On the Well-Posedness for a Class of Pseudo-Differential Parabolic Equations

In this work we study the well-posedness of the Cauchy problem for a class of pseudo-differential parabolic equations in the framework of Weyl–Hörmander calculus. We establish regularity estimates, existence and uniqueness in the scale of Sobolev spaces H(m, g) adapted to the corresponding Hörmander classes. Some examples are included for fractional parabolic equations and degenerate parabolic equations.


Introduction
The research on degenerate parabolic equations has been very active in the last decades and developed into multiple directions. A good account on the local structure of solutions and other interests can be found in [11]. On the other hand, fractional diffusion equations have a wide variety of applications e.g. fluid mechanics [6,22], mathematical finance [7] and fractional dynamics [18,19,26].
Here we will establish the well-posedness of the Cauchy problem for a class of pseudo-differential parabolic equations in the setting of Weyl-Hörmander calculus. We will give some examples of applications for fractional degenerate parabolic equations and fractional diffusion equations. The analysis of PDEs under the scrutiny of pseudo-differential calculus has been largely exploited. Some of the main ideas to study parabolic equations in this work correspond to an adaptation of the methods used for the classical classes of pseudo-differential operators as presented for instance in [5,27,28].
We now give a more precise description of the problem studied in this work. Let g be a Hörmander metric [15,20] on the phase space R n × R n and m a g-weight, we denote by S(m, g) the associated class of symbols and by OpS(m, g) the corresponding class of pseudodifferential operators. The Kohn-Nirenberg quantization of a symbol a(x, ξ) will be denoted by a(x, D) and the Weyl quantization by a w . We will study the Cauchy problem associated to a parabolic equation of the form where for every t ∈ R, K(t, x, D x ) is a suitable pseudo-differential operator of order 2 relative to a Hörmander metric g (cf. [14,15,20]) and a suitable positivity condition is imposed on ReK(t, x, ξ). We will establish regularity estimates, existence and uniqueness with respect to the Sobolev spaces H(m, g) adapted to the S(m, g) classes. The simplest case corresponds to the heat equation, which is associated to an operator of the form ∂ ∂t − K, where K is a suitable elliptic second order differential operator and the considered symbol classes are the ones introduced by Kohn and Nirenberg (cf. [16]). Degenerate hyperbolic pseudo-differential equations have been recently studied in [9] as well as some preliminary results in the parabolic case.
The Planck function or the uncertainty parameter h(x, ξ) associated to a Hörmander metric g is defined by h(x, ξ) = sup (y,η) =0 g (x,ξ) (y,η) g σ (x,ξ) (y,η) where g σ is the dual metric of g with respect to the canonical symplectic form [(y, η), (z, ζ)] = z · η − y · ζ. This function can be used to extend the notion of operators of order ν for a real number ν (cf. Sect. 2). Indeed, the weight h(x, ξ) −ν will correspond to the order ν.
The main theorem of this work can be described as follows. Given a Hörmander metric g, we study the following Cauchy problem for K(t, x, ξ) ∈ S(h −ν , g) with ν ∈ {1, 2} and write K(t) = K(t, x, D x ): , g)) for some s ∈ R and a suitable positivity condition is imposed on K. Under such assumptions we will prove the existence and uniqueness of a solution v ∈ C([0, T ], H(h −s , g)).
An important example of an operator of order 2 with respect to g is given by Δ (g) ∈ OpS(h −2 , g) of the form where R ∈ OpS(h −1 , g). A basic example is the one of the metric g = g 1,0 = dx 2 + dξ 2 ξ 2 , where ξ 2 = 1 + |ξ| 2 . In that case one has h −2 (x, D) = −Δ + I, where Δ is the Laplacian on R n and I the identity operator. By choosing K(t) = −Δ (g) = Δ − I and R(x, D) = I in (1.2), (1.1) becomes the classical Cauchy problem for the heat equation. In general, the operator Δ (g) is not elliptic and there is where the difficulty arises. Indeed this operator does not even have a parametrix as a classical pseudodifferential operator. In this situation the Weyl-Hörmander calculus is appropriate to deal with it. The type of estimates here obtained for the problem (1.1) are well known when the operator Δ (g) is replaced by a second order partial differential operator P (t, x, D x ) such that ∂ ∂t + P is strictly parabolic. By choosing a suitable Hörmander metric one can consider for instance −K of the form: (i) K(D) = −(−Δ + I) for 0 < ≤ 1. A further analysis of this example will allow us to study fractional diffusion equations. Strictly speaking the fractional Laplacian (−Δ) s 0 < s < 1 has not a symbol in a class of symbols for pseudo-differential operators. However we can decompose it in the form (−Δ) s = σ(x, D)+E so that the main term σ(x, D) is a pseudodifferential operator and E has some good properties allowing to extend our analysis to the fractional Laplacian. The underlying ideas for such splitting has been recently applied in [12].
(ii) K(x, D) = −((−Δ) 2 + |x| 4 ); higher order parabolic equations have been investigated for the last fifty years and it is well known that they exhibit different pathologies with respect to the classical case. For instance the kernels of the heat operators depends on the space dimension and they are not expressible in a simple form. Another pathology is the lack of the positivity preserving property (cf. [1,13,21] and the references therein). ( where L is a second order differential operator with index of sub-ellipticity τ as in (3.6) and J is the Bessel potential of order 1 on R n , i.e. corresponding to the symbol ξ . The operators in (iii) are subelliptic with order of subellipticity τ = 1 since the associated vector fields to the sum of squares satisfy the Hörmander condition of order 2. They have been studied in [29] C-J. Xu and X. Zhu, where the authors prove that K(x, D) − C is invertible in a suitable nonhomogeneous class for C > 0 large enough. The operator K(x, D) can be seen as a particular case of Q = n i=1 ∂ 2 xi + |x| 2k m i=1 ∂ 2 ti − C on R n+m for which, Beals, Gaveau and Greiner have found exact fundamental solutions (cf. [2]).
We also point out that parabolic equations (possibly completely degenerate) in the case of a constant Weyl-Hormander metric (hence corresponding to the symbol classes S 0 0,0 ) were also addressed in [23], where a representation of the propagator as pseudodifferential operator with symbol in the same class was proved as well. We hope to consider such a representation in a second paper.

