Anisotropic Maximal $L^p$-regularity Estimates for a Hypoelliptic Operator

Let us consider the Ornstein-Uhlenbeck operator in R N :

where A = (a ij ) is an N × N constant matrix that is symmetric positive semi-definite and B = (b ij ) is an N × N constant matrix. The operator A is the infinitesimal generator of the Uhlenbeck-Ornstein semigroup, which is the Markov semigroup associated to a stochastic differential equation that describes a random motion of a particle in a fluid.

Hörmander in [8] studied when L := ∂ t -A has hypoellipticity. Roughly speaking, a differential operator P is called hypoelliptic if the smoothness of P u implies the smoothness of u itself. In [8], Hörmander showed that the operator L is hypoelliptic if and only if t 0 exp(-sB ⊺ )A exp(-sB) ds > 0 for all t > 0. Here, B ⊺ denotes the transpose of B. Lanconelli and Polidoro [11] proved that if L is hypoelliptic, then we can set

A 0 0 0 0 with a p 0 × p 0 constant matrix A 0 (p 0 ≤ N ) that is symmetric and positive definite and where B j is a p j-1 × p j block with rank p j , p 0 ≥ p 1 ≥ • • • ≥ p r ≥ 1 and p 0 + p 1 + • • • + p r = N for some basis of R N . We say that L is degenerate if p 0 < N and say that L is non-degenerate otherwise.

In this paper, we study a special case of the Ornstein-Uhlenbeck operators, N = 2d and A = 0 0 0 I , and B = 0 -I 0 0 .

Here I is the d×d identity matrix. Then L is hypoelliptic and degenerate. To focus on this case, we introduce the notations

x = (x 1 , . . . , x d ), y = (y 1 , . . . , y d ) = (x d+1 , . . . , x 2d ),

Then A is written as

To state our main results, we give the definitions of fundamental concepts.

This is equivalent to the definition by the Fourier transform if 1 ≤ p ≤ 2. See Kwaśincki [10], for example.

Definition 3. For p, q ≥ 1, the L q y L p x,t -norm of a function f is defined by

The L q y L p x -norm is also defined in the similar way. As a main result of this paper, we shall prove the following global estimates in the anisotropic L p -spaces.

Then there exists a weak solution of

The key step of the proof of Theorem 1 is the following estimates for the non-stationary problem, which has its own interest.

Then there exists a weak solution of

for some constant C = C(p, q, d) > 0. Moreover, the corresponding weak (1,1) result also holds:

Here, C ′ is a positive constant depending only on the dimension.

When p = q = 2, the estimate (1) is proved by Bouchut [1] as the a priori estimates. The proof relies on Hörmander’s commutator

and the energy method based on the integration by parts. Moreover, its method is also valid for the solution of Au = f because

The L p -estimate of ∆ y u for general hypoelliptic degenerate Ornstein-Uhlenbeck operators is proved by Bramanti, Cupini, Lanconelli and Priola [2]. However, since they take the case tr B ̸ = 0 into consideration, they are forced to take L p -norms only on the strip S = R N × [-1, 1], rather than on R N × R. We extend the result of [2] to the time-global case in exchange for specialization.

It is a classical result proved in [13] that if Au = f and f belongs to the Sobolev spaces W α,p with p ∈ (1, ∞) and α ≥ 0, then u is in W α+2/3,p loc . This result is very general because it is independent of the domain of the functions. When ∆ y in L is replaced by the fractional Laplacian ∆ α/2 y = -(-∆ y ) α/2 with α ∈ (0, 2), the maximal regularity on both x and y is proved by Chen and Zhang [3] and Huang, Menozzi and Priola [9]. In [3] the proof is based on the Fefferman-Stein type estimate that leads to the L ∞ -BMO boundedness. This approach is used also in [4] to prove the maximal regularity for the Kolmogorov-type hypoelliptic operator with time-dependent coefficients. The proof in [9] is based on the Hörmander condition to derive the weak (1,1) estimate and similar to ours. However, our approach is based on the pointwise estimates for the fundamental solution in a more explicit form than in [9] particularly in the estimate of |∂ x | 2/3 Γ; see (2) below. As for the optimal smoothing estimates in the Hölder spaces, the Shauder estimates are proved by Da Prato and Lunardi [5] in the non-degenerate case and by Lunardi [12] in the degenerate case.

In any case, the previous studies introduced above [2,4,9,13] considered isotropic L p -estimates. On the other hand, since the operator A has an anisotropic structure with respect to x and y, it is natural to study the anisotropic estimates for solutions to Au = f . For example, Dong and Yastrzhembskiy [6] have considered anisotropic norms. (However, the estimate in [6] contains u itself in the upper bound and therefore does not cover our results.) We also expect that the anisotropic L p -estimates are useful in the study of some non-linear problem such as the triple deck equations arising from the boundary layer theory in the fluid mechanics, where the equations contain A as the principal linear term.

Our main goal is to establish the maximal regularity estimates in the anisotropic L p -spaces for the stationary problem Au = f