Schauder Estimates for Poisson Equations Associated with Non-local Feller Generators

We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation Af=g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Af=g$$\end{document} associated with the (extended) infinitesimal generator of a Feller process. The regularity of f is described in terms of Hölder–Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equations. Most prominently, we establish Schauder estimates for the Poisson equation associated with the fractional Laplacian of variable order. As a by-product, we obtain new regularity estimates for semigroups associated with stable-like processes.


Introduction
Let (X t ) t≥0 be an R d -valued Feller process with semigroup P t f (x) = E x f (X t ), x ∈ R d . In this paper, we study the regularity of functions in the abstract Hölder space B Franziska Kühn franziska.kuehn1@tu-dresden.de 1 Fachrichtung Mathematik, Institut für Mathematische Stochastik, TU Dresden, 01062 Dresden, Germany the so-called Favard space of order 1; cf. [9,14]. It is known that for any f ∈ F 1 the limit exists up to a set of potential zero (cf. [1]) and this gives rise to the extended infinitesimal generator A e which maps the Favard space F 1 into the space of bounded Borel measurable functions B b (R d ); cf. Sect. 2 for details. It is immediate from Dynkin's formula that A e extends the (strong) infinitesimal generator A of (X t ) t≥0 ; in particular, F 1 contains the domain D(A) of the infinitesimal generator. We are interested in the following questions: • What does the existence of limit (1) tell us about the regularity of f ∈ F 1 ? In particular: How smooth are functions in the domain of the infinitesimal generator of (X t ) t≥0 ? • If f ∈ F 1 is a solution to the equation A e f = g and g has a certain regularity, say g is Hölder continuous of order δ ∈ (0, 1), then what additional information do we get on the smoothness of f ?
Our aim is to describe the regularity of f in terms of Hölder spaces of variable order. More precisely, we are looking for a mapping κ : R d → (0, 2) such that where C κ(·) b (R d ) denotes the Hölder-Zygmund space of variable order equipped with the norm cf. Sect. 2 for details. If A e f = g ∈ C δ b (R d ) for some δ > 0, then it is natural to expect that f "inherits" some regularity from g, i. e.
for some constant = (δ) > 0. Moreover, we are interested in establishing Schauder estimates, i. e. estimates of the form Let us mention that the results, which we present in this paper, do not apply to Feller semigroups with a roughening effect (see e.g. [16] for examples of such semigroups); we study exclusively Feller semigroups with a smoothing effect (see below for details).
instance, if (X t ) t≥0 is an isotropic stable-like process with infinitesimal generator A = −(− ) α(•)/2 (cf. (3)), then an application of Proposition 3.1 shows where α 0 := inf x∈R d α(x), and this is much weaker than regularity (4) which we would expect. Our main result in Sect. 3 is a "localized" version of Proposition 3.1 which takes into account the local behaviour of the Feller process (X t ) t≥0 and which allows us to describe the local regularity of a function f ∈ F 1 (cf. Theorem 3.2 and Corollary 3.4). As an application, we obtain a regularity result for solutions to the Poisson equation A e f = g with g ∈ C δ b (R d ) (cf. Theorem 3.5). In the second part of the paper, Sect. 4, we illustrate the results from Sect. 3 with several examples. Applying the results to isotropic stable-like processes, we establish Schauder estimates for the Poisson equation −(− ) α(•)/2 f = g associated with the fractional Laplacian of variable order (cf. Theorem 4.1 and Corollary 4.3). Schauder estimates of this type seem to be a novelty in the literature. As a by-product of the proof, we obtain Hölder estimates for semigroups of isotropic stable-like processes which are of independent interest (see Sect. 6.1). Furthermore, we present Schauder estimates for random time changes of Lévy processes (Proposition 4.5) and solutions to Lévy-driven SDEs (Proposition 4.7) and discuss possible extensions.

Basic Definitions and Notation
We consider the Euclidean space R d with the 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 ) and closed balls B(x, r ). As usual, we set x ∧ y := min{x, y} and x ∨ y := max{x, y} for x, y ∈ R. If f is a realvalued function, then supp f denotes its support, ∇ f the gradient and ∇ 2 f the Hessian of f . For two stochastic processes (X t ) t≥0 and (Y t ) t≥0 we write (X t ) t≥0 t≥0 have the same finite-dimensional distributions.
where k ∈ N is the smallest number strictly larger than α ∞ and are the iterated difference operators. Moreover, we set Clearly, If α(x) = α is constant, then we write C α (U ) and C α b (U ) for the associated Hölder-Zygmund spaces. For U = R d and α / ∈ N, the Hölder-Zygmund space C α b (R d ) is the "classical" Hölder space C α b (R d ) equipped with the norm cf. [52,Section 2.7]. For α = 1, it is possible to show that C 1 b (R d ) is strictly larger than the space of bounded Lipschitz continuous functions (cf. [51, p. 148]), which is in turn strictly larger than C 1 b (R d ).
Feller processes: A Markov process (X t ) t≥0 is a Feller process if the associated transition semigroup P t f (x) := E x f (X t ) is a Feller semigroup (see e.g. [6,19] for details). Without loss of generality, we may assume that (X t ) t≥0 has right-continuous sample paths with finite left-hand limits. Following [14,II.5.(b)], we call the Favard space of order 1. The (strong) infinitesimal generator (A, D(A)) is defined by If D(A) is rich, in the sense that C ∞ c (R d ) ⊆ D(A), then a result by Courrège and von Waldenfels (see e.g. [6,Theorem 2.21]), shows that A| C ∞ c (R d ) is a pseudo-differential operator, wheref (ξ ) := (2π) −d R d e −i x·ξ f (x) dx is the Fourier transform of f and is a continuous negative definite symbol. If (7) holds, then we say that (X t ) t≥0 is a Feller process with symbol q. We assume from now on that q(x, 0) = 0. For each x ∈ R d , (b(x), Q(x), ν(x, dy)) is a Lévy triplet, i. e. b(x) ∈ R d , Q(x) ∈ R d×d is symmetric positive semidefinite and ν(x, ·) is a measure on R d \{0} satisfying y =0 min{1, |y| 2 } ν(x, dy) < ∞. |q(x, ξ)| < ∞.
If (X t ) t≥0 is a Feller process with symbol q, then (9) holds for an absolute constant c > 0; this maximal inequality goes back to Schilling [47] (see also [6,Theorem 5.1] or [22,Lemma 1.29]). If the symbol q(ξ ) = q(x, ξ) of a Feller process (L t ) t≥0 does not depend on x ∈ R d , then (L t ) t≥0 is a Lévy process. By [6,Theorem 2.6], this is equivalent to saying that (L t ) t≥0 has stationary and independent increments. It is natural to ask whether for a given mapping q of form (8), there is a Feller process (X t ) t≥0 with symbol q. In general, the answer is negative; see the monographs [6,19,22] for a survey on known existence results for Feller processes. In this article, we will frequently use an existence theorem from [22] which constructs Feller processes with symbol of the form q(x, ξ) = ψ α(x) (ξ ), where α : R d → I is a Hölder continuous mapping and ξ → ψ β (ξ ), β ∈ I , is a family of characteristic exponents of Lévy processes. For instance, it can be applied to the family ψ β (ξ ) = |ξ | β , β ∈ I = (0, 2], to prove the existence of isotropic stable-like processes, i.e. Feller processes with symbol q(x, ξ) = |ξ | α(x) , where α : is Hölder continuous and inf x∈R d α(x) > 0 (cf. [22,Theorem 5.2]). Later on, we will use that any Feller process (X t ) t≥0 with infinitesimal generator (A, D(A)) solves the (A, D(A))-martingale problem, i. e.
is a P x -martingale for any x ∈ R d and f ∈ D(A). Our standard reference for Feller processes are the monographs [6,19]; for further information on martingale problems, we refer the reader to [15,18].
In the remaining part of this section, we define the extended infinitesimal generator and state some results which we will need later on. Following [44], we define the extended (infinitesimal) generator A e in terms of the λ-potential operator R λ , that is, f ∈ D(A e ) and g = A e f if and only if The mapping g = A e f is defined up to a set of potential zero, i.e. up to a set B ∈ B(R d ) We will often choose a representative with a certain property; for instance if we write " A e f is continuous", this means that there exists a continuous function g such that (i),(ii) hold. In abuse of notation, we set A e f ∞ := inf{c > 0; |A e f | ≤ c up to a set of potential zero}.
Clearly, the extended infinitesimal generator (A e , D(A e )) extends the (strong) infinitesimal generator (A, D(A)). The following result is essentially due to Airault and Föllmer [1] and shows the connection to the Favard space of order 1 (cf. (6)).

