Interior Schauder estimates for elliptic equations associated with L\'evy operators

We study the local regularity of solutions $f$ to the integro-differential equation $$ Af=g \quad \text{in $U$}$$ associated with the infinitesimal generator $A$ of a L\'evy process $(X_t)_{t \geq 0}$. Under the assumption that the transition density of $(X_t)_{t \geq 0}$ satisfies a certain gradient estimate, we establish interior Schauder estimates for both pointwise and weak solutions $f$. Our results apply for a wide class of L\'evy generators, including generators of stable L\'evy processes and subordinated Brownian motions.


Introduction
Let (X t ) t≥0 be a d-dimensional Lévy process. By the Lévy-Khintchine formula, (X t ) t≥0 is uniquely characterized (in distribution) by its infinitesimal generator A, which is an integro-differential operator with representation for f ∈ C ∞ c (R d ), here (b, Q, ν) denotes the Lévy triplet of (X t ) t≥0 , cf. Section 2. In this paper, we study the local Hölder regularity of weak and pointwise solutions f to the integro-differential equation Af = g in U for open sets U ⊆ R d . We are interested in interior Schauder estimates, i.e. our aim is to describe the regularity of f on the set {x ∈ U ; d(x, U c ) > δ} for δ > 0 and to establish estimates for its Hölder norm. We will see that there is a close connection between the regularity of f and volatility of the Lévy process: the higher the volatility (caused by a non-vanishing diffusion component or a high small-jump activity), the higher the regularity of f . For the particular case that there is no jump part, i.e. ν = 0, the generator A is a second-order differential operator and interior Schauder estimates for solutions to Af = g are well studied, see e.g. Gilbarg [6]. One of the most prominent non-local Lévy operators is the fractional Laplacian −(−∆) α 2 , α ∈ (0, 2), defined by 1 y d+α dy for some normalizing constant c d,α > 0. The fractional Laplacian is the infinitesimal generator of the isotropic α-stable Lévy process and plays an important role in analysis and probability theory, see e.g. the survey paper [19] for a detailed discussion. Global regularity estimates for solutions to −(−∆) α 2 f = g go back to Stein [29], see also Bass [1]. Since then, several extensions and refinements of these estimates have been obtained. Ros-Oton & Serra [24] studied the interior Hölder regularity of solutions to equations Af = g associated with symmetric α-stable operators and established under a mild degeneracy condition on the spectral measure estimates of the form f C α+κ (B(0,1)) ≤ c f C κ (R d ) + g C κ (B(0,2)) for κ ≥ 0 such that α + κ is not an integer. In the recent paper [14], global Schauder estimates were obtained for a wide class of Lévy processes satisfying a certain gradient estimate, see (C2) below; here C β b (R d ) denotes the Hölder-Zygmund space of order β, cf. Section 2 for the definition. Moreover, there are numerous results on the regularity of functions which are harmonic with respect to a Lévy generator, see e.g. [7,8,16,18,30]. Let us mention that the regularity of solutions integrodifferential equations Af = g has been studied, more generally, for classes of Lévy-type operators, see e.g. [1,3,11,12,15,20,22]. Of course, Schauder estimates are also of interest for parabolic equations, we point the interested reader to the recent works [2,9,21] and the references therein. In this paper, we combine the global Schauder estimates from [14], cf. (1), with a truncation technique to derive local Hölder estimates for solutions to Af = g. We will assume that the Lévy process (X t ) t≥0 with Lévy triplet (b, Q, ν) satisfies the following conditions.
(C1) The characteristic exponent ψ satisfies the Hartman-Wintner growth condition (C2) There exist constants M > 0 and α > 0 such that the transition density p t , t > 0, satisfies the gradient estimate It follows from (C1),(C2) that the global Schauder estimate (1) holds, and we will use (C3) to localize these estimates. Before stating our results, let us give some remarks on (C1)-(C3).
Next we state our main results; see Section 2 for the definition of the notation used in the statements.
1.2. Theorem. Let (X t ) t≥0 be a Lévy process with infinitesimal generator (A, D(A)) satisfying (C1)-(C3), and denote by α ∈ (0, 2] the constant from (C2). Let f be a weak solution to the equation (i) If f ∈ L ∞ (R d ) and g ∈ L ∞ (U ), then f has a modificationf which is continuous on U and satisfies the interior Schauder estimate for every δ > 0. The constant C δ does not depend on f , g.
(ii) If f ∈ C κ b (R d ) and g ∈ C κ b (U ) for some κ > 0, then there exists for every δ > 0 a constant C δ > 0 (independent of f , g) such that Theorem 1.2 applies for a wide class of Lévy processes. It generalizes the interior Schauder estimates for stable processes obtained in [24], and for the particular case that there is no jump part, i.e. ν = 0, we recover the classical regularity estimates for second-order differential operators with constant coefficients, see Section 4 for details and further examples.
The weak solution f ∈ L ∞ (R d ) to Af = g is only determined up to a Lebesgue null set. The interior Schauder estimate (2) implies continuity off on U , and so (2) cannot hold for any representativef of f but only for a suitably chosen representative.
Consequently, the interior Schauder estimates in Theorem 1.3 hold for any Lévy process (X t ) t≥0 with generator (A, D(A)) satisfying the global Schauder estimate (4) and the balance condition (C3). This means that (the proof of) Theorem 1.3 actually gives a general procedure to localize Schauder estimates. (iii) If we interpret the constant α from (C2) as a measure for the volatility of (X t ) t≥0 , cf. Remark 1.1(iii), then Theorem 1.2 shows that a high volatility of the Lévy process (X t ) t≥0 results in a high regularity of f U . This is a natural result, and we believe the regularity estimates to be optimal for many Lévy processes. In some cases, certain properties of the Lévy process or the Lévy triplet may lead to an additional smoothing effect; for instance, Grzywny & Kwaśnicki [7, Theorem 1.7] studied the regularity of harmonic functions f (i.e. Af = 0) associated with unimodal Lévy processes and showed that the regularity of the density of the Lévy measure ν carries over to f ; this regularity of f is not related to the volatility of the process. (iv) In general, the assumption f ∈ C κ b (R d ) in Theorem 1.2(ii) cannot be relaxed to f ∈ C κ b (U ); for stable processes a counterexample can be found in [24, Proposition 6.1].
(v) If f is in the domain of the strong infinitesimal generator of (X t ) t≥0 , then [14,Theorem 1.1] gives f ∈ C α b (R d ), and so the assumption f ∈ C κ b (R d ) in Theorem 1.2(ii) is automatically satisfied for κ ≤ α, see also Corollary 1.5 below.
Our second main result gives interior Schauder estimates for pointwise solutions to the equation Af = g.
t exists for all x ∈ U and assume that for any compact set K ⊆ U . Denote by α ∈ (0, 2] the constant from (C2).
. As an immediate consequence, we obtain local Schauder estimates for functions in the domain in the strong infinitesimal generator. They extend in a natural way the global Schauder estimates from [14].
1.5. Corollary. Let (X t ) t≥0 be a Lévy process satisfying (C1)-(C3), and denote by α ∈ (0, 2] the constant from (C2). Let f be a function in the domain of the strong infinitesimal generator, i.e. f is a continuous function vanishing at infinity and the limit The proof of Corollary 1.5 shows that the interior Schauder estimates (i),(ii) actually hold for any Lévy process (X t ) t≥0 with generator (A, D(A)) satisfying (C3) and the global Schauder estimate This paper is organized as follows. In Section 2 we introduce basic definitions and notation. Our main results are proved in Section 3, and examples are presented in Section 4.

Definitions
We consider the d-dimensional Euclidean space R d with the canonical scalar product x ⋅ y ∶= ∑ d j=1 x j y j and the Borel σ-algebra B(R d ) generated by the open balls B(x, r). If f is a real-valued function, then supp f denotes its support, ∇f the gradient and ∇ 2 f the Hessian of f . For α ≥ 0 we set The smooth functions with compact support are denoted by C ∞ c (R d ). Superscripts k ∈ N are used to denote the order of differentiability, e.g. f ∈ C k b (R d ) means that f and its derivatives up to order k are bounded continuous functions. For U ⊆ R d we set

For every α ≥ 0 and every open set
is the distance of x from the complement of U , k ∈ N is the smallest natural number strictly larger than α and are iterated difference operators. For U = R d it is known that replacing k by an arbitrary number j strictly larger than α gives an equivalent norm, i.e.
cf. [ has independent and stationary increments and t ↦ X t (ω) is right-continuous with finite left-hand limits for almost all ω ∈ Ω. The Lévy-Khintchine formula shows that every Lévy process is uniquely determined in distribution by The characteristic exponent ψ has a Lévy-Khintchine representation ∞ (Lévy measure). Our standard reference for Lévy processes is the monograph [25] by Sato. By the independence and stationarity of the increments, every Lévy process is a time-homogeneous Markov process, i.e. P t f (x) ∶= Ef (x + X t ) defines a Markov semigroup on B b (R d ). We denote by (A, D(A)) the (weak) infinitesimal generator, and Restricted to C ∞ c (R d ), the infinitesimal generator is a pseudo-differential operator with symbol ψ, Given an open set U ⊆ R d and g ∈ L ∞ (U ), a function f is called a weak solution to Af = g in U if It is implicitly assumed that the integral on the left-hand side exists; a sufficient condition is

