Interior Schauder estimates for elliptic equations associated with Lévy operators
IINTERIOR SCHAUDER ESTIMATES FOR ELLIPTIC EQUATIONSASSOCIATED WITH L´EVY OPERATORS
FRANZISKA K ¨UHN
Abstract.
We study the local regularity of solutions f to the integro-differential equation Af = g in U associated with the infinitesimal generator A of a L´evy process ( X t ) t ≥ . Under the assumptionthat the transition density of ( X t ) t ≥ satisfies a certain gradient estimate, we establish interiorSchauder estimates for both pointwise and weak solutions f . Our results apply for a wide classof L´evy generators, including generators of stable L´evy processes and subordinated Brownianmotions. Introduction
Let ( X t ) t ≥ be a d -dimensional L´evy process. By the L´evy–Khintchine formula, ( X t ) t ≥ is uniquelycharacterized (in distribution) by its infinitesimal generator A , which is an integro-differentialoperator with representation Af ( x ) = b ⋅ ∇ f ( x ) +
12 tr ( Q ⋅ ∇ f ( x )) + ∫ R d /{ } ( f ( x + y ) − f ( x ) − ∇ f ( x ) ⋅ y ( , ) (∣ y ∣)) ν ( dy ) for f ∈ C ∞ c ( R d ) , here ( b, Q, ν ) denotes the L´evy triplet of ( X t ) t ≥ , cf. Section 2. In this paper,we study the local H¨older regularity of weak and pointwise solutions f to the integro-differentialequation Af = g in U for open sets U ⊆ R d . We are interested in interior Schauder estimates, i.e. our aim is to describethe regularity of f on the set { x ∈ U ; d ( x, U c ) > δ } for δ > f and volatility of theL´evy process: the higher the volatility (caused by a non-vanishing diffusion component or a highsmall-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-orderdifferential operator and interior Schauder estimates for solutions to Af = g are well studied, seee.g. Gilbarg [6]. One of the most prominent non-local L´evy operators is the fractional Laplacian −(− ∆ ) α / , α ∈ ( , ) , defined by −(− ∆ ) α / f ( x ) = c d,α ∫ y ≠ ( f ( x + y ) − f ( x ) − ∇ f ( x ) ⋅ y ( , ) (∣ y ∣)) ∣ y ∣ d + α dy for some normalizing constant c d,α >
0. The fractional Laplacian is the infinitesimal generator ofthe isotropic α -stable L´evy 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 solutionsto −(− ∆ ) α / f = g go back to Stein [29], see also Bass [1]. Since then, several extensions andrefinements of these estimates have been obtained. Ros-Oton & Serra [24] studied the interiorH¨older regularity of solutions to equations Af = g associated with symmetric α -stable operatorsand established under a mild degeneracy condition on the spectral measure estimates of the form ∥ f ∥ C α + κ ( B ( , )) ≤ c (∥ f ∥ C κ ( R d ) + ∥ g ∥ C κ ( B ( , )) ) for κ ≥ α + κ is not an integer. In the recent paper [14], global Schauder estimates ∥ f ∥ C α + κb ( R d ) ≤ c (∥ f ∥ ∞ + ∥ g ∥ C κb ( R d ) ) , κ ≥ , (1) Mathematics Subject Classification.
Primary 60G51; Secondary 45K05, 60J35.
Key words and phrases.
L´evy process; integro-differential equation; Schauder estimate; H¨older space; gradientestimate. a r X i v : . [ m a t h . P R ] A p r NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 2 were obtained for a wide class of L´evy processes satisfying a certain gradient estimate, see (C2)below; here C βb ( R d ) denotes the H¨older–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 respectto a L´evy generator, see e.g. [7, 8, 16, 18, 30]. Let us mention that the regularity of solutions integro-differential equations Af = g has been studied, more generally, for classes of L´evy-type operators,see e.g. [1, 3, 11, 12, 15, 20, 22]. Of course, Schauder estimates are also of interest for parabolicequations, 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 truncationtechnique to derive local H¨older estimates for solutions to Af = g . We will assume that the L´evyprocess ( X t ) t ≥ with L´evy triplet ( b, Q, ν ) satisfies the following conditions.(C1) The characteristic exponent ψ satisfies the Hartman–Wintner growth conditionlim ∣ ξ ∣→∞ ∣ Re ψ ( ξ )∣ log ( + ∣ ξ ∣) = ∞ . (C2) There exist constants M > α > p t , t >
0, satisfiesthe gradient estimate ∫ R d ∣∇ p t ( x )∣ dx ≤ M t − / α , t ∈ ( , ) . (C3) Either α > Q =
0. Moreover, α + > γ for a constant γ ∈ ( , ] with ∫ ∣ y ∣≤ ∣ y ∣ γ ν ( dy ) < ∞ .It follows from (C1),(C2) that the global Schauder estimate (1) holds, and we will use (C3) tolocalize these estimates. Before stating our results, let us give some remarks on (C1)-(C3).1.1. Remark. (i) If α >
1, we may choose γ = ∫ ∣ y ∣≤ ∣ y ∣ ν ( dy ) < ∞ holds forany L´evy measure.(ii) The Hartman–Wintner condition (C1) implies that the law of X t , t >
0, has a density p t ∈ C ∞ b ( R d ) with respect to Lebesgue measure, see [13] for a detailed discussion.(iii) The constant α in (C2) is always less or equal than 2. This follows from the fact thatthe growth condition ∣ ψ ( ξ )∣ ≤ c ( + ∣ ξ ∣ (cid:37) ) , ξ ∈ R d , implies α ≤ (cid:37) , cf. [14, Remark 3.2(ii)].Roughly speaking, α is a measure for the volatility of the process: if α > α is close to 2, then ( X t ) t ≥ has a high volatility (many small jumps or non-vanishingdiffusion part).(iv) If the diffusion matrix Q is positive definite, then (C1)-(C3) hold with γ = α =
2, cf.Example 4.2. Further classes of L´evy processes satisfying (C1)-(C3) will be presented inSection 4.(v) Condition (C3) is essentially a balance condition on the growth of Re ψ at infinity. If weset β = max { Q ≠ , γ } , then ∣ Re ψ ( ξ )∣ ≤ C ( + ∣ ξ ∣ β ) . On the other hand, (C2) means,roughly, that Re ψ ( ξ ) ≥ c ( + ∣ ξ ∣ α ) . Since (C3) is equivalent to ∣ β − α ∣ <
1, this shows thatthe lower and upper growth rate of Re ψ at infinity should be sufficiently close to eachother, e.g. ψ ( ξ, η ) ∶= ∣ ξ ∣ α + ∣ η ∣ β does not satisfy (C3) is ∣ β − α ∣ >
1, cf. Example 4.5.Next we state our main results; see Section 2 for the definition of the notation used in the state-ments.1.2.
Theorem.
Let ( X t ) t ≥ be a L´evy process with infinitesimal generator ( A, D ( A )) satisfying (C1) - (C3) , and denote by α ∈ ( , ] the constant from (C2) . Let f be a weak solution to theequation Af = g in U for an open set U ⊆ R d . Set U δ ∶= { x ∈ U ; d ( x, U c ) > δ } for δ > .(i) If f ∈ L ∞ ( R d ) and g ∈ L ∞ ( U ) , then f has a modification ˜ f which is continuous on U andsatisfies the interior Schauder estimate ∥ ˜ f ∥ C αb ( U δ ) ≤ C δ (∥ f ∥ L ∞ ( R d ) + ∥ g ∥ L ∞ ( U ) ) (2) for every δ > . The constant C δ does not depend on f , g . NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 3 (ii) If f ∈ C κb ( R d ) and g ∈ C κb ( U ) for some κ > , then there exists for every δ > a constant C δ > (independent of f , g ) such that ∥ f ∥ C κ + αb ( U δ ) ≤ C δ (∥ f ∥ C κb ( R d ) + ∥ g ∥ C κb ( U ) ) . (3)Theorem 1.2 applies for a wide class of L´evy processes. It generalizes the interior Schauder estimatesfor 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 withconstant coefficients, see Section 4 for details and further examples.1.3.
