U;H∞)→C∞(U;H∞) defines an automorphism, because eα(t,A)A is invertible for every t∈Ω, which extends to another U:C∞(U;H-∞)→C∞(U;H-∞), just by taking its adjoint.
From the definition of U it is just a calculation to get, for j=1,2,…,n, the equality (28)LjUut=ULj,0ut,for u∈C∞U;H∞,t∈U.
If we define, for u=∑|J|=puJdtJ, (29)Uu≔∑J=pUuJdtJequality (28) tells us that (30)LUu=UL0u,for u∈C∞U;H∞. As the same equality above it is true for u∈C∞(U;H-∞); 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 ϕ0∈C∞(Ω), consider the Cauchy problem(31)t′s=-∇Reϕ0ts,s≥0,t0=t0∈Ω,and let E≔{t∗∈Ω:∇Reϕ0(t∗)=0} be the set of all equilibrium points of it.
If, for each t0∈Ω, ω(t0)>0 indicates the maximal time of existence of the solution T(s)t0, s>0, of this problem, then, for each t0∈Ω0≔Ω∖E and δ>0 with d(t0,E)>2δ, there exist an open set U⊂Ω with t0∈U and τ>0, such that
ω(t)≥τ for every t∈U,
T(s)U⊂Oδ(E∪∂Ω) whenever s≥τ (when X⊂Ω, the symbol Oδ(X) stands for the union of all open balls with radius δ>0 and center in some point of X),
T(s)U⊂Ω0 when 0≤s≤τ,
U∩Oδ(E∪∂Ω)=⌀.
As we have seen in Lemma 2, we just need to study the complex generated by L0. That fact will be implicit in the results we establish below.
Theorem 4.
In the conditions above, given t0∈Ω∖E, there exists an open set U⊂Ω∖E, with t0∈U, such that L is hypoelliptic in U.
Proof.
Indeed, given t0∈Ω0=Ω∖E and δ>0 with d(t0,E)>2δ, let U and τ>0 be the ones given by the lemma above.
Also, let {e-sA:s≥0} be the analytic semigroup generated by the minus sectorial operator -A. As we well know, e-Asu∈H∞ for every u∈H-∞ whenever s>0 (see [6]).
Now, for ω∈Λ1C∞(U;H∞) (or ω∈Λ1C∞(U;H-∞)) and for t∈U, inspired in work [9], we define the linear operator(32)Kωt≔-∫γteReϕ0z-ϕ0tAωzdz,where the integration path is γt(s)≔T(s)t, s∈[0,τ].
In the same way, we can define K in each open subset W of U.
We have to say that the value (Kω)(t) is well defined because the function Reϕ0 is a Lyapunov function for Cauchy problem (31), so Reϕ0(T(s)t)≤Reϕ0(t) for every s∈[0,τ] and t∈U; hence we may apply the semigroup {e-As:s≥0} in s=-Reϕ0(T(s)t)-ϕ0(t)≥0 and, for the case when ω∈Λ1C∞(U;H-∞), H-∞, endowed with the weak star topology, is complete.
Besides, it is not hard to see that K maps Λ1C∞(U′;H∞) into C∞(U′;H∞) and Λ1C∞(U′;H-∞) into C∞(U′;H-∞), for every open subset U′⊂U.
On the other hand, let g∈Cc∞(U;H-∞), consider L0g∈Λ1C∞(U;H-∞), and define K(L0g).
From this, for every t∈U we have, by Lemma 3, that T(τ)t∉U; hence T(τ)t∉supp(g), so, integrating by parts and using the fact that T(s)t is the solution of (31), we see that for t∈U(33)KL0gt=-∫γteReϕ0z-ϕ0tAL0gzdz=-∫γteReϕ0z-ϕ0tAdtgzdz-∫γteReϕ0z-ϕ0tAReω0z∧Agzdz=-eReϕ0Tst-ϕ0tAgzs=0τ+∫γteReϕ0z-ϕ0tAReω0z∧Agzdz-∫γteReϕ0z-ϕ0tAReω0z∧Agzdz=-eReϕ0Tτt-ϕ0tAgTτt-eReϕ0t-ϕ0tAgt=gt.In resume(34)KL0gt=gt,for every t∈U.Thus, if u∈C∞(U;H-∞) has L0u=f∈Λ1C∞(U;H∞), for each t′∈U we may choose φ∈Cc∞(U;R), with φ=1 in some neighborhood of U′ of t′. Then, g≔φu∈Cc∞(U;H-∞) and we have (35)L0φu=φL0u+∑j=1n∂φ∂tjtutdtj=φf+∑j=1n∂φ∂tjtutdtj.