Theorem 2.1
Let (X t ) t≥0 be a Feller process with semigroup (P t ) t≥0 and extended generator (A e , D(A e )). The associated Favard space F 1 of order 1 satisfies and, moreover, Dynkin's formula holds for any x ∈ R d and any stopping time τ such that E x τ < ∞.
The next corollary shows how the Favard space can be defined in terms of the stopped process X t∧τ x r . Since we will frequently use stopping techniques, it plays an important role in our proofs .

Corollary 2.2
Let (X t ) t≥0 be a Feller process with semigroup (P t ) t≥0 , extended generator (A e , D(A e )) and symbol q. Denote by the exit time of (X t ) t≥0 from the closed ball B(x, r ). If q has bounded coefficients, then the following statements are equivalent for any f ∈ B b (R d ): (ii) There exists r > 0 such that

If one (hence both) of the conditions is satisfied, then
up to a set of potential zero for any r > 0. In particular, A e f ∞ ≤ K r ( f ) for r > 0.
For the proof of Theorem 2.1 and Corollary 2.2 and some further remarks, we refer to Appendix A.

Main Results
Let (X t ) t≥0 be a Feller process with semigroup (P t ) t≥0 . Throughout this section, is the Favard space of order 1 associated with (X t ) t≥0 . By Theorem 2.1, we have where A e denotes the extended infinitesimal generator. The results which we present in this section will be proved in Sect. 5. Our first result, Proposition 3.1, shows how regularity estimates for the semigroup (P t ) t≥0 can be used to obtain Schauder estimates of the form Since the domain D(A) of the (strong) infinitesimal generator of (X t ) t≥0 is contained in F 1 , Proposition 3.1 gives, in particular, . Proposition 3.1 is a useful tool, but it does not, in general, give optimal regularity results. Since Feller processes are inhomogeneous in space, the regularity of f ∈ F 1 will, in general, depend on the space variable x, e.g.
and therefore it is much more natural to use Hölder-Zygmund spaces of variable order to describe the regularity; this is also indicated by the results obtained in [30]. Our second result, Theorem 3.2, shows how Hölder estimates for Feller semigroups can be used to establish local Hölder estimates (14). Before stating the result, let us explain the idea. Let (X t ) t≥0 be a Feller process with symbol q and Favard space F X 1 , and fix x ∈ R d . Let (Y t ) t≥0 be another Feller process which has the same behaviour as (X t ) t≥0 in a neighbourhood of x, in the sense that its symbol p satisfies for some δ > 0. The aim is to choose (Y t ) t≥0 in such a way that its semigroup (T t ) t≥0 satisfies a "good" regularity estimate here "good" means that κ is large. Because of (15), it is intuitively clear that for z close to x and "small" t. (16) (We will use stopping to specify what "small" means; see Lemma 5.2.) If χ is a truncation function such that 1 B(x,ε) ≤ χ ≤ 1 B(x,2ε) for small ε > 0, then it is, because of (16), natural to expect that for any f ∈ F X 1 the truncated mapping g : , and g = f in a neighbourhood of x, this entails that f (·) is κ-Hölder continuous in a neighbourhood of x. Since κ = κ(x) depends on the point x ∈ R d , which we fixed at the beginning, this localizing procedure allows us to obtain local Hölder estimates (14) for f . Theorem 3.2 Let (X t ) t≥0 be a Feller process with extended generator (A e , D(A e )) and Favard space F X 1 such that for a continuous negative definite symbol q (cf. (7)). Let x ∈ R d and δ ∈ (0, 1) be such that there exists a Feller process (Y (x) t ) t≥0 with the following properties: p (x) has bounded coefficients, and then 4δ)) is an a priori estimate on the regularity of f . If the semigroup (P t ) t≥0 of (X t ) t≥0 satisfies a regularity estimate of form (13), then such an a priori estimate can be obtained from Proposition 3.1. Note that, by (18), there is a trade-off between the required a priori regularity of f and the roughness of the measures ν (x) (z, dy), z ∈ B(x, 4δ). If the measures ν (x) (z, dy) only have a weak singularity at y = 0, in the sense that sup |z−x|≤4δ |y|≤1 |y| ν (x) (z, dy) < ∞, then we can choose (x) = 0, i. e. it suffices that f is continuous. In contrast, if (at least) one of the measures has a strong singularity at y = 0, then we need a higher regularity of f (in a neighbourhood of x).
(ii) It is not very restrictive to assume that (Y (x) t ) t≥0 has bounded coefficients since (Y (x) t ) t≥0 is only supposed to mimic the behaviour of (X t ) t≥0 in a neighbourhood of x (cf. (17)). We are, essentially, free to choose the behaviour of the process far away from x. In dimension d = 1, it is, for instance, a natural idea to consider note that p (x) has bounded coefficients even if q has unbounded coefficients.
If additionally then C b (U ) and there exists a constant C > 0 such that in particular, the domain D(A) of the (strong) infinitesimal generator A satisfies C (21) holds for any f ∈ C In many examples (see e.g. Sect. 4), it is possible to choose the mapping in such a way that b (U )) and the Schauder estimate (21) holds for any function f ∈ F 1 . In our applications, we will even have f , and therefore (21) becomes In Sect. 4, we will apply Corollary 3.4 to isotropic stable-like processes, i. e. Feller processes with symbol of the form q(x, ξ) = |ξ | α(x) . The study of the domain D(A) of the infinitesimal generator A is particularly interesting since A is an operator of variable order. We will show that any function f ∈ D(A) satisfies the Hölder estimate of variable order for ε > 0 (cf. Theorem 4.1) for the precise statement. Our final result in this section is concerned with Schauder estimates for solutions to the equation A e f = g for Hölder continuous mappings g. To establish such Schauder estimates, we need additional assumptions on the regularity of the symbol and improved regularity estimates for the semigroup of the "localizing" Feller process Theorem 3.5 Let (X t ) t≥0 be a Feller process with extended generator (A e , D(A e )) and Favard space F X 1 such that for a continuous negative definite symbol q. Assume that there exists δ ∈ (0, 1) such that for any x ∈ R d there exists a Feller process (Y 1 − e iy·ξ + iy · ξ 1 (0,1) (|y|) ν (x) (z, dy), (22) satisfying (C1)-(C3) in Theorem 3.2. Assume additionally that the following conditions hold for absolute constants C 1 , C 2 > 0: (S1) For any x, z ∈ R d , there exists α (x) (z) ∈ (0, 2) such that and the following statement holds for every r ∈ (0, 1) and every x, z ∈ R d : If u : R d → R is a measurable mapping such that for some c u > 0, then there exist C 3,r > 0 and H r > 0 (not depending on u, x, z) such that for all |h| ≤ H r . (25) for any x ∈ R d and λ ∈ [0, ]; here M(x), κ(x) and β(x) denote the constants from (C3). (S4) The mapping κ : R d → (0, ∞) is uniformly continuous and bounded away from zero, i. e.
Let : R d → [0, 2] be a uniformly continuous function satisfying and Moreover, the Schauder estimate holds for any ε ∈ (0, κ 0 ) and some finite constant C ε which does not depend on f , g.