Remark. (i) The weak solution f ∈ L ∞ ( R d ) to Af = g is only determined up to aLebesgue null set. The interior Schauder estimate (2) implies continuity of ˜ f on U , and so(2) cannot hold for any representative ˜ f of f but only for a suitably chosen representative.(ii) In the proof of Theorem 1.3 we use (C1),(C2) only to obtain from [14, Theorem 1.1] theglobal Schauder estimate ∥ f ∥ C αb ( R d ) ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ) , f ∈ C b ( R d ) . (4)Consequently, the interior Schauder estimates in Theorem 1.3 hold for any L´evy process ( X t ) t ≥ with generator ( A, D ( A )) satisfying the global Schauder estimate (4) and thebalance condition (C3). This means that (the proof of) Theorem 1.3 actually gives ageneral procedure to localize Schauder estimates.(iii) If we interpret the constant α from (C2) as a measure for the volatility of ( X t ) t ≥ , cf.Remark 1.1(iii), then Theorem 1.2 shows that a high volatility of the L´evy process ( X t ) t ≥ results in a high regularity of f ∣ U . This is a natural result, and we believe the regularityestimates to be optimal for many L´evy processes. In some cases, certain properties of theL´evy process or the L´evy triplet may lead to an additional smoothing effect; for instance,Grzywny & Kwa´snicki [7, Theorem 1.7] studied the regularity of harmonic functions f (i.e. Af =
0) associated with unimodal L´evy processes and showed that the regularity ofthe density of the L´evy measure ν carries over to f ; this regularity of f is not related tothe volatility of the process.(iv) In general, the assumption f ∈ C κb ( R d ) in Theorem 1.2(ii) cannot be relaxed to f ∈ C κb ( 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 ≥ , then [14, Theorem 1.1]gives f ∈ C αb ( R d ) , and so the assumption f ∈ C κb ( R d ) in Theorem 1.2(ii) is automaticallysatisfied 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 .1.4. Corollary.
Let ( X t ) t ≥ be a L´evy process satisfying (C1) - (C3) , and let U ⊆ R d be an openset. Let f ∈ B b ( R d ) be a function such that g ( x ) ∶= lim t → E f ( x + X t ) − f ( x ) t exists for all x ∈ U and assume that sup x ∈ K sup t ∈( , ) ∣ E f ( x + X t ) − f ( x ) t ∣ < ∞ (5) for any compact set K ⊆ U . Denote by α ∈ ( , ] the constant from (C2) .(i) If g ∈ L ∞ ( U ) then there exists for any δ > a constant C δ > (independent of f , g ) suchthat ∥ f ∥ C αb ( U δ ) ≤ C δ (∥ f ∥ L ∞ ( R d ) + ∥ g ∥ L ∞ ( U ) ) . (ii) If f ∈ C κb ( R d ) and g ∈ C κb ( U ) for some κ > then there exists for any δ > a constant C δ > (independent of f , g ) such that ∥ f ∥ C κ + αb ( U δ ) ≤ C δ (∥ f ∥ C κb ( R d ) + ∥ g ∥ C κb ( U ) ) . As an immediate consequence, we obtain local Schauder estimates for functions in the domain inthe strong infinitesimal generator. They extend in a natural way the global Schauder estimatesfrom [14].
NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 4
Corollary.
Let ( X t ) t ≥ be a L´evy process satisfying (C1) - (C3) , and denote by α ∈ ( , ] theconstant 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 Af ( x ) ∶= lim t → E f ( x + X t ) − f ( x ) t exists uniformly in x ∈ R d .(i) For each δ > there exists a finite constant C δ (not depending on f ) such that ∥ f ∥ C αb ( B ( x,δ )) ≤ C δ (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( x, δ ) ) for all x ∈ R d .(ii) If Af ∈ C κb ( R d ) for some κ > , then f ∈ C κ + αb ( R d ) and ∥ f ∥ C α + κb ( B ( x,δ )) ≤ C δ (∥ f ∥ ∞ + ∥ Af ∥ ∞ + ∥ Af ∥ C κb ( B ( x, δ )) ) , x ∈ R d , δ > , for a finite constant C δ , which does not depend on f , g . The proof of Corollary 1.5 shows that the interior Schauder estimates (i),(ii) actually hold for anyL´evy process ( X t ) t ≥ with generator ( A, D ( A )) satisfying (C3) and the global Schauder estimate ∥ f ∥ C αb ( R d ) ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ) , f ∈ C b ( R d ) , see Remark 1.3(ii).This paper is organized as follows. In Section 2 we introduce basic definitions and notation. Ourmain results are proved in Section 3, and examples are presented in Section 4.2. Definitions
We consider the d -dimensional Euclidean space R d with the canonical scalar product x ⋅ y ∶=∑ dj = 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-valuedfunction, then supp f denotes its support, ∇ f the gradient and ∇ f the Hessian of f . For α ≥ ⌊ α ⌋ ∶= max { k ∈ N ; k ≤ α } . Function spaces: B b ( R d ) is the space of bounded Borel measurable functions f ∶ R d → R . Thesmooth functions with compact support are denoted by C ∞ c ( R d ) . Superscripts k ∈ N are used todenote the order of differentiability, e.g. f ∈ C kb ( R d ) means that f and its derivatives up to order k are bounded continuous functions. For U ⊆ R d we set ∥ f ∥ ∞ ,U ∶= sup x ∈ U ∣ f ( x )∣ and ∥ f ∥ ∞ ∶= ∥ f ∥ ∞ , R d . For every α ≥ U ⊆ R d we define the H¨older–Zygmund space C αb ( U ) by C αb ( U ) ∶= ⎧⎪⎪⎨⎪⎪⎩ f ∈ C b ( U ) ; ∥ f ∥ C αb ( U ) ∶= sup x ∈ U ∣ f ( x )∣ + sup x ∈ U sup <∣ h ∣< min { ,r x / k } ∣ ∆ kh f ( x )∣∣ h ∣ α < ∞⎫⎪⎪⎬⎪⎪⎭ where r x ∶= d ( x, U c ) is the distance of x from the complement of U , k ∈ N is the smallest naturalnumber strictly larger than α and∆ h f ( x ) ∶= f ( x + h ) − f ( x ) ∆ (cid:96)h f ( x ) ∶= ∆ h ( ∆ (cid:96) − h f )( x ) , (cid:96) ≥ 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. ∥ f ∥ C αb ( R d ) = ∥ f ∥ ∞ + sup x ∈ R d sup <∣ h ∣< ∣ ∆ kh f ( x )∣∣ h ∣ α ≍ ∥ f ∥ ∞ + sup x ∈ R d sup <∣ h ∣< ∣ ∆ jh f ( x )∣∣ h ∣ α , (6)cf. [31, Theorem 2.7.2.2]. We need a localized version of this result.2.1. Lemma.