Preliminaries
In this section we review some elements of the Weyl-Hörmander calculus. For a comprehensive study on this important theory we refer the reader to [4,15,20].
The Kohn-Nirenberg and Weyl quantizations are recalled below: For a(x, ξ) ∈ S (R n × R n ) (x ∈ R n and ξ ∈ R n ), we define the Kohn-Nirenberg or classic quantization as the operator a(x, D) : The Weyl quantization has a fundamental relationship with the symplectic structure of R n × R n = T * R n as we will see below. The Weyl quantisation of a(x, ξ), is given by the operator a w : The operation # is useful in order to describe the composition a w • b ω , indeed one has a w • b ω = (a#b) ω . Moreover, from the formula for the composition we notice how the symplectic form naturally appears in the integral expression above.
We shall now recall the definition of Hörmander metrics on the phasespace. A Hörmander metric is a special Riemannian metric which carries basic information of the differential operator target into the phase space. Definition 2.3. For X ∈ R n × R n let g X (·) be a positive definite quadratic form on R n × R n , we say that g(·) is a Hörmander's metric if the following three conditions are satisfied: 1. Continuity-There exist constants C, c, c ∈ R such that g X (Y ) ≤ C, for X, Y ∈ R n × R n , implies c · g X+Y (T ) ≤ g X (T ) ≤ c · g X+Y (T ) for every T ∈ R n × R n . 2. Uncertainty principle-For Y = (y, η) and Z = (z, ζ) we define σ(Y, Z) = z · η − y · ζ, and We say that g satisfies the uncertainty principle if Let g be a Hörmander's metric, the uncertainty parameter or the Planck function associated to g is defined by The uncertainty principle can be translated then by the condition where a i (X) and b i (X) are positive functions, one can prove (ii) A special case of (i) is the one of symmetrically split metric i.e. of type The metric (3.7) is an example of a such type. (iii) If g is a split metric one can prove the following formula for λ g The classical weight ξ m is generalised in the following way for a corresponding Hörmander metric. Definition 2.5. We say that a strictly positive function M is a g-weight or g-continuous if there existsC > 0 and N ∈ N such that Definition 2.6. For a Hörmander metric g and a g-weight M , we denote by S(M, g) the set of all smooth functions a on R n ×R n such that for any integer k there exists C k ∈ R, such that for all X, For a ∈ S(M, g) we denote by a k,S(M,g) the minimum C k satisfying the above inequality. The class S(M, g) becomes a Frechet space endowed with the family of seminorms · k,S(M,g) .
Remark 2.7. Given a Hörmander metric g. Instead of use the Planck function for the formulation of the statements we can equivalently employ the weight λ g defined by λ g = h −1 . The function λ g is a g-weight for the metric g (cf. [15]). Given a g-weight M , it is possible to construct an equivalent smooth weight M such that M ∈ S(M, g) (cf. [15,20]). In particular, for λ g there exists an equivalent smooth weight λ g such that λ g ∈ S(λ g , g). Hence, λ g ∈ S( λ g , g). A weight M such that M ∈ S(M, g) is called regular. Thus the weight λ g and consequently the Planck function h can be assumed to be regulars.
In the special case ρ = 1, δ = 0 then one has λ g (X) = ξ . The weight λ g can be seen as an extension of the basic one ξ , for the (ρ, δ) classes. The symbols in S(λ μ g , g) for μ ∈ R can be seen as the symbols of order μ with respect to the metric g. In particular, λ μ g is a symbol of order μ with respect to g.
We recall the following theorem which gives the asymptotic expansion for the composition of two symbols. The proof can also be found in [4] or [20].
We now recall the definition of Sobolev spaces adapted to the Weyl-Hörmander calculus. Here we adopt the Beals's definition for simplicity in the presentation of the basic theory. Comprehensive treatments on Sobolev spaces in this setting can be found in [3,20]. Definition 2.9. Let g be a Hörmander metric and M a g-weight. We will call Sobolev space relative to M and it will be denoted by H(M, g), the set of tempered distributions u on R n such that Remark 2. 10. We observe that the definition above requires a test over all the symbols in S(M, g). In contrast, we note that the classical Sobolev spaces H m = H( ξ m , g 1,0 ) are defined by the condition on the tempered distribution u: This means that, in the classical case the Sobolev space is defined by a fixed symbol. This property can be generalised to the Sobolev spaces H(M, g). Indeed, that is a consequence of the existence (cf. [3]), for every g-weight M of a one parameter group (for the operation #) of symbols b t ∈ S(M t , g). (2.5) The characterisation (2.5) is very useful for the analysis carried out to obtain the main results.
The action of the Weyl quantization on the Sobolev spaces is determined by the following theorem (cf. [3]).
Theorem 2.11. Let g be a Hörmander metric, M and M 1 be two g-weights.