Remark 3.6 (i)
In our examples in Sect. 4, we will be able to choose in such a way that α (x) (z) − (z) is arbitrarily small for x ∈ R d and z ∈ B(x, 4δ), and therefore the constant σ in (26) will be close to 1. Noting that θ ≤ 1, it follows that we can discard σ in (27) and (28) i. e. we get We would like to point out that it is, in general, not possible to improve this estimate and to obtain that f ∈ C κ(·)+λ−ε b (R d ), ε ∈ (0, κ 0 ). To see this, consider a Feller process then A e f = b f = 1 is smooth. However, the regularity of f clearly depends on the regularity of b, which means that f is less regular than A e f . (ii) It suffices to check (25) for λ = ; for λ ∈ (0, ), the inequality then follows from the interpolation theorem (see e.g. [ where 0 < r < R < 1, x, z ∈ R d and γ ∈ (0, 3). (v) In (S2), we assume that θ ≤ 1; this assumption can be relaxed. To this end, we have to replace in (23) and (24) the differences of first order, by iterated differences of higher order (cf. (5)). This makes the proof more technical, but the idea of the proof stays the same.
The proofs of the results stated in this section will be presented in Sect. 5.

Applications
In this section, we apply the results from the previous section to various classes of Feller processes. We will study processes of variable order (Theorem 4.1 and Corollary 4.3), random time changes of Lévy processes (Proposition 4.5) and solutions to Lévy-driven SDEs (Proposition 4.7). Our aim is to illustrate the range of applications, and therefore, we do not strive for the greatest generality of the examples; we will, however, point the reader to possible extensions of the results which we present. We remind the reader of the notation The first part of this section is devoted to isotropic stable-like processes, i. e. Feller processes (X t ) t≥0 with symbol of the form q(x, ξ) = |ξ | α(x) ; they appeared first in papers by Bass [3]. A sufficient condition for the existence of such a Feller process is that α : R d → (0, 2] is Hölder continuous and bounded from below (cf. [22,Theorem 5 , which means that A is a fractional Laplacian of variable order, i.e. A = −(− ) α(·)/2 . This makes A-and hence the stable-like process (X t ) t≥0 -an interesting object of study. To our knowledge, there are no Schauder estimates for the Poisson equation A f = g available in the existing literature. Using the results from the previous section, we are able to derive Schauder estimates for functions f in the Favard space F 1 (and, hence in particular, for f ∈ D(A)) (cf. Theorem 4.1), as well as Schauder estimates for solutions to A f = g (cf. Corollary 4.3).

Theorem 4.1 Let (X t ) t≥0 be a Feller process with symbol q(x, ξ) = |ξ | α(x) for a Hölder continuous function
The associated Favard space F 1 of order 1 (cf. (6)) satisfies where A e denotes the extended generator of (X t ) t≥0 . In particular, (30) holds for any

Remark 4.2 (i) Theorem 4.1 allows us to obtain information on the regularity of the transition density
for a finite constant C = C(ε, α, T ). Some related results on the regularity of the transition density were recently obtained in [10].
and that is the so-called carré du champ operator (cf. [8,12]) and ν(x, dy) = c d,α(x) |y| −d−α(x) dy is the family of Lévy measures associated with the symbol |ξ | α(x) via the Lévy-Khintchine representation. (iii) Theorem 4.1 can be generalized to a larger class of "stable-like" Feller processes, e.g. relativistic stable-like processes and tempered stable-like processes (cf. [22,Section 5.1] or [25,Example 4.7]) for the existence of such processes. In order to apply the results from Sect. 3, we need two key ingredients: general existence results-which ensure the existence of a "nice" Feller process (Y t ) t≥0 whose symbol is "truncated" in a suitable way (cf. Step 1 in the proof of Theorem 4.1)and certain heat kernel estimates needed to establish Hölder estimates for the semigroup; in [22], both ingredients were established for a wide class of stablelike processes.
As a corollary of Theorem 4.1 and Theorem 3.5, we will establish the following Schauder estimates for the elliptic equation A f = g associated with the infinitesimal generator A of the isotropic stable-like process.

Corollary 4.3 Let (X t ) t≥0 be a Feller process with infinitesimal generator (A, D(A))
and symbol q( It is possible to extend Corollary 4.3 to a larger class of "stable-like" processes (see also Remark 4.2(ii)). Let us give some remarks on the assumption that
for γ > 1, we can apply Corollary 4.3 with γ = 1, but this gives a weaker regularity estimate for f than we would expect; this is because we lose some information on the regularity of α. The reason why we have to restrict ourselves to γ ∈ (0, 1) is that two tools which we need for the proof (Theorem 3.5 and Proposition 6.2) are only available for γ ∈ (0, 1). However, we believe that both results are valid for γ > 0, and hence that that the assumption γ ∈ (0, 1) in Corollary 4.3 can be dropped.
Since the proofs of Theorem 4.1 and Corollary 4.3 are quite technical, we defer them to Sect. 6. The idea is to apply Theorem 3.2 and Theorem 3.5. As "localizing" process (Y (x) t ) t≥0 , we will use a Feller process with symbol for fixed x ∈ R d and small ε > 0. In order to apply the results from the previous section, we need suitable regularity estimates for the semigroup (P t ) t≥0 associated with an isotropic stable-like process (Y t ) t≥0 . We will study the regularity of x → P t u(x) using the parametrix construction of (the transition density of) (Y t ) t≥0 in [22]; the results are of independent interest, we refer the reader to Sect. 6.1.
Next we study Feller processes with symbols of the form q(x, ξ) = m(x)|ξ | α . They can be constructed as random time changes of isotropic α-stable Lévy processes (see e.g. [6, Section 4.1] and [26] for further details). This class of Feller processes includes, in particular, solutions to SDEs driven by a one-dimensional isotropic α-stable Lévy process (L t ) t≥0 , α ∈ (0, 2]; for instance if σ > 0 is continuous and at most of linear growth, then there exists a unique weak solution to the SDE, and the solution is a Feller process with symbol q(x, ξ) = |σ (x)| α |ξ | α (cf. [23,Example 5.4]).