Let α ∈ ( , ∞) , and let U ⊆ R d be open. The following statements hold for any j ≥ k ∶= ⌊ α ⌋ + : NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 5 (i) There exists a constant c > such that sup <∣ h ∣≤ r ∣ ∆ kh f ( x )∣∣ h ∣ α ≤ cr − α ∥ f ∥ ∞ ,U + c sup <∣ h ∣≤ r / j sup z ∈ B ( x,r ( k + )) ∣ ∆ jh f ( z )∣∣ h ∣ α for all f ∈ C b ( U ) , r > , and x ∈ U with B ( x, r ( k + )) ⊆ U .(ii) If α > then there exists a constant c > such that ∣∇ f ( x )∣ ≤ cr − α ∥ f ∥ ∞ ,U + c sup <∣ h ∣≤ r / j sup z ∈ B ( x,r ( k + )) ∣ ∆ jh f ( z )∣∣ h ∣ α for all f ∈ C b ( U ) , r > and x ∈ U with B ( x, r ( k + )) ⊆ U . We defer the proof of Lemma 2.1 to the appendix. For α ∈ ( , ∞)/ N the H¨older–Zygmund space C αb ( R d ) coincides with the “classical” H¨older space C αb ( R d ) , cf. [31, Theorem 2.7.2.1]. If α = k ∈ N ,then the inclusion C kb ( R d ) ⊆ C kb ( R d ) is strict. L´evy processes:
Let ( Ω , A , P ) be a probability space. A stochastic process X t ∶ Ω → R d , t ≥
0, isa ( d -dimensional) L´evy process if X = ( X t ) t ≥ has independent and stationaryincrements and t ↦ X t ( ω ) is right-continuous with finite left-hand limits for almost all ω ∈ Ω. TheL´evy–Khintchine formula shows that every L´evy process is uniquely determined in distribution byits characteristic exponent ψ ∶ R d → C satisfying E exp ( iξ ⋅ X t ) = exp (− tψ ( ξ )) , t ≥ , ξ ∈ R d . The characteristic exponent ψ has a L´evy–Khintchine representation ψ ( ξ ) = − ib ⋅ ξ + ξ ⋅ Qξ + ∫ y ≠ ( − e iy ⋅ ξ + iy ⋅ ξ ( , ) (∣ y ∣)) ν ( dy ) , ξ ∈ R d , where the L´evy triplet ( b, Q, ν ) consists of a vector b ∈ R d ( drift vector ), a symmetric positive semi-definite matrix Q ∈ R d × d ( diffusion matrix ) and a measure ν on R d /{ } with ∫ y ≠ min { , ∣ y ∣ } ν ( dy ) <∞ ( L´evy measure ). Our standard reference for L´evy processes is the monograph [25] by Sato. Bythe independence and stationarity of the increments, every L´evy process is a time-homogeneousMarkov process, i.e. P t f ( x ) ∶= E f ( x + X t ) defines a Markov semigroup on B b ( R d ) . We denote by ( A, D ( A )) the (weak) infinitesimal generator , D ( A ) ∶= { f ∈ B b ( R d ) ; ∃ g ∈ B b ( R d ) ∀ x ∈ R d ∶ lim t → E f ( x + X t ) − f ( x ) t = g ( x )} ,Af ( x ) ∶= lim t → E f ( x + X t ) − f ( x ) t , f ∈ D ( A ) . If f ∈ C b ( R d ) then f ∈ D ( A ) and Af ( x ) = b ⋅ ∇ f ( x ) +
12 tr ( Q ⋅ ∇ f ( x )) + ∫ y ≠ ( f ( x + y ) − f ( x ) − ∇ f ( x ) ⋅ y ( , ) (∣ y ∣)) ν ( dy ) . (7)Restricted to C ∞ c ( R d ) , the infinitesimal generator is a pseudo-differential operator with symbol ψ , Af ( x ) = − ∫ R d ψ ( ξ ) e ix ⋅ ξ ˆ f ( ξ ) dξ, f ∈ C ∞ c ( R d ) , where ˆ f ( ξ ) = ( π ) − d ∫ R d f ( x ) e − ix ⋅ ξ dx is the Fourier transform of f . The pseudo-differential A ∗ with symbol ψ ( ξ ) = ψ (− ξ ) is the adjoint of A , in the sense that, ∀ f, g ∈ C ∞ c ( R d ) ∶ ∫ R d Af ( x ) g ( x ) dx = ∫ R d f ( x ) A ∗ g ( x ) dx. 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 ∀ ϕ ∈ C ∞ c ( U ) ∶ ∫ R d f ( x ) A ∗ ϕ ( x ) dx = ∫ U g ( x ) ϕ ( x ) dx. (8)It is implicitly assumed that the integral on the left-hand side exists; a sufficient condition is f ∈ L ∞ ( R d ) , see e.g. [4, Lemma 2.1] and [16, Proposition 2.1] for milder growth conditions on f . NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 6 Proofs
In this section we present the proofs of our main results. Corollary 1.4 and Corollary 1.5 areconsequences of Theorem 1.2, and therefore the main part is to establish Theorem 1.2. The ideais to combine the global Schauder estimates from [14] with a truncation technique to establishinterior Schauder estimates. We start with the following auxiliary result.3.1.
Lemma.
Let ( X t ) t ≥ be a L´evy process with generator ( A, D ( A )) and L´evy triplet ( b, Q, ν ) .If f, g ∈ C b ( R d ) then f g ∈ D ( A ) and A ( f g ) = gAf + f Ag + Γ ( f, g ) where Γ ( f, g )( x ) ∶= ∇ f ( x ) ⋅ Q ∇ g ( x ) + ∫ y ≠ ( f ( x + y ) − f ( x ))( g ( x + y ) − g ( x )) ν ( dy ) , x ∈ R d , is the Carr´e du Champ operator .Proof.
Clearly, f ⋅ g ∈ C b ( R d ) ⊆ D ( A ) . The identity for A ( f ⋅ g ) follows by applying (7) for f ⋅ g and rearranging the terms. (cid:3) 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 = f, g need to satisfy a certain H¨older 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 1.2.3.2. Proposition.
Let ( X t ) t ≥ be a L´evy process with generator ( A, D ( A )) , characteristic exponent ψ and L´evy triplet ( b, Q, ν ) . If (C1) - (C3) hold, then there exists for every R > and δ > someconstant c > such that ∥ f ∥ C αb ( B ( x,R )) ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( x,R + δ ) ) for all f ∈ C b ( R d ) and x ∈ R d ; here α ∈ ( , ] denotes the constant from (C2) . In the proof of Proposition 3.2 we will use the elementary inequalitiessup ∣ h ∣≤ R ∣ ∆ h f ( x )∣∣ h ∣ α ≤ r − α ∥ f ∥ ∞ + sup ∣ h ∣≤ r ∣ ∆ h f ( x )∣∣ h ∣ α (9) ≤ r − α ∥ f ∥ ∞ + r ε sup ∣ h ∣≤ r ∣ ∆ h f ( x )∣∣ h ∣ α + ε , (10)which hold for any 0 < r < R ≤ ε > f ∈ B b ( R d ) and x ∈ R d . Proof of Proposition 3.2.
For simplicity of notation we consider x = R = δ =
1, i.e. we needto show ∥ f ∥ C αb ( B ( , )) ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) ) . Let χ ∈ C ∞ c ( R d ) be such that B ( , / ) ≤ χ ≤ B ( , / ) . For x ∈ B ( , ) set r x ∶= d ( x, B ( , ) c ) and χ ( x ) ( y ) ∶= χ ( r − x ( y − x )) . As r x ≤
2, it follows from the chain rule that ∥ χ ( x ) ∥ C b ( R d ) ≤ ( + r − )∥ χ ∥ C b ( R d ) ≤ r − ∥ χ ∥ C b ( R d ) . We split the proofs in several steps.