Results
In this section we prove the results of this work. First, we consider energy estimates for parabolic equations of order 1 and 2 relative to a Hörmander metric g. Second, we establish the existence, regularity and uniqueness with respect to H(m, g) spaces for the problem (1.1) and give some examples. We consider the cases ν = 1 and ν = 2 separately since the corresponding energy inequalities that we will obtain are of different nature.

Case of order ν = 1
We start with an energy estimate for parabolic equations of order 1 in the S(M, g)-calculus. Gårding inequalities in the setting of S(M, g) classes will play an essential role.   , g)) and write ω = Qv. We will consider the characterisation of Sobolev spaces H(λ s g , g) via the one parameter group b s as described by Remark 2.10. We will write the Kohn-Nirenberg quantization of b s as b s (x, D) = Λ s g . By (2.5) with M = λ g , a tempered distribution u belongs to H(λ s g , g) if and only if The term (3.2) can be written in the following way . Since 0 ≤ Re(−K(t, x, ξ)) ∈ S(λ 1 g , g) by the sharp Gårding inequality (cf. Theorem 18.6.7. [15]), it follows that . An application of the Gronwall inequality gives us the following energy inequality We can also prove an analogous estimate with v(T ) instead of v(0) on the right-hand side of the inequality (3.3). Since Re(K * ) = Re(K) one can note that the same conclusion holds for Q * .