Proposition 4.5
Let (X t ) t≥0 be a Feller process with symbol q(x, ξ) = m(x)|ξ | α for α ∈ (0, 2) and a Hölder continuous function m : (i) The infinitesimal generator (A, D(A)) and the Favard space F 1 of order 1 satisfy For any κ ∈ (0, α) there exists a finite constant C 1 > 0 such that here A e denotes the extended infinitesimal generator.
Proof It follows from [22,Theorem 3.3] that there exists a unique Feller process As in the proof of Proposition 6.1 and Proposition 6.2, it follows from the parametrix construction of the transition density p in [22] that the semigroup (P t ) t≥0 satisfies for any κ ∈ (0, α) and λ ∈ [0, θ]; for the particular case α ∈ (0, 1] the first inequality follows from [37]. Applying Proposition 3.1, we get (34); in particular [30,Example 5.4]. The Schauder estimate in (ii) follows Theorem 3.5 applied with Y (x) t := X t for all x ∈ R d (using the regularity estimates for (P t ) t≥0 from above).
continuous negative definite functions ψ, e.g. the characteristic exponent of a relativistic stable or tempered stable Lévy process (cf. [22,  Combining the results with Proposition 3.1, we can obtain Schauder estimates for functions in the domain of the infinitesimal generator of (X t ) t≥0 . Let us mention that for ν(x, y) = m(x, y)|y| −d−α dy Schauder estimates were studied in [4].
We close this section with some results on solutions to Lévy-driven SDEs.
Proposition 4.7 Let (L t ) t≥0 be a 1-dimensional isotropic α-stable Lévy process for some α ∈ (0, 2). Consider the SDE for a bounded β-Hölder continuous mapping b : R → R and a bounded Lipschitz continuous mapping σ : then there exists a unique weak solution (X t ) t≥0 to (35), and it gives rise to a Feller process with infinitesimal generator (A, D(A)). The associated Favard space F 1 of order 1 satisfies and there exists for any k ∈ N a finite constant C > 0 such that where A e denotes the extended generator. In particular, (37) holds for any f ∈ D(A) with A e f = A f .
Before giving some remarks on possible extensions of Proposition 4.7, let us mention that sufficient conditions for a function f to be in the domain D(A) were studied in [30]. For instance, if the SDE has no drift part, i. e. b = 0, then it follows from Proposition 4.7 and [30, Example 5.6] that and see (33) for the definition of C α+ ∞ (R). Intuitively, one would expect that (38) holds for α ∈ (0, 2). If we knew that the semigroup (P t ) t≥0 of the solution to (35) satisfies for some constant c = c(κ) > 0, this would immediately follow from Proposition 3.1. We could not find (40) in the literature, but we strongly believe that the parametrix construction of the transition density in [33] can be used to establish such an estimate; this is also indicated by the proof of Theorem 4.1 (see in particular the proof of Proposition 6.1). In fact, we believe that the parametrix construction in [33] entails estimates of the form for κ ∈ (0, α), λ > 0 (recall that β is the Hölder exponent of the drift b), which would then allow us to establish Schauder estimates to the equation A f = g for g ∈ C λ b (R) using Theorem 3.5.

Remark 4.8 (Possible extensions of Proposition 4.7)
(i) The gradient estimates in [36] were obtained under more general conditions, and (the proof of) Proposition 4.7 extends naturally to this more general framework. Firstly, Proposition 4.7 can be extended to higher dimensions; the assumption σ L > 0 in (36) is then replaced by the assumption that σ is uniformly nondegenerate in the sense that for some absolute constant M > 0 which does not depend on ξ ∈ R d . Secondly, Proposition 4.7 holds for a larger class of driving Lévy processes; it suffices to assume that the Lévy measure ν satisfies ν(dz) ≥ c|z| −d−α 1 {|z|≤η} for some c, η > 0 and that SDE (35) has a unique weak solution. Under the stronger balance condition β + α/2 > 1 this is automatically satisfied for a large class of Lévy processes, e.g. if (L t ) t≥0 is an relativistic stable or a tempered stable Lévy process (cf. [11]). (ii) Recently, Kulczycki et al. [32] established Hölder estimates for the semigroup associated with the solution to the SDE driven by a d-dimensional Lévy process (L t ) t≥0 , d ≥ 2, whose components are independent α-stable Lévy processes, α ∈ (0, 1), under the assumption that the coefficient σ : R d → R d×d is bounded, Lipschitz continuous and satisfies inf x det(σ (x)) > 0. Combining the estimates with Proposition 3.1, we find that the assertion of Proposition 4.7 remains valid in this framework, i.e. the Favard space F 1 associated with the unique solution (iii) Using coupling methods, Luo and Wang [41, Section 5.1] and Liang et. al [38] recently studied the regularity of semigroups associated with solutions to SDEs with additive noise for a large class of driving Lévy processes (L t ) t≥0 . The results from [38,41] and Sect. 3 can be used to obtain Schauder estimates for functions in the domain of the infinitesimal generator of (X t ) t≥0 .

Proofs of Results from Sect. 3
For the proof of Proposition 3.1, we use the following lemma which shows how Hölder estimates for a Feller semigroup translate to regularity properties of the λ-potential operator be a Feller process with semigroup (P t ) t≥0 and λ-potential Iterating the procedure, it follows easily that (41) holds.
then (41) gives that the iterated difference operator K h (cf. (5)) satisfies for any x ∈ R d and |h| ≤ 1. Since, by the linearity of the integral, On the other hand, we have R λ u ∞ ≤ λ −1 u ∞ , and therefore we get for all λ > m which proves the assertion.
We are now ready to prove Proposition 3.1.
Proof of Proposition 3.1 By Lemma 5.1(i), (41) holds with m := log(2)β/T for any For the proof of Theorem 3.2, we need two auxiliary results.

Lemma 5.2
Let (X t ) t≥0 and (Y t ) t≥0 be Feller processes with infinitesimal generator (A, D(A)) and (L, D(L)), respectively, such that (7)) and assume that the If x ∈ U and r > 0 are such that B(x, r ) ⊆ U , then for the stopping times the random variables X t∧τ X and Y t∧τ Y are equal in distribution with respect to P x for any t ≥ 0. 1

Proof
Set It follows from the well-posedness of the (A, C ∞ c (R d ))-martingale problem that the local martingale problem for U is well-posed (cf. [15,Theorem 4.6.1] or [18] for details). On the other hand, Dynkin's formula shows that both (X t∧σ X ) t≥0 and (Y t∧σ Y ) t≥0 are solutions to the local martingale problem, and therefore (X t∧σ X ) t≥0 equals in distribution (Y t∧σ Y ) t≥0 with respect to P x for any x ∈ U . If x ∈ U and r > 0 are such that B(x, r ) ⊆ U , then it follows from the definition of τ X and τ Y that τ X ≤ σ X and τ Y ≤ σ Y U ; in particular, Approximating τ X and τ Y from above by sequences of discrete-valued stopping times,