Step 1:
There exist constants ε ∈ ( , α ) , (cid:37) > C > f ) such that ∥ f χ ( x ) ∥ C αb ( R d ) ≤ C ⎛⎝ r − α − (cid:37)x ∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) + sup z ∈ B ( x, r x / ) K ( x, z )⎞⎠ (11)for all x ∈ B ( , ) , where K ( x, z ) ∶= sup ∣ h ∣≤ r x / ∣ ∆ h f ( z )∣∣ h ∣ α − ε (12) Indeed:
We fix x ∈ ( , ) and write r ∶= r x for brevity. Since f χ ( x ) is twice continuously differentiableand vanishing at infinity, it is contained in the domain of the strong infinitesimal generator, andso it follows from [14, Theorem 1.1] that ∥ f χ ( x ) ∥ C αb ( R d ) ≤ c (∥ f ∥ ∞ + ∥ A ( f χ ( x ) )∥ ∞ ) NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 7 for some constant c > f and x . Hence, by Lemma 3.1, ∥ f χ ( x ) ∥ C αb ( R d ) ≤ c (∥ f ∥ ∞ + ∥ χ ( x ) Af ∥ ∞ + ∥ f Aχ ( x ) ∥ ∞ + ∥ Γ ( f, χ ( x ) )∥ ∞ ) . As 0 ≤ χ ( x ) ≤
1, supp χ ( x ) ⊆ B ( , ) and ∥ Aχ ( x ) ∥ ∞ ≤ c ∥ χ ( x ) ∥ C b ( R d ) ≤ c r − ∥ χ ∥ C b ( R d ) , this gives ∥ f χ ( x ) ∥ C αb ( R d ) ≤ ( + c ) ( r − ∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) + ∥ Γ ( f, χ ( x ) )∥ ∞ ) . (13)It remains to estimate the supremum norm of the term involving the Carr´e du champ operator.For z ∈ R d / B ( x, r / ) we have χ ( x ) = B ( z, r / ) , and thereforeΓ ( f, χ ( x ) )( z ) = ∫ ∣ y ∣> r / ( f ( z + y ) − f ( z )) ( χ ( x ) ( z + y ) − χ ( x ) ( z )) ν ( dy ) implying ∣ Γ ( f, χ ( x ) )( z )∣ ≤ ∥ f ∥ ∞ ∫ ∣ y ∣> r / ν ( dy ) ≤ + γ r − γ ∥ f ∥ ∞ ∫ y ≠ min { , ∣ y ∣ γ } ν ( dy ) . (14)Since ∫ y ≠ min { , ∣ y ∣ γ } ν ( dy ) < ∞ , cf. (C3), and r ≤
2, this implies ∣ Γ ( f, χ ( x ) )( z )∣ ≤ c r − α − (cid:37) ∥ f ∥ ∞ , z ∈ R d / B ( x, r / ) , for suitable constants c > (cid:37) >
0, which is an estimate of the desired form. For z ∈ B ( x, r / ) we consider the local term and the non-local term separately, i.e. we write Γ ( f, χ ( x ) )( z ) = D ( z ) + I ( z ) where D ( z ) ∶= ∇ f ( z ) ⋅ Q ∇ χ ( x ) ( z ) and I ( z ) ∶= ∫ y ≠ ( f ( z + y ) − f ( z ))( χ ( x ) ( z + y ) − χ ( x ) ( z )) ν ( dy ) . For the local term it clearly suffices to consider the case Q ≠
0. If Q ≠ α > ε ∈ ( , ) such that α − ε >
1. Clearly, ∣ D ( z )∣ ≤ ∥ χ ( x ) ∥ C b ( R d ) ∥ Q ∥∣∇ f ( z )∣ ≤ r − ∥ Q ∥∥ χ ∥ C b ( R d ) ∣∇ f ( z )∣ . Since there exists a constant c > ∣∇ f ( z )∣ ≤ c r − α ∥ f ∥ ∞ + c sup ∣ y − z ∣≤ r / sup ∣ h ∣≤ ∣ ∆ h f ( y )∣∣ h ∣ α − ε , cf. Lemma 2.1(ii), it follows from (9) and (10) that ∣∇ f ( z )∣ ≤ c ′ r − θ − α ∥ f ∥ ∞ + c ′ sup { ∣ ∆ h f ( y )∣∣ h ∣ α − ε ; y ∈ B ( x, r / ) , ∣ h ∣ ≤ ( r ) θ }≤ c ′ r − θ − α ∥ f ∥ ∞ + c ′′ r θε sup { ∣ ∆ h f ( y )∣∣ h ∣ α − ε ; y ∈ B ( x, r / ) , ∣ h ∣ ≤ ( r ) θ } for any θ > z ∈ B ( x, r / ) . Choosing θ ∶= / ε we get ∣ D ( z )∣ ≤ c r − − / ε − α ∥ f ∥ ∞ + c sup { ∣ ∆ h f ( y )∣∣ h ∣ α − ε ; y ∈ B ( x, r / ) , ∣ h ∣ ≤ ( r ) θ } for all z ∈ B ( x, r / ) . As r / ≤ θ >
1, the supremum over ∣ h ∣ ≤ ( r / ) θ is less or equal thanthe supremum over ∣ h ∣ ≤ ( r / ) , and so ∣ D ( z )∣ ≤ c r − − / ε − α ∥ f ∥ ∞ + c sup y ∈ B ( x, r / ) K ( x, y ) with K defined in (12). It remains to estimate the non-local term. We consider the cases γ = γ ∈ ( , ) separately. If γ =
2, then by (C3) α >
1. Choose ε ∈ ( , ) such that α − ε > θ = / ε . By the Lipschitz continuity of χ ( x ) , ∣ I ( z )∣ ≤ ∥ χ ( x ) ∥ C b ( R d ) ∫ ∣ y ∣≤ r / ∣ f ( y + z ) − f ( z )∣ ∣ y ∣ ν ( dy ) + ∥ f ∥ ∞ ∫ ∣ y ∣> r / ν ( dy ) . NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 8
Since ∫ y ≠ min { , ∣ y ∣ γ } ν ( dy ) < ∞ and ∥ χ ( x ) ∥ C b ( R d ) ≤ r − ∥ χ ∥ C b ( R d ) , this implies that there existsa finite constant c > ∣ I ( z )∣ ≤ c r − ∥ χ ∥ C b ( R d ) sup ∣ h ∣≤ r / ∣∇ f ( z + h )∣ ∫ ∣ y ∣≤ r / ∣ y ∣ ν ( dy ) + c r − γ ∥ f ∥ ∞ . For the first term on the right-hand side we can now use a reasoning similar to that for the localterm and get the required estimate. Finally we consider the case γ <
2. Fix ε ∈ ( , ) such thatmax { , γ − } < α − ε – it exists because of (C3) – and set θ ∶= / ε . We have ∣ I ( z )∣ ≤ ∥ χ ( x ) ∥ C b ( R d ) sup ∣ h ∣≤ ∣ f ( z + h ) − f ( z )∣∣ h ∣ max { ,γ − } ∫ ∣ y ∣≤ ∣ y ∣ γ ν ( dy ) + ∥ f ∥ ∞ ∫ ∣ y ∣> ν ( dy ) . Using that ∫ min { , ∣ y ∣ γ } ν ( dy ) < ∞ and ∥ χ ( x ) ∥ C b ( R d ) ≤ r − ∥ χ ∥ C b ( R d ) , we find a finite constant c > ∣ I ( z )∣ ≤ c ∥ f ∥ ∞ + c r − sup ∣ h ∣≤ ∣ f ( z + h ) − f ( z )∣∣ h ∣ max { ,γ − } . Since max { , γ − } <
1, an application of Lemma 2.1(i) shows thatsup ∣ h ∣≤ ∣ f ( z + h ) − f ( z )∣∣ h ∣ max { ,γ − } ≤ c r − ∥ f ∥ ∞ + c sup ∣ h ∣≤ sup ∣ y − z ∣≤ r / ∣ ∆ h f ( y )∣∣ h ∣ max { ,γ − } . Hence, by (9) and (10),sup ∣ y ∣≤ ∣ f ( z + y ) − f ( z )∣∣ y ∣ max { ,γ − } ≤ c ′ r − θ ∥ f ∥ ∞ + c r θε sup { ∣ ∆ h f ( y )∣∣ h ∣ α − ε ; y ∈ B ( x, r / ) , ∣ h ∣ ≤ ( r ) θ } for all z ∈ B ( x, r / ) . Recalling that θε = r ≤ ∣ I ( z )∣ ≤ c r − θ − ∥ f ∥ ∞ + c sup { ∣ ∆ h f ( y )∣∣ h ∣ α − ε ; y ∈ B ( x, r / ) , ∣ h ∣ ≤ ( r ) θ }≤ c r − θ − ∥ f ∥ ∞ + c sup y ∈ B ( x, r / ) K ( x, y ) . Step 2:
There exists a constant C > ∣ f ∣ B ( , ) ,α,(cid:37) ≤ C (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) + ∣ f ∣ B ( , ) ,α − ε,(cid:37) ) , (15)where (cid:37) , ε are the constants from Step 1 and ∣ f ∣ B ( , ) ,σ,κ ∶= sup x ∈ B ( , ) sup ∣ h ∣≤ r x / r κ + σx ∣ ∆ h f ( x )∣∣ h ∣ σ , κ > , σ ∈ ( , ) . (16) Indeed:
For z ∈ B ( x, r x / ) and x ∈ B ( , ) we have r z ≥ r x /
8, and therefore the mapping K defined in (12) satisfies r α + (cid:37)x sup z ∈ B ( x, r x / ) K ( x, z ) ≤ α + (cid:37) sup z ∈ B ( x, r x / ) sup ∣ h ∣≤ r x / r α + (cid:37)z ∣ ∆ h f ( z )∣∣ h ∣ α − ε ≤ α + (cid:37) sup z ∈ B ( , ) sup ∣ h ∣≤ r z / r α + (cid:37)z ∣ ∆ h f ( z )∣∣ h ∣ α − ε ≤ α + (cid:37) ∣ f ∣ B ( , ) ,α − ε,(cid:37) . Consequently, it follows from Step 1 that r (cid:37) + αx ∥ f χ ( x ) ∥ C αb ( R d ) ≤ C (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) + α + (cid:37) ∣ f ∣ B ( , ) ,α − ε,(cid:37) ) for all x ∈ B ( , ) . As α ∈ ( , ] , this gives r (cid:37) + αx sup ∣ h ∣≤ sup z ∈ R d ∣ ∆ h ( f χ ( x ) )( z )∣∣ h ∣ α ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) + ∣ f ∣ B ( , ) ,α − ε,(cid:37) ) (17)for all x ∈ B ( , ) and some constant c > x and f , cf. (6). Since χ ( x ) = B ( x, r x / ) , this implies r (cid:37) + αx sup ∣ h ∣≤ r x / ∣ ∆ h f ( x )∣∣ h ∣ α ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) + ∣ f ∣ B ( , ) ,α − ε,(cid:37) ) . NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 9
On the other hand, we have r (cid:37) + αx sup r x / ≤∣ h ∣≤ r x / ∣ ∆ h f ( x )∣∣ h ∣ α ≤ r (cid:37) + αx ∥ f ∥ ∞ ( r x ) − α ≤ c ∥ f ∥ ∞ for some uniform constant c >
0. Combining both estimates yields r (cid:37) + αx sup ∣ h ∣≤ r x / ∣ ∆ h f ( x )∣∣ h ∣ α ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) + ∣ f ∣ B ( , ) ,α − ε,(cid:37) ) for all x ∈ B ( , ) , and this proves (15). Conclusion of the proof : Choose δ ∈ ( , ) sufficiently small such that C δ ≤ for the constant C from Step 2. By definition, cf. (16), ∣ f ∣ B ( , ) ,α − ε,(cid:37) = sup x ∈ B ( , ) sup h ∶∣ h / r x ∣≤ / r (cid:37)x ( r x ∣ h ∣ ) α − ε ∣ ∆ h f ( x )∣ . Since sup h ∶∣ h / r x ∣≤ δ / ε r (cid:37)x ( r x ∣ h ∣ ) α − ε ∣ ∆ h f ( x )∣ ≤ δ sup h ∶∣ h / r x ∣≤ δ / ε r (cid:37)x ( r x ∣ h ∣ ) α ∣ ∆ h f ( x )∣ ≤ δ ∣ f ∣ B ( , ) ,α,(cid:37) and sup h ∶ δ / ε <∣ h / r x ∣≤ / r (cid:37)x ( r x ∣ h ∣ ) α − ε ∣ ∆ h f ( x )∣ ≤ δ − α / ε sup h ∶ δ / ε <∣ h / r x ∣≤ / r (cid:37)x ∣ ∆ h f ( x )∣ ≤ δ − α / ε (cid:37) ∥ f ∥ ∞ , it follows that there exists a constant c > ∣ f ∣ B ( , ) ,α − ε,(cid:37) ≤ δ ∣ f ∣ B ( , ) ,α,(cid:37) + c ∥ f ∥ ∞ . (18)As C δ ≤ , we find from Step 2 and (18) that ∣ f ∣ B ( , ) ,α,(cid:37) ≤ C (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) + ∣ f ∣ B ( , ) ,α − ε,(cid:37) )≤ C (( + c )∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) ) + ∣ f ∣ B ( , ) ,α,(cid:37) , i.e. ∣ f ∣ B ( , ) ,α,(cid:37) ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ,B ( , ) ) . On the other hand, we also have ∣ f ∣ B ( , ) ,α,(cid:37) ≥ sup x ∈ B ( , ) sup ∣ h ∣≤ / ∣ ∆ h f ( x )∣∣ h ∣ α . Combining both estimates and applying Lemma 2.1 proves the assertion. (cid:3)
The seminorms ∣ f ∣ U,α,(cid:37) which we introduced in (15) are closely related to seminorms which appearin the study of Schauder estimates for second order differential operators, cf. [6]; our definition isinspired by H¨older–Zygmund norms whereas the seminorms in [6] are based on “classical” H¨oldernorms.In order to apply the a priori estimate from Proposition 3.2, we have to approximate the weaksolution f by a sequence ( f k ) k ≥ 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 ≥ . To make this approximation work, weneed to know that Af k is on U close to Af = g , and this is what the next lemma is about.3.3. Lemma.
Let ( X t ) t ≥ be a L´evy process with infinitesimal generator ( A, D ( A )) . Let f ∈ L ∞ ( R d ) be a weak solution to the equation Af = g in U for g ∈ L ∞ ( U ) and U ⊆ R d open. If χ ∈ C ∞ c ( B ( , r )) for some r > , then f ∗ χ ∈ D ( A ) and A ( f ∗ χ )( x ) = ( g ∗ χ )( x ) for all x ∈ U with B ( x, r ) ⊆ U .Proof. It is well known that u ∶= f ∗ χ ∈ C ∞ b ( R d ) and ∂ α u = f ∗ ( ∂ α χ ) , α ∈ N d , (19)see e.g. [26]. In particular, u ∈ C b ( R d ) ⊆ D ( A ) and Au ( x ) = b ⋅ ∇ u ( x ) +
12 tr ( Q ⋅ ∇ u ( x )) + ∫ y ≠ ( u ( x + y ) − u ( x ) − ∇ u ( x ) ⋅ y ( , ) (∣ y ∣)) ν ( dy ) . NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 10
Using (19) and Fubini’s theorem it follows that Au ( x ) = ( f ∗ Aχ )( x ) = ∫ R d f ( y )( Aχ )( x − y ) dy for all x ∈ R d . As ( Aχ )( x − y ) = ( A ∗ χ ( x − ⋅))( y ) for the adjoint A ∗ , we get Au ( x ) = ∫ R d f ( y )( A ∗ χ ( x − ⋅))( y ) dy. If x ∈ U is such that B ( x, r ) ⊆ U , then supp χ ( x − ⋅) ⊆ U and so it follows from the definition of theweak solution, cf. (8), that Au ( x ) = ∫ R d g ( y ) χ ( x − y ) dy = ( g ∗ χ )( x ) for any such x . (cid:3) 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 χ ∈ C ∞ c ( R d ) such that χ ≥
0, supp χ ⊆ B ( , ) , ∫ χ ( x ) dx = χ k ( x ) ∶= k d χ ( kx ) . If we define f k ( x ) ∶= ( f ∗ χ k )( x ) ∶= ∫ R d f ( y ) χ k ( x − y ) dy, then Lemma 3.3 shows that f k ∈ C b ( R d ) ⊆ D ( A ) and Af k ( x ) = g k ( x ) ∶= ( g ∗ χ k )( x ) for all x ∈ U with B ( x, / k ) ⊆ U . (20)(i) Set U δ ∶= { x ∈ U ; d ( x, U c ) > δ } for δ >
0. By Proposition 3.2, there exists a constant c = c ( δ ) such that ∥ f k ∥ C αb ( B ( x,δ / )) ≤ c (∥ f k ∥ ∞ + ∥ Af k ∥ ∞ ,U δ / ) for all x ∈ U δ and k ≥
1. Choosing k ≫ k < δ we find from (20) and ∥ f k ∥ ∞ ≤ ∥ f ∥ L ∞ ( R d ) that ∥ f k ∥ C αb ( B ( x,δ / )) ≤ c (∥ f ∥ L ∞ ( R d ) + ∥ g ∥ L ∞ ( U ) ) for all x ∈ U δ . (21)It follows from the Arzel`a-Ascoli theorem that there exists a continuous function f ( δ ) ∶ U δ → R such that a subsequence f k j converges pointwise to f ( δ ) on U δ . By (21), we have ∣ ∆ Nh f ( δ ) ( x )∣∣ h ∣ α = lim j →∞ ∣ ∆ Nh f k j ( x )∣∣ h ∣ α ≤ c (∥ f ∥ L ∞ ( R d ) + ∥ g ∥ L ∞ ( U ) ) for all x ∈ U δ and ∣ h ∣ ≤ δ / N ∈ { , , } is the smallest integer larger than α . Hence, ∥ f ( δ ) ∥ C αb ( U δ ) ≤ c (∥ f ∥ L ∞ ( R d ) + ∥ g ∥ L ∞ ( U ) ) . On the other hand, f k → f in L ( dx ) and so f = f ( δ ) Lebesgue almost everywhere on U δ . As f ( δ ′ ) = f ( δ ) on U δ for δ ′ < δ , the mapping˜ f ( x ) ∶= ⎧⎪⎪⎨⎪⎪⎩ f ( δ ) ( x ) , if x ∈ U δ , δ > ,f ( x ) , if x ∈ U c , is well defined. Clearly, ˜ f = f Lebesgue almost everywhere and the interior Schauder estimate ∥ ˜ f ∥ C αb ( U δ ) ≤ C δ (∥ f ∥ L ∞ ( R d ) + ∥ g ∥ L ∞ ( U ) ) holds for all δ > f ∈ C κb ( R d ) and g ∈ C κb ( U ) for some κ >
0. Since A ∣ C b ( R d ) commutes with the shift operator, i.e. ( Aϕ )( x + x ) = ( Aϕ (⋅ + x ))( x ) , ϕ ∈ C b ( R d ) , x, x ∈ R d , cf. (7), it follows easily by induction that A ( ∆ Nh ϕ ) = ∆ Nh ( Aϕ ) for all ϕ ∈ C b ( R d ) , N ∈ N . Hence, by (20), A ( ∆ Nh f k )( x ) = ( ∆ Nh Af k )( x ) = ∆ Nh g k ( x ) (22) NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 11 for all N ∈ N , x ∈ U δ , ∣ h ∣ < δ /( N ) and k ≫ k < δ . Let N ∈ N be the smallest number whichis strictly larger than κ . Applying part (i) to x ↦ ∆ Nh f k ( x ) , we obtain from (22) that ∥ ∆ Nh f k ∥ C αb ( U δ ) ≤ c (∥ ∆ Nh f k ∥ ∞ + ∥ ∆ Nh g k ∥ ∞ ,U δ ) for every δ ∈ ( , ) and some constant c = c ( δ ) > f , g and k . Since f ∈ C κb ( R d ) and g ∈ C κb ( U ) , this implies ∥ ∆ Nh f k ∥ C αb ( U δ ) ≤ c ∣ h ∣ κ (∥ f ∥ C κb ( R d ) + ∥ g ∥ C κb ( U ) ) for all ∣ h ∣ < δ /( N ) . If we denote by M ∈ N the smallest number strictly larger than α , then we get ∣ ∆ Mh ∆ Nh f k ( x )∣ ≤ c ∣ h ∣ κ + α (∥ f ∥ C κb ( R d ) + ∥ g ∥ C κb ( U ) ) for all x ∈ U δ and small h . The left-hand side converges to ∆ Mh ∆ Nh f ( x ) = ∆ M + Nh f ( x ) as k → ∞ because f is continuous. Applying Lemma 2.1 finishes the proof. (cid:3) For the proof of Corollary 1.4 we need another auxiliary result.3.4.
Lemma.
Let ( X t ) t ≥ be a L´evy process with characteristic exponent ψ satisfying the Hartman–Wintner condition (C1) . Let f ∈ B b ( R d ) be such that lim t → ∣ E f ( x + X t ) − f ( x )∣ = , x ∈ U, (23) for an open set U ⊆ R d . If ˜ f ∈ 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 u ∶= ˜ f U + f U c , satisfies u ∈ B b ( R d ) 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 Lebesguemeasure for t >
0, and so E u ( x + X t ) = E f ( x + X t ) for all x ∈ R d , t > . If we can show that lim t → E u ( x + x t ) = u ( x ) = ˜ f ( x ) for all x ∈ U, (24)then it follows immediately from (23) that f ( x ) = lim t → E f ( x + X t ) = lim t → E u ( x + X t ) = ˜ f ( x ) for all x ∈ U, which proves the assertion. To prove (24), fix x ∈ U . As ∣ E u ( x + X t ) − u ( x )∣ ≤ sup ∣ y ∣≤ δ ∣ u ( x + y ) − u ( x )∣ + ∥ u ∥ L ∞ ( R d ) P ( sup s ≤ t ∣ X s ∣ > δ ) , we find from the right-continuity of the sample paths and the monotone convergence theorem thatlim sup t → ∣ E u ( x + X t ) − u ( x )∣ ≤ sup ∣ y ∣≤ δ ∣ u ( x + y ) − u ( x )∣ . Since u is continuous at x , the right-hand side tends to 0 as δ →
0, and this gives (24). (cid:3)
Remark.
The proof of Lemma 3.4 shows the following statement: If ( X t ) t ≥ is a L´evy process,then lim t → E f ( x + X t ) = f ( x ) holds for any continuity point x of f ∈ B b ( R d ) ; this is a localized version of the continuity of thesemigroup T t f ( x ) ∶= E f ( x + X t ) at t = Proof of Corollary 1.4.
Let ϕ ∈ C ∞ c ( U ) . Because of the uniform boundedness assumption (5), itfollows from the dominated convergence theorem that ∫ R d g ( x ) ϕ ( x ) dx = lim t → t ( E ∫ R d f ( x + X t ) ϕ ( x ) dx − ∫ R d f ( x ) ϕ ( x ) dx ) . NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 12
Performing a change of variables we get ∫ R d g ( x ) ϕ ( x ) dx = lim t → t ( E ∫ R d f ( x ) ϕ ( x − X t ) dx − ∫ R d f ( x ) ϕ ( x ) dx )= ∫ R d f ( x ) A ∗ ϕ ( x ) dx, i.e. f is a weak solution to Af = g on U . If g ∈ L ∞ ( U ) then it follows from Theorem 1.2(i) thatthere exists a function ˜ f ∈ L ∞ ( R d ) such that f = ˜ f Lebesgue almost everywhere, ˜ f ∈ C ( U ) and ∥ ˜ f ∥ C αb ( U δ ) ≤ C δ (∥ f ∥ L ∞ ( R d ) + ∥ g ∥ ∞ ,U ) , δ > . (25)By Lemma 3.4, we have f = ˜ f on U , and consequently (25) holds with ˜ f replaced by f ; this provesthe first assertion. The second assertion follows directly from Theorem 1.2(ii). (cid:3) Proof of Corollary 1.5.
The first assertion is immediate from Corollary 1.4(i). For (ii), we notethat Af ∈ C κb ( R d ) implies f ∈ C κ + αb ( R d ) and ∥ f ∥ C κb ( R d ) ≤ ∥ f ∥ C κ + αb ( R d ) ≤ c (∥ f ∥ ∞ + ∥ Af ∥ ∞ ) , cf. [14, Theorem 1.1], and so Corollary 1.4(ii) applies. (cid:3) Examples
In this section, we present examples of L´evy processes for which interior Schauder estimates canbe obtained from Theorem 1.2, Corollary 1.4 and Corollary 1.5. We start with two tools whichare useful to construct wide classes of L´evy processes satisfying the assumptions (C1)-(C3) of ourmain results.4.1.
Proposition.