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On the Well-Posedness for a Class Page 9 of 19 3 As an application of the above estimates we will establish a theorem on the existence, uniqueness and regularity in the case of order 1. We will also required of the following spaces H +∞  Proof. We now prove the existence of a solution v of (3.4) in C([0, T ], H(λ s g , g)). The main arguments here are adapted from the corresponding part in the proof of Theorem 4.5, Chap. 6 in [5]. We write Q = ∂ ∂t − K and we introduce the space )|ϕ(T ) = 0}. We will see that we can define a linear form β on Q * E by ϕ(0, ·)).
We will now prove that v ∈ C ([0, T ], H(λ s  g , g)) and that it satisfies the energy estimate (3.1). Suppose we have sequences H(λ s  g , g)).
In the limit, we have Q v = ω, v(0, ·) = f ; consequently, the uniqueness shows that v = v.
The corresponding inequality (3.1) for v is obtained passing to the limit in this inequality applied to v j . In this way we conclude the proof of the Theorem. The uniqueness of the solution v follows from the energy inequality (3.1). Now, coming back to the problem (1.1), we first note that for a given Hörmander metric g we can choose λ g equivalent to λ g and regular. In particular, λ g is a smooth symbol and we can justify the use of quantization for the Planck function in (1.2) by writing where R ∈ OpS(λ g , g).
We note that if two g-weights M 1  (i) Let L be a second order differential operator, formally self-adjoint, where (a ij (x)) is a positive semi-definite matrix of real functions and a ij ∈ C ∞ b (R n ) (C ∞ functions which are bounded as well as all their derivatives). If L is subelliptic with order of subellipticity 1 ≤ τ ≤ 2, we associate to L the Hörmander metric g defined by a(x, ξ) is the principal symbol of L. The invertibility of L + C in the setting of S(m, g) classes has been studied in [8]. The case 0 < τ ≤ 1 has been investigated for special highly degenerate Grushin operators in [29] and [10]. The study of invertibility requires of lower bounds estimates in a crucial way [24,25]. Fundamental solutions for highly degenerate elliptic operators (or in other terms small τ ) have been obtained in [2]. For L as in (3.6) and the metric (3.7), the parameter λ g is given by λ g (x, ξ) = a(x,ξ)+ ξ τ ξ . An example of an operator operator of order μ ∈ R is given by L μ (g) of the form , g and J is the Bessel potential of order 1 on R n , i.e. corresponding to the symbol ξ and R(x, D) ∈ OpS(λ g , g). (ii) Let 0 < ≤ 1 and g be the metric defined by . (3.9) In the case = 1 it is well known that g is a Hörmander metric [20].
In the general case, the uncertainty principle follows immediately. The slowness follows from the fact that 0 < ≤ 1 and the case = 1. The temperateness is a consequence of the slowness for this type of metric (cf. Example 2, P. 378 of [14] and [20]). In this case λ g = (1+|x| 2 +|ξ| 2 ) and one can apply Theorem 3.2 to the operator K(x, D x ) = −(−Δ + |x| 2 + 1) . The point of view of order with respect to a Hörmander metric g allows us to consider some operators K(t, x, D x ) that being of order 1 with respect to g are also of order as differential operators. For instance that will be the case of metrics of the form g = dx 2 (1+|x| 2 +|ξ| 2 ) + dξ 2 (1+|x| 2 +|ξ| 2 ) from which we can consider an operator of the form K(x, D x ) = −(1 + |x| 2 − Δ) which is of order 2 but only of order 1 with respect to the metric (3.9).
We also note that the uncertainty principle is not satisfied if > 1, what makes the metric (3.9) not allowed in our setting in that case.