Lemma 5.3 Let (Y t ) t≥0 be a Feller process with infinitesimal generator (A, D(A))
and symbol min{1, |y| α } ν(z, dy) < ∞, 1 Here and below we are a bit sloppy in our notation. The Feller processes (X t ) t≥0 and (Y t ) t≥0 each come with a family of probability measures, i.e. their semigroups are of the form f (X t ) P x (dy) and f (Y t )P x (dy), respectively, for families of probability measures (P x ) x∈R d and (P x ) x∈R d . To keep the notation simple, we will not distinguish these two families. Formally written, the assertion of Lemma 3.5 reads P x (X t∧τ X ∈ ·) =P x (Y t∧τ Y ∈ ·). then there exists an absolute constant c > 0 such that the stopped process (Y t∧τ U ) t≥0 , where for all x ∈ U , t ≥ 0.
Note that (43) implies, by Jensen's inequality, that the moment estimate holds for any β ∈ [0, α], x ∈ U and t ≥ 0. If (Y t ) t≥0 has a compensated drift, in the sense that b(z) = |y|<1 y ν(z, dy) for all z ∈ U , then Lemma 5.3 holds also for α ∈ (0, 1]. Let us mention that estimates for fractional moments of Feller processes were studied in [21]; it is, however, not immediate how Lemma 5.3 can be derived from the results in [21].

Proof of Lemma 5.3 Let
, an application of Dynkin's formula shows that for all t ≥ 0. Since α > 1, there exists an absolute constant C > 0 such that for any z ∈ U . Hence, for x ∈ U . Applying Fatou's lemma twice, we conclude that We are now ready to prove Theorem 3.2.

Proof of Theorem 3.2 Since
x ∈ R d is fixed throughout this proof, we will omit the superscript x in the notation which we used in the statement of Theorem 3.2, e.g. we will write (Y t ) t≥0 instead of (Y (x) t ) t≥0 , L instead of L (x) etc. Denote by (L e , D(L e )) the extended generator of (Y t ) t≥0 , and fix a truncation To prove the assertion, it suffices by (C3) and Proposition 3.1 to show that v := f · χ is in D(L e ) and for a suitable constant C > 0. The first-and main-step is to estimate for the stopping time We consider separately the cases z ∈ B(x, 3δ) and z ∈ R d \B(x, 3δ). For fixed Hence, Applying the maximal inequality (9) for Feller processes, we find that there exists an absolute constant c 1 > 0 such that 3δ); the right-hand side is finite since p has, by assumption, bounded coefficients.
We estimate the terms separately. By (17) and (C2), it follows from Lemma 5.2 that where τ z δ (X ) is the exit time of (X t ) t≥0 from B(z, δ). As 0 ≤ χ ≤ 1 we thus find Since f ∈ F X 1 , an application of Dynkin's formula (11) shows that We turn to I 2 .
As χ ∈ C ∞ c (R d ) ⊆ D(L) we find from the (classical) Dynkin formula that A straightforward application of Taylor's formula shows that Since 0 ≤ (x) ≤ 1 and χ is chosen such that χ C 2 b (R d ) ≤ 10δ −2 , we thus get ν(z, dy) .
It remains to estimate I 3 . Because of the assumptions on the Hölder regularity of f on B(x, 4δ), we have It follows from Lemma 5.3 that there exists an absolute constant c 2 > 0 such that I 3 is bounded above by Combining the estimates and applying Corollary 2.2, we find that v = χ · f ∈ D(L e ) and  1} ν(z, dy).
This finishes the proof of (45). The continuous dependence of the constant C > 0 in (19) on the parameters β(x) follows from the fact that each of the constants in this proof depends continuously on these parameters; see also Lemma 5.1.
The remaining part of this section is devoted to the proof of Theorem 3.5. We need the following auxiliary result.
The main ingredient for the proof of Lemma 5.4 is [30, Theorem 4.2], which states that the family of measures p t (x, B) := t −1 P x (Y t −x ∈ B), t > 0, converges vaguely to ν(x, dy) as t → 0.

Proof of Lemma 5.4 By the Portmanteau theorem, it suffices to show that lim sup
χ n (y) ν(x, dy) for all n ∈ N. On the other hand, an application of Dynkin's formula yields that Since (Y t ) t≥0 has right-continuous sample paths, we have P x (τ x r ≤ t) → 0 as t → 0, and therefore we obtain that χ n (y) ν(x, dy).