So, by (34), we have (36)Kφft+K∑j=1n∂φ∂tjudtjt=KL0φut=φut,∀t∈U.
Since φf∈Λ1C∞(U;H∞), we have Kφf∈C∞(U;H∞). So if we show that (37)K∑j=1n∂φ∂tjudtjis in C∞(U′;H∞), then the theorem follows, once U′ was arbitrary.
Indeed, on one hand, since φ is constant in U′ we have ∑j=1n(∂φ/∂tj)(r)u(r)dtj=0 as long as r∈U′.
On the other, for each t′∈U′ there exist a neighborhood V′ in U′, for it, and τ1>0 such that T(s)t∈U′ whenever s∈[0,τ1] and t∈V′.
So, for t∈V′(38)K∑j=1n∂φ∂tjudtjt=-∫τ1τeReϕ0Tst-ϕ0t+ηAe-ηA∑j=1n∂φ∂tjTstuTstdTstjdsds, where η≔Re(ϕ0(t)-ϕ0(T(s)τ1))>0 and d(T(s)t)j/ds stands for the components, j=1,2,…,n, of the vector dT(s)t/ds∈Rn.
Observe that η>0, because, for t∈U fixed, we only have Reϕ0(T(s)t)=Reϕ0(t) to a finite number of s in [0,τ]. Otherwise, there exists a sequence (sj)j∈N in [0,τ] with sj→s0∈[0,τ], so ∇Reϕ0(T(s0)t)=0; that is, T(s0)t∈E, but it cannot be true, because T(s)U⊂Ω0 when 0≤s≤τ.
Finally, it is not hard to see that if α∈R is fixed, for every h∈C∞([τ1,τ];H-∞) we have that e-ηAh∈C∞([τ1,τ];Hα) and, by that, (39)eReϕ0Tst-ϕ0t+ηAe-ηAh∈C∞τ1,τ;H∞.Putting all these results together we get that for every t∈U′ holds (40)φut=Kφft-∫τ1τeReϕ0Tst-ϕ0t+ηAe-ηA∑j=1n∂φ∂tjTstuTstdTstjdsds,so the second term in the sum above defines also an element of C∞(U′;H∞); therefore φu∈C∞(U′;H∞). But φu=u in U′ 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 t∗∈E is a local minimal point for Reϕ0, then t∗ has a neighborhood V in Ω, where L is not hypoelliptic.
Proof.
Indeed, let V be an open set of Ω, where Reϕ0(t∗)≤Reϕ0(t) for all t∈V.
Take u0∈H∖H∞ and define u:V→H-∞ by (41)ut≔eReϕ0t∗-ϕ0tAu0,t∈V. It follows that u is well defined and u∈C∞(V;H-∞).
Now, it is pretty easy to see that L0u=0 in V, so L0u∈Λ1C∞(V;H∞). However, since u(t∗)=u0∉H∞, we do not have u∈C∞(V;H∞), and the claim is true.
Remark 6.
It is easy to see that when t∗∈E is an isolated saddle point, then Reϕ0 is an open map in the same neighborhood of t∗.
We finish this section restricting us to the case where the operator A:D(A)⊂H→H and the Hilbert space H are A=1-Δ, D(A)=H2(RN), and H=L2(RN), 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 L2 situation allows us to use the Fourier transform to get the regularity of the solutions of the equation Lu=f 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 Reϕ0 is an analytic function.
Let t∗∈E and let B be an open ball contained in Ω such that B∩E is connected by piecewise smooth paths and take t0∈B∩E. Then there exist
an open neighborhood B∗⊂B of t∗;
a constant K>0 and ɛ>0;
a family (γt)t∈B∗ of piecewise smooth paths γt:[0,1]→B, such that one has the following:
γt(0)=t, for every t∈B∗;
Reϕ0γt(s)≤Reϕ0(t), for all s∈[0,1] and all t∈B∗;
the length l(γt) of γt is such that l(γt)≤K for all t∈B∗;
if t∈B∗, then one of the following properties holds:
γt(1)=t0,
Reϕ0γt(1)≤Reϕ0(t)-ɛ.
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 “B∩E is connected by piecewise smooth paths” instead of the one stating that “B∩E is connected,” only, as the authors consider there. We made this because our data Reϕ0 need not be constantly equal to zero on E, as they have there, but the fact that “B∩E is connected by piecewise smooth paths” allows us to get that Reϕ0 is constant on B∩E, an alteration which does not change the proof that we have in [5].