Let ( X ( i ) t ) t ≥ , i = , , be independent R d i -valued L´evy processes.(i) ( d = d ) If ( X ( ) t ) t ≥ satisfies (C1) and (C2) for some α > , then the L´evy process Y t ∶= X ( ) t + X ( ) t , t ≥ , satisfies (C1) and (C2) for the same constant α .(ii) If ( X ( i ) t ) t ≥ satisfies (C1) and (C2) for a constant α i > , i = , , then Z t ∶= ( X ( ) t X ( ) t ) , t ≥ , satisfies (C1) and (C2) with α ∶= min { α , α } .Proof. (i) Set d ∶= d = d . The characteristic exponent of ( Y t ) t ≥ equals ψ ∶= ψ ( ) + ψ ( ) where ψ ( i ) is the characteristic exponent of X ( i ) t , i = ,
2. In particular, Re ψ ≥ Re ψ ( ) , and therefore theHartman–Wintner condition (C1) for ψ ( ) implies that ψ satisfies (C1). Consequently, the law of Y t has a density p t with respect to Lebesgue measure for t >
0, and it satisfies p t ( x ) = ∫ R d p ( ) t ( x − y ) µ t ( dy ) , x ∈ R d , t > , where p ( ) t is the density of X ( ) t and µ t is the law of X ( ) t . As p ( ) t ∈ C ∞ b ( R d ) , cf. Remark 1.1(ii),it follows easily from the differentiation lemma for parametrized integrals, see e.g. [26], that ∇ p t ( x ) = ∫ R d ∇ x p ( ) t ( x − y ) µ t ( dy ) , x ∈ R d , t > . Hence, by Tonelli’s theorem and the gradient estimate (C2) for p ( ) t , ∫ R d ∣∇ p t ( x )∣ dx ≤ ∫ R d ∫ R d ∣∇ x p ( ) t ( x − y )∣ dx µ t ( dy ) ≤ ct − / α , t ∈ ( , ) . (ii) It is obvious that the characteristic exponent ( ξ, η ) ↦ ψ ( ) ( ξ )+ ψ ( ) ( η ) of ( Z t ) t ≥ satisfies (C1)whenever ψ ( ) and ψ ( ) satisfy (C1). Since the density of Z t is given by p t ( x, y ) = p ( ) t ( x ) p ( ) t ( y ) , x ∈ R d , y ∈ R d , t > NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 13 we can follow the reasoning from the first part, i.e. apply the differentiation lemma and Tonelli’stheorem, to find that ∫ R d + d ∣∇ p t ( z )∣ dz ≤ ct − / min { α ,α } , t ∈ ( , ) . (cid:3) Corollary.
Let ( X t ) t ≥ be a L´evy process with L´evy triplet ( b, Q, ν ) . If Q is positive definite,then the interior Schauder estimates in Theorem 1.2, Corollary 1.4 and Corollary 1.5 hold with α = . If there is no jump part, i.e. ν =
0, then the infinitesimal generator associated with ( X t ) t ≥ is givenby Af ( x ) = b ⋅ ∇ f ( x ) +
12 tr ( Q ⋅ ∇ f ( x )) , f ∈ C ∞ c ( R d ) , 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]. Proof of Corollary 4.2.
The L´evy process ( X t ) t ≥ has a representation of the form X t = QW t + J t , t ≥ , where ( B t ) t ≥ is a Brownian motion and ( J t ) t ≥ is a L´evy process with L´evy triplet ( b, , ν ) . Sincethe transition density of QB t is of Gaussian type and Q is positive definite, it is straightforward tocheck that ( QB t ) t ≥ satisfies (C1) and (C2) with α =
2. The L´evy process ( J t ) t ≥ is independent of ( QB t ) t ≥ , see e.g. [10, Theorem II.6.3], and therefore Proposition 4.1(i) shows that ( X t ) t ≥ satisfies(C1) and (C2) with α =
2. If we choose γ =
2, then ∫ ∣ y ∣≤ ∣ y ∣ γ ν ( dy ) < ∞ and the balance condition(C3) is trivial. Hence, the assumptions (C1)-(C3) of Theorem 1.2, Corollary 1.4 and Corollary 1.5are satisfied for α = γ = (cid:3) Our next result applies to a large class of jump L´evy processes, including stable L´evy processes.It is a direct consequence of the gradient estimates obtained in [27].4.3.
Corollary.
Let ( X t ) t ≥ be a pure-jump L´evy process. Assume that its L´evy measure ν satisfies ν ( B ) ≥ ∫ r ∫ S d − B ( rθ ) r − − (cid:37) µ ( dθ ) dr + ∫ ∞ r ∫ S d − B ( rθ ) r − − β µ ( dθ ) dr, B ∈ B ( R d /{ }) (26) for some constants r > , (cid:37) ∈ ( , ) , β ∈ ( , ∞] and a finite measure µ on the unit sphere S d − ⊆ R d which is non-degenerate, in the sense that its support is not contained in S d − ∩ V for some lower-dimensional subspace V ⊆ R d . If ∫ ∣ y ∣≤ ∣ y ∣ γ ν ( dy ) < ∞ (27) for some γ < + (cid:37) , then the interior Schauder estimates from Theorem 1.2, Corollary 1.4 andCorollary 1.5 hold with α = (cid:37) . If the L´evy measure ν equals the right-hand side of (26), then the assumption (27) is triviallysatisfied; this is, for instance, the case if ( X t ) t ≥ is isotropic α -stable or relativistic α -stable. Inparticular, Corollary 4.3 generalizes [24, Theorem 1.1].The next corollary gives a criterion for (C1)-(C3) in terms of the growth of the characteristicexponent of ( X t ) t ≥ .4.4. Corollary.
Let ( X t ) t ≥ be a L´evy process with infinitesimal generator ( A, D ( A )) . If thecharacteristic exponent ψ satisfies the sector condition, ∣ Im ψ ( ξ )∣ ≤ c Re ψ ( ξ ) , and Re ψ ( ξ ) ≍ ∣ ξ ∣ (cid:37) as ∣ ξ ∣ → ∞ (28) for some (cid:37) ∈ ( , ) , then the interior Schauder estimates in Theorem 1.2, Corollary 1.4 and Corol-lary 1.5 hold with α = (cid:37) .Proof. The Hartman–Wintner condition (C1) is trivially satisfied. It follows from [27] that thegradient estimate ∫ R d ∣∇ p t ( x )∣ dx ≤ ct − / (cid:37) holds for t ∈ ( , ) , i.e. α = (cid:37) in (C2). Moreover, (28)implies that the L´evy measure ν satisfies ∫ ∣ y ∣≤ ∣ y ∣ β ν ( dy ) < ∞ for all β > (cid:37) , cf. [17, Lemma A.2],and by choosing β close to (cid:37) , we find that the balance condition (C3) holds. (cid:3) NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 14
Corollary 4.4 covers many important and interesting examples, e.g. ● isotropic stable, relativistic stable and tempered stable L´evy processes, ● subordinated Brownian motions with characteristic exponent of the form ψ ( ξ ) = f (∣ ξ ∣ ) for a Bernstein function f satisfying f ( λ ) ≍ λ (cid:37) / for large λ , cf. [28] for details. ● L´evy processes with symbol of the form ψ ( ξ ) = ∣ ξ ∣ (cid:37) + ∣ ξ ∣ β , ξ ∈ R d , for β ∈ ( , (cid:37) ) .Our final example in this section is concerned with the operator Af ( x, y ) = − (− ∂ ∂x ) β / f ( x, y ) − (− ∂ ∂y ) β / f ( x, y ) , f ∈ C ∞ c ( R ) , x, y ∈ R , which arises as infinitesimal generator of a L´evy process ( X t ) t ≥ of the form X t = ( X ( ) t , X ( ) t ) where ( X ( i ) t ) t ≥ are independent one-dimensional isotropic stable L´evy processes with index β i ∈ ( , ] , i = ,
2, see e.g. [23] for more information. The difference ∣ β − β ∣ measures how much the behaviourof the first coordinate ( X ( ) t ) t ≥ differs from the behaviour of the second coordinate ( X ( ) t ) t ≥ . If ∣ β − β ∣ is large (i.e. close to 2), we are dealing with a highly anisotropic process.4.5. Example.