Proof of Theorem 3.5 For fixed
Step 1-3 of this proof, we will often omit the superscript x in our notation, i.e. we will write Step 1 Show that v := χ · f is in the domain D(L e ) of the extended generator of (Y t ) t≥0 and determine L e (v).
First of all, we note that (X t ) t≥0 , (Y t ) t≥0 and f satisfy the assumptions of Theorem 3.2. Since we have seen in the proof of Theorem 3.2 that v = χ · f is in the Favard space F Y 1 of order 1 associated with (Y t ) t≥0 , it follows that v ∈ D(L e ) and L e (v) ∞ < ∞. Applying Corollary 2.2, we find that (up to a set of potential zero), where On the other hand, the proof of Theorem 3.2 shows that where here τ z δ (X ) denotes the exit time of (X t ) t≥0 from B(z, δ). Since χ ∈ C ∞ c (R d ) is in the domain of the (strong) infinitesimal generator L of (Y t ) t≥0 and f is the Favard space F X 1 associated with (X t ) t≥0 , another application of Corollary 2.2 shows that for all z ∈ R d , where ν(z, dy) = ν (x) (z, dy) denotes the family of Lévy measures associated with (22)). Once we have shown this, it follows that To prove (48), we fix a function ϕ ∈ C ∞ c (R d ) such that 1 B(0,1) ≤ ϕ ≤ 1 B(0,2) and set ϕ ε (y) := ϕ(ε −1 y) for ε > 0, y ∈ R d . Since y → (1 − ϕ ε (y)) is zero in a neighbourhood of 0, we find from Lemma 5.4 that If z ∈ R d \B(x, 3δ), then χ = 0 on B(z, δ), and therefore the integrand on the right hand side equals zero for |y| < δ. By dominated convergence, the right-hand side converges to ( f , χ)(z), defined in (48), as ε → 0. For z ∈ B(x, 3δ) we note that (26); it now follows from (S1) and dominated convergence that the right-hand side converges to ( f , χ)(z) as ε → 0. To prove (48), it remains to show that By (26) and (S1), there exists some constant γ > 0 such that Indeed: On { ≥ 1} this inequality holds since α is bounded away from 2 (cf. (S1)), and on { < 1} this is a direct consequence of (26). Now fix some z ∈ B(x, 3δ). As supp ϕ ε ⊆ B(0, 2ε), it follows from f ∈ C with γ from (50) and some constant c 1 > 0 (not depending on f , x, z). An application of Lemma 5.3 now yields which is finite because of (S1) and (S5). Hence, lim sup If z ∈ R d \B(x, 3δ), then it follows from χ | B(z,δ) = 0 and supp ϕ ⊆ B(0, 2ε) that Applying the maximal inequality (9) for Feller processes, we conclude that Step 2 If : R d → [0, 2] is a uniformly continuous function satisfying (26) and for any λ ∈ [0, ] where χ = χ (x) is the truncation function chosen at the beginning of the proof; see (S2), (S3) and (26) for the definition of θ , and σ .
As θ ≤ 1 we have 0 ∧ λ ∧ θ ∧ σ ≤ 1, and therefore it suffices to estimate Estimate of I 1 = f Lχ : First we estimate the Hölder norm of Lχ . As χ ∈ C ∞ c (R d ) a straight-forward application of Taylor's formula shows that for all z, h ∈ R d . To estimate the first two terms on the right-hand side we use the Hölder continuity of b (cf. (S2)) and the fact that χ ∈ C 2 b (R d ). For the third term, we use cf. [4, Theorem 5.1] for details, and noting that we can estimate the fourth term for small h by applying (S2). Hence, for small h > 0. Hence, Estimate of I 2 = χ A e f : By assumption, it follows from the definition of ( f , χ) (cf. (48)) that 3δ), then y χ(z) = 0 for all |y| ≤ δ, and so for all z ∈ R d \B(x, 3δ). Combining both estimates and using (26), (S1) and (S5), we get for some constant c 2 > 0 not depending on x, z and f . To study the regularity of ( f , χ) we consider separately the cases ∞ ≤ 1 and ∞ > 1. We start with the case ∞ ≤ 1; see the end of this step for the other case. To estimate h ( f , χ), we note that where We estimate the terms separately, and start with J 1 . Fix ε ∈ (0, min{ 0 , σ }/2) (cf. (26) for the definition of σ ). Since is uniformly continuous, there exists r ∈ (0, 1) such that For |h| ≤ r and |y| ≤ r it then follows from f ∈ C (Here we use ∞ ≤ 1; otherwise we would need to replace (z) by (z) ∧ 1 etc.) On the other hand, we also have for all y ∈ R d . Combining both estimates yields for |h| ≤ r . It is now not difficult to see from (S1) and (S5) that there exists a constant c 3 > 0 (not depending on x, z, f ) such that By the very definition of σ (cf. (26)), this implies that sup z∈B(x,3δ) If z ∈ R d \B(x, 3δ), then y χ(z + h) = 0 for |h| ≤ δ/2 and |y| ≤ δ/2. Using (52), we get Invoking once more (S1) and (S5), we obtain that sup z∈R d \B(x,3δ) for some constant c 4 not depending on x, z and f . In summary, we have shown that To estimate J 2 , consider again separately the cases z ∈ B(x, 3δ) and z ∈ R d \B(x, 3δ). If z ∈ R d \B(x, 3δ), then y χ(z + h) = 0 = y χ(z) for all |y| ≤ δ/2 and |h| ≤ δ/2. Since we also have we find that for |h| ≤ δ/2. Because of (S1) and (S5), this gives the existence of a constant c 6 > 0 (not depending on f , x and z) such that For z ∈ B(x, 3δ), we combine with (53) to get min{|y| (z) , 1} min{|y|, |h|} ν(z, dy), which implies, by (S1), (S5) and (26), that sup z∈B(x,3δ) We conclude that It remains to estimate J 3 . By the uniform continuity of there exists r ∈ (0, 1) such that | h (z)| ≤ σ/2 for all |h| ≤ r . Since f ∈ C and thus, by (26) and our choice of r ∈ (0, 1), for all |z − x| ≤ 3δ and |h| ≤ r . On the other hand, if z ∈ R d \B(x, 3δ), then χ = 0 on B(z, δ) and so Consequently, there exists a constant c 9 = c 9 (δ, r ) > 0 such that for all z ∈ R d , y ∈ R d and |h| ≤ min{r , δ}/2. Applying (S2), we thus find Combining the above estimates, we conclude that provided that ∞ ≤ 1. In the other case, i. e. if takes values strictly larger than one, then we need to consider second differences 2 h ( f , χ)(z) in order to capture the full information on the regularity of f . The calculations are very similar to the above ones but quite lengthy (it is necessary to consider nine terms separately) and so we do not present the details here.

Conclusion of
Step 2 For any small ε > 0 there exists a finite constant K 1,ε > 0 such that The constant K 1,ε does not depend on x, z and f . Step for for μ sufficiently large and some constant K = K (μ) > 0. This is a direct consequence of (S3) and Lemma 5.

Conclusion of the proof
Without loss of generality, we may assume that 0 := inf x (x) > 0. Indeed: It follows from Corollary 3.4 that f ∈ C κ(·)−ε b (R d ) for ε := κ 0 /2 := inf x κ(x)/2 > 0, and therefore we may replace bỹ which is clearly bounded away from zero and satisfies the assumptions of Theorem 3.5.

Proof of Schauder Estimates for Isotropic Stable-Like Processes
In this section we present the proof of the Schauder estimates for isotropic stable-like processes which we stated in Theorem 4.1 and Corollary 4.3. Throughout this section, (X t ) t≥0 is an isotropic stable-like process, i. e. a Feller process with symbol of the form q(x, ξ) = |ξ | α(x) , x, ξ ∈ R d , for a mapping α : R d → (0, 2]. We remind the reader that such a Feller process exists if α is Hölder continuous and bounded away from zero. We will apply the results from Sect. 3 to establish the Schauder estimates. To this end, we need regularity estimates for the semigroup (P t ) t≥0 associated with (X t ) t≥0 . The results, which we obtain, are of independent interest and we present them in Sect. 6.1. Once we have established another auxiliary statement in Sect. 6.2, we will present the proof of Theorem 4.1 and Corollary 4.3 in Sect. 6.3.