Another thing, the hypothesis that “B∩E is connected by piecewise smooth paths” is always satisfied when E is discrete, just taking B with radius as small as it needs to be B∩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 Reϕ0 if we suppose, for example, that the second derivative of Reϕ0 in t∗∈E is an isomorphism, as we can see in [11].
We are now in position to prove our final theorem.
Theorem 8.
Suppose that Reϕ0 is an analytic function.
Let A=1-Δ:H2(RN)⊂L2(RN)→L2(RN), u∈C∞(Ω;H-∞) with L0u=f∈Λ1C∞(Ω;H∞), t∗∈E, and suppose that one of the following properties holds:
Reϕ0 is an open map at t∗; that is, Reϕ0 transforms neighborhoods of t∗ in neighborhoods of Reϕ0(t∗).
There is t0∈B∩E such that u(t0,·)∈H∞, where B is taken from Lemma 7.
Then, u∈C∞(B∗×RN) for some neighborhood B∗⊂B of t∗.
Proof.
Well, applying the Fourier transform in variable x∈RN to the equality L0u=f we get(42)dtu^+Reω0t∧aξu^=f^,for t∈B,where the “hat” stands for the Fourier transform in the variable x, a(ξ)=1+4π2|ξ|2 is the symbol of the operator 1-Δ, and B∗ is the one obtained in the last lemma.
Multiplying equality (42) by ea(ξ)Reϕ0(t) and using the product rule we may write (43)dteaξReϕ0tu^t,ξ=eaξReϕ0tf^t,ξ,∀t∈B, ξ∈RN.
Also by Lemma 7, considering the family of paths (γt)t∈B∗ and integrating the equality above along γt, for t∈B∗ and ξ∈RN, we get (44)eaξReϕ0γt1u^γt1,ξ-eaξReϕ0tu^t,ξ=∫γtdteaξReϕ0zu^z,ξ=∫γteaξReϕ0zf^z,ξ, so, for all t∈B∗ and ξ∈RN holds (45)u^t,ξ=eaξReϕ0γt1-Reϕ0tu^γt1,ξ-∫γteaξReϕ0z-Reϕ0tf^z,ξdz, and hence(46)u^t,ξ≤eaξReϕ0γt1-Reϕ0tu^γt1,ξ+∫γteaξReϕ0z-Reϕ0tf^z,ξdz.
At this point, we divide the proof into two cases.
Case 1. The conclusion (IV)1 of Lemma 7 holds.
In this case, we use Theorem 8 hypothesis (ii); therefore for every s∈R we have that(47)1+ξ2s/2u^t0,·∈L2RN.
Thanks to the fact that f∈Λ1C∞(Ω;H∞), for every s∈R we also have (48)1+ξ2s/2f^jt,·∈L2RNfor all t∈Ω (in particular, for t∈B∗), and the map Ω∋t↦fj(t,·)∈H∞ is C∞, for all j, where we have written f=∑j=1nfjdtj.
Thus, using these facts and conclusion (III) from Lemma 7 in inequality (46) we obtain, for each real s, all ξ∈RN and t∈B∗(49)1+ξ2s/2u^t,ξ≤1+ξ2s/2u^t0,ξ+∫γt1+ξ2s/2f^z,ξdz.Now, observe that, by the Minköwski inequality for integrals, we have (50)∫RN∫γt1+ξ2s/2f^z,ξdz2dξ1/2≤∫γt∫RN1+ξ2sf^z,ξ2dξ1/2dz≤Ksupz∈Bfz,·Hs<∞. This and (47) give us that (1+|ξ|2)s/2|u^(t,·)|∈L2(RN) for all real s.
Case 2. The conclusion (IV)2 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 s,(51)1+ξ2s/2u^t,ξ≤1+ξ2s/2e-ɛaξu^γt1,ξ+∫γt1+ξ2s/2f^z,ξdz.
From that we see that, to take care of ∫γt(1+|ξ|2)s/2f^(z,ξ)dz, we may use the same method we have used in Case 1 and since (52)1+ξ2α/2u^γt1,·∈L2RNfor the same real α, the exponential decay of e-ɛa(ξ) gives us that, for every real s, (53)1+ξ2s/2e-ɛaξu^γt1,·∈L2RN,and hence(54)ut,·Hs≤1+ξ2s/2e-ɛaξu^γt1,·L2+Ksupz∈Bfz,·Hs<∞ for all t∈B∗ and s∈R, completing the proof of this case.
From the cases we have studied above, we conclude that u(t,·)∈H∞⊂C∞(RN) for all t∈B∗.
Finally, differentiating with respect to tk the equation L0u=f we get (55)∂∂tk∂u∂tjt+∂Reϕ0∂tktA∂u∂tjt=∂fj∂tkt-∂2Reϕ0∂tk∂tjtAut, so we can repeat the procedure we have made above to conclude that (56)∂u∂tjt,·∈H∞⊂C∞RNfor all t∈B∗; thus the induction will show us u∈C∞(B∗×RN), 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 [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 A 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 Floc2(Rn), topological dual space of the space Fc2(Rn), the one which is an inductive limit of Hilbert spaces.
There, the author does a systematic study of the problem dtu+b(t,Dx)∧u=f, for u∈Floc2(Rn), where b(t,Dx):Floc2(Rn)→Floc2(Rn), t∈Ω, is a pseudodifferential operator which has no need to be in the same class as our operator Reϕ0(t)A:D(A):H→H, t∈Ω.
Another situation we must point out is that if the operator A:D(A)⊂H→H fulfills all the properties we have made above to prove Theorem 4 and, besides these, H is separable and A-1 is compact, as we well know, in this case, the operator A admits the spectral resolution (57)Au=∑j=1∞λjPju,u∈DA,where λj’s are the eigenvalues of A and Pj:H→Ej are the sequence of projections into the eigenspaces Ej corresponding and the semigroup analytic is written like this: (58)e-Asu=∑j=1∞e-λjsPju,u∈H.
In this situation, for s≥0, the spaces Hs admit the characterization(59)Hs=u∈H:λjsPjuHj∈N∈l2Nand are equipped with the norm(60)Hs∋u⟼us≔∑j=1∞λj2sPjuH21/2.Also, for s<0 the space Hs is the topological dual space of H-s or even the completion of the set Hs defined in the same way as (59) with respect to the norm ·s defined just as (60).
For each j∈N, it is possible to extend the projection Pj:H→Ej to a new projection P~j:H-∞→Ej. Therefore, in these conditions, considering the differential operator L associated with the operator A, we see that to get the regularity of the solutions of the equation Lu=f we just have to study the decay behavior of the sequences λjsP~ju(t)Hj∈N in the same way as we have done in Theorem 8, that is, to prove that this sequence is in l2(N) for every real s. This way, the same proof we gave for Theorem 8 applies to this case and we can state the following.
Theorem 9.
Besides the hypothesis one has made for the operator A:D(A)⊂H→H, suppose also that H is separable and A-1 is compact.
If u∈C∞(Ω;H-∞) verify Lu=f with f∈Λ1C∞(Ω;H∞); for t∗∈E suppose that one of the following properties holds:
Reϕ0 is an open map at t∗.
There is t0∈B∩E such that u(t0,·)∈H∞.
Then, u∈C∞(Ω;H∞).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is partially supported by 2011/52052-9 and 2014/02899-3 from São Paulo Research Foundation (FAPESP), Brazil.
TrèvesF.Concatenations of second-order evolution equations applied to local solvability and hypoellipticity19732620125010.1002/cpa.3160260206MR0340804TrevesF.Study of a model in the theory of complexes of pseudodifferential operators1976104226932410.2307/1971048MR0426068YamaokaL. C.2011IME-USP Tese de DoutoradoHanZ.Local solvability of analytic pseudodifferential complexes in top degree199787112810.1215/S0012-7094-97-08701-9MR1440061ZBL0873.470312-s2.0-0039377676BergamascoA. P.CordaroP. D.MalaguttiP. A.Globally hypoelliptic systems of vector fields1993114226728510.1006/jfan.1993.1068MR1223704ZBL0777.580412-s2.0-0003290878HenryD.1981840Berlin, GermanySpringerLecture Notes in MathematicsMR610244HörmanderL.1963New York, NY, USASpringer10.1007/978-3-642-46175-0TrevesF.1967New York, NY, USAAcademic PressMR0225131HounieJ.Globally hypoelliptic and globally solvable first order evolution equations197925223324810.2307/1998087MR534120PazyA.1963New York, NY, USASpringer10.1007/978-1-4612-5561-1MR710486Aragao-CostaE. R.CarvalhoA. N.Marín-RubioP.PlanasG.Gradient-like nonlinear semigroups with infinitely many equilibria and applications to cascade systems20134223453762-s2.0-84893364721