The Liouville theorem and linear operators satisfying the maximum principle

A result by Courr\`ege says that linear translation invariant operators satisfy the maximum principle if and only if they are of the form $\mathcal{L}=\mathcal{L}^{\sigma,b}+\mathcal{L}^\mu$ where $$ \mathcal{L}^{\sigma,b}[u](x)=\text{tr}(\sigma \sigma^{\texttt{T}} D^2u(x))+b\cdot Du(x) $$ and $$ \mathcal{L}^\mu[u](x)=\int \big(u(x+z)-u-z\cdot Du(x) \mathbf{1}_{|z| \leq 1}\big) \,\mathrm{d} \mu(z). $$ This class of operators coincides with the infinitesimal generators of L\'evy processes in probability theory. In this paper we give a complete characterization of the translation invariant operators of this form that satisfy the Liouville theorem: Bounded solutions $u$ of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$ are constant. The Liouville property is obtained as a consequence of a periodicity result that completely characterizes bounded distributional solutions of $\mathcal{L}[u]=0$ in $\mathbb{R}^d$. The proofs combine arguments from PDE and group theories. They are simple and short.


Introduction and main results
The classical Liouville theorem states that bounded solutions of ∆u = 0 in R d are constant. The Laplace operator ∆ is the most classical example of an operator L : C ∞ c (R d ) → C(R d ) satisfying the maximum principle in the sense that (1) L[u](x) ≤ 0 at any global maximum point x of u.
In the class of linear translation invariant 1 operators (which includes ∆), a result by Courrège [13] 2 says that the maximum principle holds if and only if 2. For general L, we show that all bounded solutions of L[u] = 0 in R d are periodic and we identify the set of admissible periods. Let us now state our results. For a set S ⊆ R d , we let G(S) denote the smallest additive subgroup of R d containing S and define the subspace V S ⊆ G(S) by V S := g ∈ G(S) : tg ∈ G(S) ∀t ∈ R .
The above Liouville result is a consequence of a periodicity result for bounded solutions of L[u] = 0 in R d . For a set S ⊆ R d , a function u ∈ L ∞ (R d ) is a.e. S-periodic if u(· + s) = u(·) in D ′ (R d ) ∀s ∈ S. Our result is the following: Theorem 1.2 (General periodicity). Assume (A σ,b ), (A µ ), and u ∈ L ∞ (R d ). Let L be given by (2)-(3)- (4). Then the following statements are equivalent: (b) u is a.e. G µ + W σ,b+cµ -periodic. 3 The representation (2)-(3)-(4) is unique up to the choice of a cut-off function in (4) and a square root σ of a = σσ T . In this paper we always use 1 |z|≤1 as a cut-off function.
This result characterizes the bounded solutions for all operators L in our class, also those not satisfying the Liouville property. Note that if G µ + W σ,b+cµ = R d , then u is constant and the Liouville result follows. Both theorems are proved in Section 2.
We give examples in Section 3. Examples 3.2 and 3.5 provide an overview of different possibilities, and Examples 3.7 and 3.8 are concerned with the case where card (supp(µ)) < ∞. The Liouville property holds in the latter case if and only if card (supp(µ)) ≥ d − dim W σ,b+cµ + 1 with additional algebraic conditions in relation with Diophantine approximation. The Kronecker theorem (Theorem 3.6) is a key ingredient in this discussion and a slight change in the data may destroy the Liouville property.
To prove that solutions of L[u] = 0 are G µ -periodic, we rely on propagation of maximum points [10,14,11,15,16,22,6,7] and a localization techniqueà la [10,3,29,7]. As far as we know, Choquet and Deny [10] were the first to obtain such results. They were concerned with the equation u * µ − u = 0 for some bounded measure µ. This is a particular case of our equation since u * µ − u = L µ [u] +´z1 |z|≤1 dµ(z) · Du. For general µ, the drift´z1 |z|≤1 dµ(z) · Du may not make sense and the identification of the full drift b + c µ relies on a standard decomposition of closed subgroups of R d , see e.g. [24]. The idea is to establish G µ -periodicity of solutions of L[u] = 0 as in [10], and then use that G µ = V µ ⊕ Λ for the vector space V µ previously defined and some discrete group Λ. This will roughly speaking remove the singularity z = 0 ∈ V µ in the computation of c µ because´V µ z1 |z|≤1 dµ(z) · Du = 0 for any G µ -periodic function. See Section 2 for details.