Regularity Estimates for the Semigroup of Stable-Like Processes
Let (P t ) t≥0 be the semigroup of an isotropic stable-like process (X t ) t≥0 with symbol q(x, ξ) = |ξ | α(x) . In this subsection, we study the regularity of the mapping x → P t u(x). We will see that there are several parameters which influence the regularity of P t u: the larger these quantities are, the higher the regularity of P t u. The regularity estimates we present rely on the parametrix construction of (the transition density of) (X t ) t≥0 in [22]. We mention that there are other approaches to obtain regularity estimates for the semigroup. Using coupling methods, Luo and Wang [40] showed that for any κ ∈ (0, α L ) there exists c > 0 such that For α L > 1, this estimate is not good enough for our purpose; we need a higher regularity of P t u.
x, ξ ∈ R d , for a mapping α : R d → (0, 2) bounded away from zero, i.e. α L := inf x∈R d α(x) > 0, and γ -Hölder continuous for γ ∈ (0, 1). For any T > 0 and κ ∈ (0, α L ) there exists a constant C > 0 such that the semigroup (P t ) t≥0 satisfies In particular, (P t ) t≥0 has the strong Feller property. The constant C > 0 depends continuously on . For the proof of Proposition 6.1, we use a representation for the transition density p which was obtained in [22] using a parametrix construction; see also [25]. For ∈ (0, 2), denote by p (t, x) the transition density of an isotropic -stable Lévy process and set The transition density p of (X t ) t≥0 has the representation where is the time-space convolution and is a suitable function satisfying for some constant λ > 0 and C 1 = C 1 (T ) > 0. For further details, we refer the reader to "Appendix B" where we collect the material from [22] which we need in this article.
Combining both estimates, we obtain that there exists a constant c 4 = c 4 (T , α L , α ∞ ) such that for cf. Lemma C.1 with r := t 1/α L . Hence, for any x, h ∈ R d and t ∈ (0, T ). Since and so we conclude that It remains to establish the Hölder estimate for P (1) t . By (62), Integrating with respect to y ∈ R d , it follows from (59) and (64) that for suitable constants c 6 and c 7 . Combining the estimates, (57) holds for some finite constant C > 0. The continous dependence of C on the parameters α L − κ ∈ (0, α L ), α L ∈ (0, 2), α C γ b > 0 and T > 0 follows from the fact that each of the constants in this proof depends continuously on these parameters. In Proposition 6.1, we studied the regularity of x → P t u(x) for measurable functions u. The next result is concerned with the regularity of P t u(·) for Hölder continuous functions u. It is natural to expect that P t u "inherits" some regularity from u. Proposition 6.2 Let (X t ) t≥0 be a Feller process with symbol q(x, ξ) = |ξ | α(x) , x, ξ ∈ R d , for a mapping α : For any T > 0, κ ∈ (0, α L ) and ε ∈ (γ 0 , min{γ, α L }), there exists a constant C > 0 such that the semigroup (P t ) t≥0 of (X t ) t≥0 satisfies for all δ > 0 and t ∈ (0, T ]. The constant C > 0 depends continuously on α L ∈ (0, 2), . For the proof of the Schauder estimates, Corollary 4.3, we will apply Proposition 6.2 for an isotropic stable-like process (X t ) t≥0 with symbol q(x, ξ) = |ξ | α(x) for a "truncated" function α of the form where x 0 ∈ R d is fixed and δ > 0 is a constant which we can choose as small as we like; in particular γ 0 := α ∞ − α L ≤ 2δ is small, and so the assumptions ε > γ 0 and γ > γ 0 in Proposition 6.2 are not a restriction. Let us mention that both assumptions, i. e. ε > γ 0 and γ > γ 0 , come into play when estimating one particular term in the proof of Proposition 6.2; see (76); a more careful analysis of this term would probably allow us to relax these two conditions. Proof of Proposition 6.2 Fix ε ∈ (γ 0 , γ ∧ α L ), κ ∈ (0, α L ) and T > 0. First of all, we note that it clearly suffices to show (65) for u ∈ C δ b (R d ) with δ ≤ γ ≤ 1. Throughout the first part of this proof, we will assume that Under (66), the assertion follows if we can show that for all x ∈ R d , |h| ≤ 1 and t ∈ (0, T ], where 2 h denotes as usual the iterated difference operator (cf. (5)). For the proof of this inequality, we use again the parametrix construction of the transition density p of (X t ) t≥0 , where see Appendix B for details. Since x, y)) dy, We estimate the terms separately. For fixed h ∈ R d , |h| ≤ 1, define an auxiliary function v by v(y) := h u(y). Proposition 6.1 gives and so, by the definition of v and the Hölder continuity of u, It remains to establish the corresponding estimate for J 2 , and to this end we use representation (67) for the transition density p.
Step 1 There exists a constant c 1 > 0 such that Indeed: If we denote by p the transition density of the d-dimensional isotropicstable Lévy process, ∈ (0, 2), then there is a constant c 2 > 0 such that for all t ∈ (0, T ], j ∈ {1, . . . , d} and ∈ [α L , α ∞ ] ⊆ (0, 2] (cf. Lemma B.1). To shorten the notation, we fix x, h ∈ R d and t ∈ (0, T ], and write q(y) for the function defined in (69). By definition of p 0 (cf. (68)), we have and so, by the fundamental theorem of calculus and the mean-value theorem, Integrating with respect to y and using (70), we obtain that On the other hand, it follows from (71) and the Hölder continuity of α that Hence, by (70), Combining (73) and (74), we find that the reasoning is very similar to the proof of Lemma C.1. Alternatively, we can use an interpolation theorem.
Step 2 There exists a constant c > 0 such that recall that ε ∈ (γ 0 , α L ∧ γ ) has been fixed at the beginning of the proof.
Indeed: Because of decomposition (67), we have J 2 = J 2,1 + J 2,2 for with q defined in (69). It follows from Step 1 that It remains to estimate J 2,2 . By the definition of the time-space convolution, Integrating with respect to y and applying Tonelli's theorem, the parametrix representation of the transition density p of (X t ) t≥0 (cf. Appendix C). For any T > 0 and any ε ∈ (γ 0 , γ ∧ α L ), there exist finite constants C > 0 and λ > 0 such that for all x ∈ R d , |h| ≤ 1 and t ∈ (0, T ]. The constant C > 0 depends continuously on Proof Fix ε ∈ (γ 0 , α L ∧γ ). To keep the calculations as simple as possible, we consider T := 1. To prove the assertion, we will use that where F i := F F (i−1) denotes the ith convolution power of cf. Appendix C.
By Lemma C.2, there exists a constant c 1 > 0 such that for all r ≥ 0, x, y ∈ R d and |h| ≤ 1. By [22, (proof of) Theorem 4.7], this implies that there is a constant c 2 > 0 such that for all x, y ∈ R d , t ∈ (0, 1) and |h| ≤ 1. Splitting up the domain of integration into three parts we find that R d |D 1 (t, x, y)| dy is bounded by To estimate the second term, note that From [22,Theorem 4.7] and the Hölder continuity of α, there exists a constant c 4 > 0 such that Now we can proceed exactly as in the first part of this step to conclude that for all x ∈ R d , |h| ≤ 1 and t ∈ (0, 1) and suitable constants c 5 , c 5 , λ 2 > 0; for the second estimate, we used that γ > γ 0 = α ∞ − α L .
Step 2 For any ε ∈ (γ 0 , min{γ, α L }) there exist constants C > 0 and λ > 0 such that Indeed Fix ∈ (γ 0 , min{γ, α}). There exist constants C > 0 and λ > 0 such that for all x ∈ R d , i ≥ 1 and t ∈ (0, 1) (cf. Appendix C). Without loss of generality, we may assume that C > 0 and λ > 0 are such that (78) holds (otherwise increase C > 0 and decrease λ > 0). We claim that (79) holds for this choice of C > 0 and λ > 0, and prove this by induction. For i = 1 the estimate is a direct consequence of (78). Now assume that (79) holds for some i ≥ 1. By the definition of the time-space convolution, From first (80) and then (78), for all x ∈ R d , |h| ≤ 1 and t ∈ (0, 1). To estimate the second term, we use (80) with i = 1 and our induction hypothesis to find that for all x ∈ R d , |h| ≤ 1 and t ∈ (0, 1). Combining these, By a change of variables s tr and Euler's formula for the Beta function, Plugging this identity in the previous estimate shows that (79) holds for i + 1, and this finishes the proof of Step 2.
Conclusion of the proof Fix ε ∈ (γ 0 , γ ∧ α L ). Since, by (77), So, by Step 2, for all x ∈ R d , |h| ≤ 1 and t ∈ (0, 1) and suitable constants C > 0 and λ > 0 (not depending on x, h, t). It is not difficult to see that the series on the right-hand side converges, and consequently, we have proved the desired estimate.

Auxiliary Result for the Proof of Theorem 4.1
Let (X t ) t≥0 be an isotropic stable-like process with symbol q(x, ξ) = |ξ | α(x) for a Hölder continuous mapping α : R d → (0, 2) with α L := inf x α(x) > 0. By Proposition 6.1 and Proposition 3.1, any function f in the Favard space F 1 associated with (X t ) t≥0 satisfies the a priori estimate for κ ∈ (0, α L ); in particular, For the proof of Theorem 4.1, we need the following auxiliary result, which will give us an improved a priori estimate once we have shown that f ∈ F 1 is sufficiently regular on {x ∈ R d ; α(x) ≤ 1}. Lemma 6.4 Let (X t ) t≥0 be a Feller process with extended infinitesimal generator (A e , D(A e )), Favard space F 1 and symbol q(x, ξ) = |ξ | α(x) for a Hölder continuous mapping α : Let f ∈ F 1 be such that for any ε ∈ (0, α L ) there exists a constant M(ε) > 0 such that for any x ∈ {α ≤ 1}. Then there exists for any θ ∈ (0, 1) a constant C = C(α, θ ) such that for any x ∈ {α ≥ 1}.
Step 2 Applying Corollary 2.2 we find that v = f · κ is in the Favard space F Y 1 of order 1 associated with (Y t ) t≥0 and Since Proposition 6.1 shows that the semigroup (T t ) t≥0 associated with (Y t ) t≥0 satisfies the Hölder estimate for c 6 = c 6 (α, θ ) > 0, Proposition 3.1 gives for some constant c 7 > 0 which does not depend on f . Finally, we note that for any x ∈ {α ≥ 1} we have κ(z) = 1 for z ∈ B(x, δ), and so for all |h| ≤ δ/2 for some constant c 4 (not depending on f and x) and Proof of Corollary 4. 3 We are going to apply Theorem 3.5 to prove the assertion. To this end, we first need to construct for each x ∈ R d a Feller process (Y (x) t ) t≥0 which satisfies (C1)-(C3) from Theorem 3.2, as well as (S1)-(S5) from Theorem 3.5. Recall that α L = inf x α(x) > 0 and that γ ∈ (0, 1) is the Hölder exponent of α.

and, moreover, Dynkin's formula
holds for any x ∈ R d and any stopping time τ such that E x τ < ∞.
Proof Denote by (R λ ) λ>0 the λ-potential operators of (X t ) t≥0 , and set First we prove F 1 ⊆ D. Let f ∈ F 1 . Airault and Föllmer [1, p. 320-322] showed that the limit g(x) = lim t→0 t −1 (P t f (x) − f (x)) exists outside a set of potential zero, and that is a P x -martingale for any x ∈ R d ; we set g = 0 on the set of potential zero where the limit does not exist. Clearly, g ∞ ≤ K ( f ) < ∞, and so it is obvious that R λ (|g|) is bounded for any λ > 0. It remains to check A.1(ii). Since the martingale (M t ) t≥0 has constant expectation, we have P t f = f + this limit exists up to a set of potential zero (cf. [1]). (ii) The weak infinitesimal generatorÃ in the sense of Dynkin [13] is the linear operatorÃ : By (the proof of) Theorem A.2, the extended generator (A e , D(A e )) is an extension of the weak generator (Ã, D(Ã)). In view of the previous remark, this is not only true for Feller processes but also for general Markov processes.
Corollary A.4 Let (X t ) t≥0 be a Feller process with semigroup (P t ) t≥0 , extended generator (A e , D(A e )) and symbol q. Denote by τ x r := inf{t > 0; |X t − x| > r } the exit time of (X t ) t≥0 from the closed ball B(x, r ). If the symbol q has bounded coefficients, then the following statements are equivalent for any f ∈ B b (R d ).
(ii) There exists r > 0 such that (iii) There exists r > 0 such that

If one (hence all) of the conditions is satisfied, then
up to a set of potential zero for any r ∈ (0, ∞]. In particular, A e f ∞ ≤ K (i) r ( f ) for i ∈ {1, 2} and r ∈ (0, ∞]. The proof of Corollary A.4 shows that the implications (i) ⇒ (ii), (i) ⇒ (iii) and (i) ⇒ (95) remain valid if the symbol q has unbounded coefficients.
Proof of Corollary A.4 (i) ⇒ (ii): If f ∈ F 1 , then it follows from Dynkin's formula (94) that (ii) ⇒ (iii): This is obvious because E x (t ∧ τ x r ) ≤ t. (iii) ⇒ (i): Fix t ∈ (0, 1). Clearly, By assumption, the first term on the right-hand side is bounded by K (2) r ( f )t. For the second term, we note that The maximal inequality (9) for Feller processes shows that there exists an absolute constant c > 0 such that note that the right-hand side is finite because q has bounded coefficients. Combining both estimates gives (i). Proof of (95): For r = ∞, this follows from [1]; see the proof of Theorem A.2. Fix r ∈ (0, ∞). By Dynkin's formula (94), we find ≤ A e f ∞ P x (τ x r ≤ t).
The right-continuity of the sample paths of (X t ) t≥0 gives P x (τ x r ≤ t) → 0 as t → 0, and so t .
Since the right-hand side equals A e f (x) up to a set of potential zero (see the proof of Theorem A.2), this proves the first equality in (95). Similarly, it follows from Dynkin's formula that As P x (τ x r ≤ t) → 0 we find that the right-hand side converges to 0 as t → 0, and this proves the second equality in (95).
where F i := F F (i−1) denotes the ith convolution power of It is possible to show that Because of representation (98), the following estimates are a useful tool to derive estimates for the transition density p. 0, 2). For all T > 0 and k ∈ N 0 , there exists a constant C > 0 such that the following estimates hold for any ∈ [a, b], x ∈ R d , t ∈ (0, T ), and any multiindex β ∈ N d 0 with |β| = k: Proof We only prove (101); for the pointwise estimate (100), see [22,Theorem 4.12]. Denote by p = p ,d the transition density of the d-dimensional isotropic -stable It is well known (see e.g. [35]) that for closed sets F, G ⊆ R d satisfying (109), there exists f ∈ C ∞ (R d ), 0 ≤ f ≤ 1, satisfying (110); however, we could not find a reference for the fact that (109) implies boundedness of the derivatives of f . It is not difficult to see that boundedness of the derivatives fails, in general, to hold if d(F, G) = 0; consider for instance F := R × (−∞, 0] and G := {(x, y); y ≥ e x }.
Proof of Lemma D. 1 As d(F, G) > 0, we can choose ε > 0 such that the sets F ε : = F + B(0, ε), G ε : = G + B(0, ε) are disjoint. It is known (see e.g. [35,) that there exists h ∈ C ∞ (R d ), , ϕ ≥ 0, such that supp ϕ = B(0, ε) and R d ϕ(y) dy = 1, and set Since f is the convolution of a bounded continuous function with a smooth function with compact support, it follows that f is smooth and its derivatives are given by for any multi-index α ∈ N d 0 (see e.g. [48]). In particular, ∂ α f ∞ ≤ ∂ α ϕ L 1 < ∞, and so f ∈ C ∞ b (R d ). Moreover, as supp ϕ ⊆ B(0, ε), it is obvious that f (x) = 0 for any x ∈ F and f (x) = 1 for x ∈ G. It remains to check that 0 < f (x) < 1 for any x ∈ (F ∪ G) c .
Case 2: x ∈ F ε \F. We have B(x, ε) ∩ F c = ∅, and so there exist y ∈ R d and r > 0 such that B(y, r ) ⊆ F c ∩ B(x, ε).