Proofs
In this section we present the proofs of our main results. Corollary 1.4 and Corollary 1.5 are consequences of Theorem 1.2, and therefore the main part is to establish Theorem 1.2. The idea is to combine the global Schauder estimates from [14] with a truncation technique to establish interior Schauder estimates. We start with the following auxiliary result.
The identity for A(f ⋅ g) follows by applying (7) for f ⋅ g and rearranging the terms.
Let us mention that the regularity assumptions in Lemma 3.1 can be relaxed. Roughly speaking, the identity holds whenever f, g ∈ D(A) are sufficiently smooth to make sense of Γ(f, g); e.g. if Q = 0 then f, g need to satisfy a certain Hölder condition, see [14,Theorem 4.3] and [17].
The following a priori estimate is the core of the proof of our first main result, Theorem 1.2.
Step 2: There exists a constant C 2 > 0 such that where , ε are the constants from Step 1 and Indeed: For z ∈ B(x, 7r x 8) and x ∈ B(0, 2) we have r z ≥ r x 8, and therefore the mapping K defined in (12) satisfies
The seminorms f U,α, which we introduced in (15) are closely related to seminorms which appear in the study of Schauder estimates for second order differential operators, cf. [6]; our definition is inspired by Hölder-Zygmund norms whereas the seminorms in [6] are based on "classical" Hölder norms.
In order to apply the a priori estimate from Proposition 3.2, we have to approximate the weak solution f by a sequence (f k ) k≥1 of twice differentiable functions; it is a natural idea to consider f k ∶= f * χ k for a suitable sequence of mollifiers (χ k ) k≥1 . To make this approximation work, we need to know that Af k is on U close to Af = g, and this is what the next lemma is about.

Using (19) and Fubini's theorem it follows that
If x ∈ U is such that B(x, r) ⊆ U , then supp χ(x − ⋅) ⊆ U and so it follows from the definition of the weak solution, cf. (8), that for any such x.
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2. Let f ∈ L ∞ (R d ) and g ∈ L ∞ (U ) be such that Af = g weakly in U . Pick for all x ∈ U δ and k ≥ 1. Choosing k ≫ 1 sufficiently large such that 1 k < δ 4 we find from (20) and It follows from the Arzelà-Ascoli theorem that there exists a continuous function f (δ) ∶ U δ → R such that a subsequence f kj converges pointwise to f (δ) on U δ . By (21), we have for all x ∈ U 2δ and h ≤ δ 3 where N ∈ {1, 2, 3} is the smallest integer larger than α. Hence, . On the other hand, f k → f in L 1 (dx) and so f = f (δ) Lebesgue almost everywhere on U δ . As is well defined. Clearly,f = f Lebesgue almost everywhere and the interior Schauder estimate commutes with the shift operator, i.e.
x, x 0 ∈ R d , cf. (7), it follows easily by induction that Hence, by (20), for all N ∈ N, x ∈ U δ , h < δ (2N ) and k ≫ 1 with 1 k < δ 2 . Let N ∈ N be the smallest number which is strictly larger than κ. Applying part (i) to x ↦ ∆ N h f k (x), we obtain from (22) that for every δ ∈ (0, 1) and some constant c 1 = c 1 (δ) > 0 not depending on f , g and k. Since for all h < δ (2N ). If we denote by M ∈ N the smallest number strictly larger than α, then we get for all x ∈ U 3δ and small h. The left-hand side converges to for an open set U ⊆ R d . Iff ∈ C b (U ) is such that f =f Lebesgue almost everywhere on U , then f =f on U ; in particular, f U is continuous.
Proof. The function and f = u Lebesgue almost everywhere on R d . Since the characteristic exponent ψ satisfies the Hartman-Wintner condition, the law of X t has a density p t with respect to Lebesgue measure for t > 0, and so If we can show that lim then it follows immediately from (23) that which proves the assertion. To prove (24), fix x ∈ U . As we find from the right-continuity of the sample paths and the monotone convergence theorem that Since u is continuous at x, the right-hand side tends to 0 as δ → 0, and this gives (24).

3.5.
Remark. The proof of Lemma 3.4 shows the following statement: If (X t ) t≥0 is a Lévy process, then lim holds for any continuity point x of f ∈ B b (R d ); this is a localized version of the continuity of the semigroup T t f (x) ∶= Ef (x + X t ) at t = 0.
Proof of Corollary 1.4. Let ϕ ∈ C ∞ c (U ). Because of the uniform boundedness assumption (5), it follows from the dominated convergence theorem that we can follow the reasoning from the first part, i.e. apply the differentiation lemma and Tonelli's theorem, to find that If there is no jump part, i.e. ν = 0, then the infinitesimal generator associated with (X t ) t≥0 is given by , and Corollary 4.2 yields the classical interior Schauder estimates for solutions to the equation Af = g associated with the second order differential operator A, see e.g. [6].
Our next result applies to a large class of jump Lévy processes, including stable Lévy processes. It is a direct consequence of the gradient estimates obtained in [27].
If the Lévy measure ν equals the right-hand side of (26), then the assumption (27) is trivially satisfied; this is, for instance, the case if (X t ) t≥0 is isotropic α-stable or relativistic α-stable. In particular, Corollary 4.3 generalizes [24, Theorem 1.1].
The next corollary gives a criterion for (C1)-(C3) in terms of the growth of the characteristic exponent of (X t ) t≥0 .