Our approach then combines PDEs and group arguments, extends the results of [10] to Courrège/Lévy operators, yields necessary and sufficient conditions for the Liouville property, and provides short and simple proofs.
Outline of the paper. Our main results (Theorems 1.1 and 1.2.) were stated in Section 1. They are proved in Section 2 and examples are given in Section 3.
Notation and preliminaries. The support of a measure µ is defined as where B r (z) is the ball of center z and radius r. To continue, we assume (A σ,b ), (A µ ), and L is given by (2)-(3)-(4).
The above distribution is well-defined since L * : The following technical result will be needed to regularize distributional solutions of L[u] = 0 and a.e. periodic functions. Let the mollifier ρ ε (x) :

Proofs
This section is devoted to the proofs of Theorems 1.1 and 1.2. We first reformulate the classical Liouville theorem for local operators in terms of periodicity, then study the influence of the nonlocal part.
2.1. W σ,b -periodicity for local operators. Let us recall the Liouville theorem for operators of the form (3), see e.g. [26,25]. In the result we use the set Note that span R {σ 1 , . . . , σ P } equals the span of the eigenvectors of σσ T corresponding to nonzero eigenvalues.
. Then the following statements are equivalent: Let us now reformulate and prove this classical result as a consequence of a periodicity result, a type of argument that will be crucial in the nonlocal case. We will consider C ∞ b (R d ) solutions, which will be enough later during the proofs of Theorem 1.1 and 1.2, thanks to Lemma 1.5.
Then the following statements are equivalent: Note that part (b) implies that u is constant in the directions defined by the vectors σ 1 , . . . , σ P , b. If their span then covers all of R d , Theorem 2.1 follows trivially. To prove Proposition 2.2, we adapt the ideas of [25] to our setting.
Since v(x, ·, ·) is bounded, we conclude by uniqueness of the heat equation that for any s < t, where K P is the standard heat kernel in R P . But then and since ∆ y K P (·, t − s) L 1 → 0 as s → −∞, we deduce that ∆ y v = 0 for all x, y, t. By the classical Liouville theorem (see e.g. [26]), v is constant in y. It is also constant in t by (6) and W σ,b = {σy − bt : y ∈ R P , t ∈ R}.
2.2. G µ -periodicity for general operators. Proposition 2.2 might seem artificial in the local case, but not so in the nonlocal case. In fact we will prove our general Liouville result as a consequence of a periodicity result. A key step in this direction is the lemma below.
, the Arzelà-Ascoli theorem implies that there exists v ∞ such that v n → v ∞ locally uniformly (up to a subsequence). Taking another subsequence if necessary, we can assume that the derivatives up to second order converge and pass to the limit in the equation A similar argument shows that there is a u ∞ such that u n → u ∞ as n → ∞ locally uniformly. Taking further subsequences if necessary, we can assume that u n and v n converge along the same sequence. Then by construction By Lemma 2.4 and an iteration, we find that M = v ∞ (mz) = u ∞ ((m + 1)z) − u ∞ (mz) for any m ∈ Z. Then by another iteration, But since u ∞ is bounded, the only choice is M = 0 and thus v ≤ M = 0. A similar argument shows that v ≥ 0, and hence, 0 = v(x) = u(x +z) − u(x) for anȳ z ∈ supp(µ) and all x ∈ R d .
We can give a more general result than Lemma 2.3 if we consider groups.
(b) The subgroup generated by a set S ⊆ R d , denoted G(S), is the smallest additive group containing S. Now we return to a key set for our analysis: .
This set appears naturally because of the elementary result below.
Proof. It suffices to show that G := {g ∈ R d : w(· + g) = w(·)} is a closed subgroup of R d . It is obvious that it is closed by continuity of w. Moreover, for any g 1 , g 2 ∈ R d and x ∈ R d , By Lemmas 2.3 and 2.6, we have proved that: , and G µ by (7). Then any solution 2.3. The role of c µ . Propositions 2.2 and 2.7 combined may seem to imply that L[u] = 0 gives (G µ + W σ,b )-periodicity of u, but this is not true in general. The correct periodicity result depends on a new drift b + c µ , where c µ is defined in (9) below. To give this definition, we need to decompose G µ into a direct sum of a vector subspace and a relative lattice. (b) A full lattice is a subgroup Λ ⊆ R d of the form Λ = ⊕ d n=1 a n Z for some basis {a 1 , . . . , a d } of R d . A relative lattice is a lattice of a vector subspace of R d . Theorem 2.9 (Theorem 1.1.2 in [24]). If G is a closed subgroup of R d , then G = V ⊕ Λ for some vector space V ⊆ R d and some relative lattice Λ ⊆ R d such that V ∩ span R Λ = {0}.