Case of order ν = 2
We shall now consider parabolic equations of order 2. We start by proving a version of Theorem 3.2 when K is of order 2 and scalar. We first establish a corresponding energy estimate. In the following theorems we will be considering operators P (x, D x ) ∈ OpS(λ ν g , g) satisfying the following condition with respect to a Hörmander metric g and ν ∈ R: (E, g, ν) There exists a positive constant C ν such that We note that λ ν g − 1 ≥ 0 due to the uncertainty principle. The condition (E, g, ν) is an extension of the notion of strongly ellipticity in the classes S ν 1,0 . For the metric g = g 1,0 and ν = 2, we have λ 2 g = ξ 2 , and the ellipticity of order 2 for K ∈ S 2 1,0 is equivalent to the condition (E, g 1,0 , 2). Some other examples are given at the end of this section. Theorem 3.3. Let g be a Hörmander metric, s ∈ R, T > 0, K(t, x, ξ) ∈ S(λ 2 g , g) depending smoothly on t and −K satisfying the condition (E, g, 2).
). Then, there exists a constant C > 0 such that v satisfies the energy estimate: for all t ∈ [0, T ]. The same conclusion holds for the operator Q * .
Proof. First we will obtain some L 2 estimates. We write Qv = ω, as in the proof of Theorem 3.1 one can prove that The term (3.12) can be estimated by applying the Fefferman-Phong inequality ( [15], Theorem 18.6.8) to the nonnegative symbol −K(t) + C 2 − C 2 λ 2 g . We have (3.14) For (3.13), we have Then, by (3.14), (3.15) we obtain .

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On the Well-Posedness for a Class Page 13 of 19 3 Integrating between 0 and t the above inequality we get (3.17) We now observe that Λ −1 ). Thus, applying this identity to the second term in the integral of the right-hand side of (3.17) we get Now by Lemma 4.4, p. 363 of [5] we obtain v(t) 2 We have proved the estimate (3.11) assuming C 2 > 1 2 . If 0 < C 2 ≤ 1 2 then K = C −1 2 K satisfies the condition (E, g, 2) with constant C 2 = 1 and consequently the estimate (3.11). for the operator Q = ∂ ∂t − K(t, x, D x ) holds. A time re-scaling argument shows that the energy estimate also holds for Q = ∂ ∂t − K(t, x, D x ) when 0 < C 2 ≤ 1 2 . This concludes the proof of the estimate (3.11). The fact that the estimate also holds for Q * follows analogously taking into account that K * also satisfies the condition (E, g, 2).
Reasoning analogously as in the proof of Theorem 3.2 by using instead the above energy inequality we obtain the following theorem. Moreover, the solution v satisfies the energy estimate (3.11).
Example 2. One can also consider order 2 versions of operators K as in Examples 1 and some others such that −K satisfies the condition (E, g, 2) with C 2 ≥ 1: (i) For the metric g in Examples 1 (i), an example of an operator operator of order 2 is given by L 2 (g) : One can then consider −K of the form (L 2 + J 2τ )J −2 . (ii) For g as in Examples 1 (ii) we can consider for instance K of order 2 defined by where 0 < ≤ 1.
The case of the fractional Laplacian requires further analysis and it will be treated in the following section.

The fractional Laplacian
We shall now consider the case of fractional diffusion as has been announced in the introduction. This example will require of some preliminary adaptation since the fractional Laplacian can not be directly treated as a pseudodifferential operator. Indeed, for 0 < s < 1 the fractional Laplacian (−Δ) s 0 < s < 1 has not a symbol in a class of symbols for pseudo-differential operators. We recall that (−Δ) s is defined by (3.21) There are several equivalent definitions of the fractional Laplacian. We refer the reader to [17] for a recent review on the most common ones and the equivalence between them. For more details on the following decomposition we refer to [12]. In order to apply the pseudo-differential calculus to the fractional Laplacian we choose a cut-off function ϕ : R n → [0, 1] such that ϕ(ξ) = 0 for |ξ| < 1 and ϕ(ξ) = 1 for |ξ| > 2.
We will now take care of the operator E. A close look to the proof of the energy estimate given by Theorem 3.1 shows to us that we can extend the energy inequality to a(D) + E as a consequence of the boundedness (3.24). Indeed, one has the following mild lemma. Lemma 3.5. Let E be the operator as in (3.23). Then Proof. We first note that from (3.24) for γ = 0 we have On the other hand, since E and Λ γ commute and by (3.24) we have The lemma above allows to handle the additional term E without losing the previous energy estimate. Now, if 1 2 < s < 1 we can consider the metric (3.25). By writing s = ρ−δ 2 we cover s in the interval ( 1 2 , 1) and we can extend Theorem 3.3: