Schauder estimates on bounded domains for KFP operators with coefficients measurable in time and Hölder continuous in space

1 Introduction and main results

In his 1936 seminal paper on the Theory of gases [12], Kolmogorov introduced the parabolic operator (nowadays called the Kolmogorov operator)

This operator K enjoys the following interesting features:

1. K is ultra-parabolic, since the diffusive term Δ x is missing;

2. K possesses an explicit, global fundamental solution p , found by Kolmogorov himself: more precisely, introducing the notation

   X = ( v , x ) , Y = ( w , y ) , Z = ( ω , ξ ) ,

   for the points of R 2 n , we have

   p t , s ( X , Y ) = p ˆ t − s ( v − w , x − ( s − t ) w − y ) , where p ˆ τ ( Z ) = c n τ 2 n exp − 1 τ ∣ ω ∣ 2 − 3 τ ⟨ ω , ξ + τ ω ⟩ + 3 τ 2 ∣ ξ + τ ω ∣ 2 1 { τ > 0 } .

   In particular, since p ˆ is clearly smooth outside the origin of R 2 n × R , the operator K is C ∞ -hypoelliptic (despite its degeneracy).

3. K can be written as a sum of squares (of smooth vector fields) plus a drift:

   K u = ∑ i = 1 n X i 2 u + Y u ,

   where X i = ∂ v i (with 1 ≤ i ≤ n ) and Y u = ⟨ v , ∇ x u ⟩ − u t .

In view of properties (i)–(iii), the operator K is presented by Hörmander as the main inspiration for his study in his 1967 celebrated paper [10]; in fact, we see that K is a degenerate and hypoelliptic sum of squares plus a drift, but it does not satisfy the sufficient conditions for hypoellipticity established by Hörmander himself in his previous work [9]. This example led Hörmander to formulate the so-called (Hörmander) Rank Condition for hypoellipticity.

Inspired by [10], at the beginning of the 90s, a large class of degenerate and hypoelliptic operators (of which the Kolmogorov operator K is the main prototype) has been deeply studied by Lanconelli and Polidoro [13]: more precisely, they considered parabolic operators ℒ of the form

which satisfy the following structural assumptions.

1. Assumption on the principal part of ℒ :

   1. 1 ≤ q ≤ N ;

   2. the constant matrix A 0 = ( a i j ) i , j = 1 q is symmetric and positive definite.

2. Assumption on the drift Y:

   1. the N × N matrix B takes the block form

      (1.2) B = ⋆ ⋆ … ⋆ ⋆ B 1 ⋆ … … … O B 2 … ⋆ ⋆ ⋮ ⋮ ⋱ ⋮ ⋮ O O … B k ⋆ ,

      where the entries of the blocks denoted by ⋆ are arbitrary, while every block B j is an m j × m j − 1 matrix of rank m j (for j = 1 , 2 , … , k ), with

      m 0 = q , m 0 ≥ m 1 ≥ … ≥ m k and m 0 + m 1 + ⋯ + m k = N .

These operators ℒ are usually called Kolmogorov-Fokker-Planck (KFP) operators. We explicitly note that the Kolmogorov operator K actually fits into this class, since it can be obtained by choosing

Under the above assumptions, Lanconelli and Polidoro [13] proved that any KFP operator ℒ satisfies the Hörmander Rank Condition, so that ℒ is C ∞ -hypoelliptic; moreover, it possesses a rich underlying geometric structure:

1. ℒ is left-invariant on the non-commutative Lie group G = ( R N + 1 , ∘ ) , where the composition law ∘ is defined as follows:

   ( y , s ) ∘ ( x , t ) = ( x + E ( t − s ) y , t + s )

   and E ( t ) = e − t B . This means that, defining τ y , s ( x , t ) = ( y , s ) ∘ ( x , t ) , for every u ∈ C ∞ ( R N + 1 ) and every ( y , s ) ∈ R N + 1 , we have

   ℒ ( u ∘ τ y , s ) = ( ℒ u ) ∘ τ y , s .

2. If, in addition, the ⋆ blocks in the matrix B vanish, ℒ is also homogeneous of degree 2 w.r.t. the non-isotropic dilations (automorphisms of G )

   (1.3) D ( λ ) ( x , t ) ≡ ( D 0 ( λ ) ( x ) , λ 2 t ) = ( λ q 1 x 1 , … , λ q N x N , λ 2 t ) ,

   where the N -tuple ( q 1 , … , q N ) is given by

   ( q 1 , … , q N ) = ( 1 , … , 1 ︸ m 0 , 3 , … , 3 ︸ m 1 , … , 2 k + 1 , … , 2 k + 1 ︸ m k ) .

   This means that, for every u ∈ C ∞ ( R N + 1 ) and every λ > 0 , we have

   ℒ ( u ∘ D λ ) = λ 2 ( ℒ u ) ∘ D λ .

After [13], more general families of (degenerate) KFP operators have been deeply studied; in particular, a great attention has been gained by KFP operators with variable coefficients a i j ( x , t ) , which play an important role in the theory of Stochastic Differential Equations (e.g., [2, Section 2.1]), as well as in collisional kinetic theory (e.g., [19,21]). In this perspective, recent research studies in the field of Stochastic Differential Equations suggest the importance of developing a theory allowing the coefficients a i j to be rough in t (say, L ∞ ), and more regular only w.r.t. the space variables.

Motivated by this fact, in [1], we studied a suitable class of KFP operators (1.1) on R N + 1 with variable coefficients a i j ( x , t ) , which are L ∞ in t and Hölder continuous in x (in a sense which will be made precise in a moment). For these operators, we proved partial Schauder estimates on unbounded strips S T = R N × ( − ∞ , T ) , i.e., a control of the supremum in t of the Hölder norms in space of u x i x j , assuming the finiteness of the analogous quantity for the coefficients a i j and the right-hand side of the equation. We also proved a local Hölder continuity estimate in the joint variables ( x , t ) for the derivatives u x i x j , without assuming any continuity in t on the coefficients nor on ℒ u .

On the other hand, since the drift Y = ⟨ B x , ∇ ⟩ − ∂ t has unbounded coefficients, requiring the global boundedness of ℒ u (which is encoded in the boundedness assumption of the full Hölder norm of ℒ u ) seems a quite demanding assumption. Therefore, it can be interesting also to establish a version of the aforementioned a priori estimates on bounded domains of R N + 1 , in the spirit of the classical interior Hölder estimates in cylinders. However, these local estimates cannot be easily deduced directly by the global bounds proved in [1], but require some nontrivial work, which is exactly the content of the present study. We will prove partial Schauder estimates for u x i x j (with 1 ≤ i , j ≤ q ) and for Y u on a bounded domain of R N + 1 (Theorem 1.6), as well as a Hölder continuity estimate in ( x , t ) of u x i x j (Theorem 1.7).

Comparison with the existing literature. Schauder estimates for KFP operators have been proved first by Manfredini [18] and later, under more general assumptions on B , by Di Francesco and Polidoro [5], on bounded domains, assuming the coefficients a i j Hölder continuous (in ( x , t ) ) with respect to the intrinsic distance induced in R N + 1 by the vector fields ∂ x 1 , … ∂ x q , Y . For related issues about Schauder estimates for KFP operators, we point out also the research by Lunardi [17], Priola [20], Imbert and Mouhot [11], Henderson and Snelson [8], Lorenzi [14], and references therein.

Partial Schauder estimates for degenerate KFP operators, of the kind dealt in our previous study [1], have been proved also by Chaudru de Raynal et al. [4], with different techniques and without obtaining the Hölder control in time of second-order derivatives. Concerning the same class of KFP operators with coefficients Hölder continuous in space and L ∞ in time, we quote the research by Lucertini et al. [15], dealing with the construction of a fundamental solution for these operators, with consequent results about the Cauchy problem, and the research by Henderson and Wang [7], containing partial Schauder estimates for a special class of KFP operators, with applications to the Landau equation. The results in [15] and [7] are independent from and do not contain our results. We also point out the preprint by Dong and Yastrzhembskiy [6] dealing with kinetic KFP operators, and the more recent preprint by Lucertini et al. [16], where the authors prove optimal global Schauder estimates, which improve some of the results in [1].

Main results. We now turn to give the precise statements of our main results; to this end we first need to introduce the appropriate metric and functional setting.

We begin by recalling that, if ℒ 0 is a KFP operator as in (1.1) with constant coefficients and with vanishing ⋆ blocks in the “drift” matrix B , see (1.2), Lanconelli and Polidoro [13] proved the following facts:

1. ℒ 0 is C ∞ -hypoelliptic;

2. ℒ 0 is left-invariant w.r.t. the composition law

   ( y , s ) ∘ ( x , t ) = ( x + E ( t − s ) y , t + s ) ;

3. ℒ 0 is 2-homogeneous w.r.t. the dilations

   D ( λ ) ( x , t ) ≡ ( D 0 ( λ ) ( x ) , λ 2 t ) = ( λ q 1 x 1 , … , λ q N x N , λ 2 t ) .

Since we aim to study KFP operators ℒ with variable coefficients a i j ( x , t ) such that the corresponding model operator ℒ 0 with constant a i j satisfies (a)–(c), it is convenient to endow R N + 1 with a structure of homogeneous quasi-metric space adapted to ∘ and to { D λ } λ > 0 .

First of all, we consider in R N + 1 , the following D λ -homogeneous norm:

and we also set

Accordingly, we define a translation-invariant quasi-distance as follows:

It is easy to see that ρ and d satisfy the following properties:

1. ρ ( ξ ) ≥ 0 for every ξ ∈ R N + 1 , and ρ ( ξ ) = 0 ⇔ ξ = 0 ;

2. ρ ( D ( λ ) ξ ) = λ ρ ( ξ ) for every ξ ∈ R N + 1 , λ > 0 ;

3. d ( ξ , η ) ≥ 0 for every ξ , η ∈ R N + 1 , and d ( ξ , η ) = 0 ⇔ ξ = η ;

4. there exists κ > 0 such that, for every ξ , η , ζ ∈ R N + 1 , we have

   d ( ξ , η ) ≤ κ ( d ( ξ , ζ ) + d ( η , ζ ) ) and d ( ξ , η ) ≤ κ d ( η , ξ ) ;

5. d ( ξ , η ) = d ( η − 1 ∘ ξ , 0 ) for every ξ , η ∈ R N + 1 .

Moreover, by the explicit expression of ∘ , we have

It should be noted that ρ ( ⋅ ) and ‖ ⋅ ‖ are not the ones defined in [1]; however, they are globally equivalent to them, and have the advantage of being smooth outside the origin.

Now, we have defined a good structure in R N + 1 , we can introduce the appropriate function spaces allowing to state our main results. Throughout the following, Ω ⊂ R N + 1 is a fixed bounded domain; if ξ = ( x , t ) ∈ Ω , we set

Moreover, given any ξ = ( x , t ) , η = ( y , s ) ∈ Ω , we also set

When the open set Ω is understood, we simply write d ξ and d ξ , η .

We are now ready to state the main results of this note. Our first result is a version of the Schauder estimates on the domain Ω , controlling the local Hölder continuity w.r.t. x of u x i x j (and the full Hölder continuity of u x i and u ) with the local Hölder continuity w.r.t. x of ℒ u and sup ∣ u ∣ .

Under the same assumptions, we will also improve our estimate on u x i x j controlling their local Hölder continuity also w.r.t. time.

2 Proof of the main results

We are now ready to begin the proofs of Theorems 1.6 and 1.7.