In this decomposition the space V is unique and can be represented by (8)  Proof. If the lemma does not hold, there exists v n + λ n → λ as n → ∞ where v n ∈ V , λ n ∈ Λ, λ n = λ. Since the projection V ⊕ span R Λ → span R Λ is continuous, λ n → λ and this contradicts the fact that each point of Λ is isolated. Lemma 2.11. Let G, V and Λ be as in Theorem 2.9. Then For any t ∈ R, tg = tv + tλ ∈ G and thus tλ ∈ G since tv ∈ V ⊆ G. Let B be an open ball containing λ such that B ∩ G = B ∩ (V + λ). Choosing t such that t = 1 and tλ ∈ B, we infer that tλ =ṽ + λ for someṽ ∈ V . Hence λ = (t − 1) −1ṽ ∈ V and this implies that λ = 0. In other words V G ⊆ V , and the proof is complete.
Remark 2.12. Any G-periodic function w ∈ C 1 (R d ) is such that z · Dw(x) = lim t→0 w(x+tz)−w(x) t = 0 for any x ∈ R d and z ∈ V G . By Theorem 2.9 and Lemma 2.11, we decompose the set G µ in (7) into a lattice and the subspace V µ := V Gµ . The new drift can then be defined as (9) c µ = −ˆ{ |z|≤1}\Vµ z dµ(z).
Proposition 2.13. Assume (A µ ) and c µ is given by (9). Then c µ ∈ R d is welldefined and uniquely determined by µ.
Proof. Using that supp(µ) for some open ball B containing 0 given by Lemma 2.10. This integral is finite by (A µ ) which completes the proof.
Proof. Using that´f dµ =´s upp(µ) f dµ, we have which by Proposition 2.2 shows that u is also W σ,b+cµ -periodic. It is now easy to see that u is G µ + W σ,b+cµ -periodic.
We now prove Theorem 1.1 on necessary and sufficient conditions for L to satisfy the Liouville property. We will use the following consequence of Theorem 2.9. Proof of Theorem 1.1. (a) ⇒ (b) Assume (b) does not hold and let us construct a nontrivial G µ + W σ,b+cµperiodic L ∞ -function. By Corollary 2.15, for some c ∈ R d and codimension 1 subspace H ⊂ R d . We can assume c / ∈ H since otherwise (10) will hold if we redefine c to be any element in H c . This means and for any x ∈ R d there exists a unique pair (x H , λ x ) ∈ H × R such that Now let U (x) := cos(2πλ x ) and note that for any h ∈ H and n ∈ Z, This proves that U is (H + cZ)-periodic and thus also G µ + W σ,b+cµ -periodic. By Theorem 1.2, L[U ] = 0, and we have a nonconstant counterexample of (a). Note indeed that u ∈ L ∞ (R d ) since it is everywhere bounded by construction and C ∞ (thus measurable) because the projection x → λ x is linear. We therefore conclude that (a) implies (b) by contraposition.

Examples
Let us give examples for which the Liouville property holds or fails. We will use Theorem 1.1 or the following reformulation:   (b) Even if supp(µ) has an empty interior, (12) may fail and Liouville still hold. This is e.g. the case for the mean value operator where S denotes the d − 1-dimensional surface measure. (c) We may have the Liouville property with just a finite number of points in the support of µ, see Example 3.7. (d) The way we have defined the nonlocal operator, if L = L µ with general µ, (11) reduces to (14) supp(µ) ⊆ H + cZ and c µ ∈ H, for some H of codimension 1 and c. We can have (12) without (14) as e.g. for the 1-d measure µ = δ −1 + 2δ 1 . Indeed supp(µ) ⊂ Z but c µ = 1 = 0. The associated operator L µ then has the Liouville property even though it would not for any symmetric measure with the same support. (e) A general operator L = L σ,b + L µ may satisfy the Liouville property even though each part L σ,b and L µ does not. A simple 3-d example is given by In the 1-d case, the general form of the operators which do not satisfy the Liouville property is very explicit.