Let ( X t ) t ≥ be a two-dimensional L´evy process with characteristic exponent ψ ( ξ, η ) = ∣ ξ ∣ β + ∣ η ∣ β , ξ, η ∈ R for some constants β i ∈ ( , ] , i = , ∣ β − β ∣ < ∣ β − β ∣ < α = min { β , β } .Let us mention a further class of processes illustrating the role of the balance condition (C3).Farkas [5, Example 2.1.15] showed that for every 0 < β < α < ψ oscillates for ∣ ξ ∣ → ∞ between ∣ ξ ∣ β and 2 ∣ ξ ∣ α . Since thegrowth of ψ at infinity is closely linked to existence of fractional moments ∫ ∣ y ∣≤ ∣ y ∣ γ ν ( dy ) , it standsto reason that ∫ ∣ y ∣≤ ∣ y ∣ γ ν ( dy ) < ∞ only for γ > β and ∫ R d ∣∇ p t ( x )∣ dx ≤ M t − / α . In particular, thebalance condition (C3) fails if α − β > Appendix
A.For the proof of our results we used the following lemma, which was already stated in Section 2.A.1.
Lemma.
Let α ∈ ( , ∞) , and let U ⊆ R d be open. The following statements hold for any j ≥ k ∶= ⌊ α ⌋ + :(i) There exists a constant c > such that sup <∣ h ∣≤ r ∣ ∆ kh f ( x )∣∣ h ∣ α ≤ cr − α ∥ f ∥ ∞ ,U + c sup <∣ h ∣≤ r / j sup z ∈ B ( x,r ( k + )) ∣ ∆ jh f ( z )∣∣ h ∣ α (29) for all f ∈ C b ( U ) , r > , and x ∈ U with B ( x, r ( k + )) ⊆ U .(ii) If α > then there exists a constant c > such that max i = ,...,d ∣ ∂ x i f ( x )∣ ≤ cr − α ∥ f ∥ ∞ ,U + c sup <∣ h ∣≤ r / j sup z ∈ B ( x,r ( k + )) ∣ ∆ jh f ( z )∣∣ h ∣ α (30) for all f ∈ C b ( U ) , r > and x ∈ U with B ( x, r ( k + )) ⊆ U .Proof. First of all, we note that it suffices to prove both statements for f ∈ C b ( U ) ; the inequalitiescan be extended using a standard approximation technique, i.e. by considering f i ∶= f ∗ ϕ i for asequence of mollifiers ( ϕ i ) i ≥ .Denote by τ h f ( x ) ∶= f ( x + h ) the shift operator. A straight-forward computation shows that∆ nh ( u ⋅ v ) = n ∑ (cid:96) = ( n(cid:96) ) ∆ (cid:96)h u ⋅ ∆ n − (cid:96)h τ (cid:96)h v (31)holds for any n ∈ N , h ∈ R d and any two functions u, v . NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 15
To prove (i) we note that the assertion is obvious for j = k , and so it suffices to consider j > k . Wewill first establish the following auxiliary statement: There exists a constant C > <∣ t ∣≤ ∣ ∆ kt g ( )∣∣ t ∣ α ≤ C ∥ g ∥ ∞ , (− k − ,k + ) + C sup <∣ t ∣≤ / j sup ∣ y ∣≤ k + ∣ ∆ jt g ( y )∣∣ t ∣ α (32)for any twice differentiable bounded function g ∶ (− k − , k + ) → R . To this end, pick χ ∈ C ∞ c ( R ) such that [− k,k ] ≤ χ ≤ (− k − / ,k + / ) . Clearly,sup <∣ t ∣≤ ∣ ∆ kt g ( )∣∣ t ∣ α = sup <∣ t ∣≤ ∣ ∆ kt ( gχ )( )∣∣ t ∣ α . Using the equivalence of the norms on C αb ( R ) , cf. (6), we getsup <∣ t ∣≤ ∣ ∆ kt g ( )∣∣ t ∣ α ≤ ∥ gχ ∥ C αb ( R ) ≤ c ∥ gχ ∥ ∞ + c sup ∣ t ∣≤ sup y ∈ R ∣ ∆ jt ( gχ )( y )∣∣ t ∣ α ≤ c ′ ∥ g ∥ ∞ , (− k − ,k + ) + c sup ∣ t ∣≤ /( j ) sup y ∈ R ∣ ∆ jt ( gχ )( y )∣∣ t ∣ α for some constants c and c ′ . As χ = R /(− k − / , k + / ) , we have ∆ jt g ( y ) = ∣ y ∣ > k + and ∣ t ∣ ≤ j . Consequently,sup <∣ t ∣≤ ∣ ∆ kt g ( )∣∣ t ∣ α ≤ c ′ ∥ g ∥ ∞ , (− k − ,k + ) + c sup ∣ t ∣≤ /( j ) sup ∣ y ∣≤ k + / ∣ ∆ jt ( gχ )( y )∣∣ t ∣ α . (33)Since ∣ ∆ (cid:96)t g ( y )∣ = ∣ ∆ (cid:96) − jt ∆ jt g ( y )∣ ≤ (cid:96) − j sup ∣ z − y ∣≤( (cid:96) − j ) t ∣ ∆ jt g ( z )∣ , (cid:96) ≥ j, and ∥ ∆ (cid:96)t χ ∥ ∞ ≤ c ∣ t ∣ k ∥ χ ∥ C kb ( R ) , (cid:96) ≥ j ≥ k, an application of the product formula (31) gives ∣ ∆ jt ( gχ )( y )∣ ≤ c ∥ g ∥ ∞ , (− k − ,k + ) j − ∑ (cid:96) = ∥ ∆ j − (cid:96)t χ ∥ ∞ + c ∥ χ ∥ ∞ j ∑ (cid:96) = j ∣ ∆ (cid:96)t g ( y )∣≤ c ∥ g ∥ ∞ , (− k − ,k + ) ∣ t ∣ k ∥ χ ∥ C kb ( R ) + c sup ∣ z − y ∣≤ tj ∣ ∆ jt g ( z )∣ for all ∣ y ∣ ≤ k + and ∣ t ∣ ≤ j . Combining this estimate with (33) and noting that α ≤ k proves (32).Now if f ∈ C b ( U ) , then we apply (32) with g ( t ) ∶= f ( x + rth ) for fixed ∣ h ∣ = α ∈ ( , ) and j =
2. The auxiliary inequalitywhich we need is ∣ g ′ ( )∣ ≤ C ∥ g ∥ ∞ , (− , ) + C sup ∣ t ∣≤ / sup ∣ y ∣≤ ∣ ∆ t g ( y )∣∣ t ∣ α (34)for a uniform constant C > g ∶ R → R is differentiable on (− , ) . To this end, choose χ ∈ C c ( R ) with [− , ] ≤ χ ≤ (− / , / ) . By the equivalence of the norms on the H¨older–Zygmundspace C αb ( R ) , cf. (6), we get ∣ g ′ ( )∣ = ∣( gχ ) ′ ( )∣ ≤ ∥ gχ ∥ C b ( R ) ≤ ∥ gχ ∥ C αb ( R ) ≤ c ∥ gχ ∥ ∞ + c sup ∣ t ∣≤ sup y ∈ R ∣ ∆ t ( gχ )( y )∣∣ t ∣ α for some finite constant c >
0. Following the reasoning in the first part of the proof (with k = j =
2) yields (34). Applying (34) for g ( t ) ∶= f ( x + rte j ) , where e j is the j -th unit vector in R d , gives (ii) for α ∈ ( , ) and j =
2. In combination with (i), this yields the desired inequality forevery α > j ≥ ⌊ α ⌋ + (cid:3) NTERIOR SCHAUDER ESTIMATES FOR EQUATIONS ASSOCIATED WITH L´EVY OPERATORS 16
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TU Dresden, Fachrichtung Mathematik, Institut f¨ur Mathematische Stochastik, 01062 Dres-den, Germany.
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