On the Cauchy problem for integro-differential equations with space-dependent operators in generalized Hölder classes
aa r X i v : . [ m a t h . P R ] O c t ON THE CAUCHY PROBLEM FORINTEGRO-DIFFERENTIAL EQUATIONS WITHSPACE-DEPENDENT OPERATORS IN GENERALIZEDH ¨OLDER CLASSES
FANHUI XU
Abstract.
Parabolic integro-differential Kolmogorov equations withdifferent space-dependent operators are considered in H¨older-type spacesdefined by a scalable L´evy measure. Probabilistic representations areused to prove continuity of the operator. Existence and uniqueness ofthe solution are established and some regularity estimates are obtained.
Contents
1. Introduction 22. Notation and Function Spaces 72.1. Basic Notation 72.2. Function Spaces of Generalized Smoothness 83. Continuity of the Operator 103.1. Operators with Space-Dependent Kernels 153.2. Operators with Space-Dependent Coefficients 193.3. Lower Order Operators 264. Proof of the Main Result 314.1. Auxiliary Results 314.2. Proof of Theorem 1.1 335. Appendix 38Acknowledgments 38References 38
Date : September 22, 2018.1991
Mathematics Subject Classification.
Key words and phrases.
Generalized H¨older smoothness, non-local parabolic Kol-mogorov equations, L´evy processes, strong solutions. Introduction
Let (Ω , F , P ) be a complete probability space and ν be a L´evy measureon R d = R d \{ } that is of order α , i.e. α := inf { σ ∈ (0 ,
2) : Z | y |≤ | y | σ ν ( dy ) < ∞} . We denote by J ( ds, dy ) a Poisson random measure on (Ω , F , P ) such that E [ J ( ds, dy )] = ν ( dy ) ds , and denote by Z νt the L´evy process Z νt = Z t Z R d χ α ( y ) y ˜ J ( ds, dy ) + Z t Z R d (1 − χ α ( y )) yJ ( ds, dy ) . (1.1)Here χ α ( y ) := 1 α ∈ (1 , + 1 α =1 | y |≤ , and˜ J ( ds, dy ) := J ( ds, dy ) − ν ( dy ) ds is the compensated Poisson measure.This work is a continuation of [15], in which we studied the Cauchy prob-lem for the following parabolic-type Kolmogorov equations in generalizedH¨older spaces ˜ C β (cid:0) R d (cid:1) endowed with norms | · | β (see Section 2.2): ∂ t u ( t, x ) = L ν u ( t, x ) − λu ( t, x ) + f ( t, x ) , λ ≥ , (1.2) u (0 , x ) = 0 , ( t, x ) ∈ [0 , T ] × R d , where L ν is the infinitesimal generator of Z νt . Namely, for any ϕ ∈ C ∞ (cid:0) R d (cid:1) , L ν ϕ ( x ) := Z [ ϕ ( x + y ) − ϕ ( x ) − χ α ( y ) y · ∇ ϕ ( x )] ν ( dy ) . (1.3)A notion of scaling functions was utilized in [15] to include some recentpopular models of ν (cf. [7, 8, 18]). Definition 1.
A continuous function w : (0 , ∞ ) → (0 , ∞ ) is called a scalingfunction if lim r → w ( r ) = 0 , lim R →∞ w ( R ) = ∞ and if there is a nondecreasing continuous function l ( ε ) , ε > such that lim ε → l ( ε ) = 0 and (1.4) w ( εr ) ≤ l ( ε ) w ( r ) , ∀ r, ε > .l is called the scaling factor of w . For any L´evy measure ν , any R > ∀ B ∈ B (cid:0) R d (cid:1) , ν R ( B ) := Z B ( y/R ) ν ( dy ) , (1.5) ˜ ν R ( dy ) := w ( R ) ν R ( dy ) . (1.6)We can always normalize w by a constant so that w (1) = 1 and ˜ ν ( dy ) = ν ( dy ). It was imposed in [15] for ν : HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 3
A(w,l) (i) (Non-degeneracy) Suppose ˜ ν R ( dy ) ≥ µ ( dy ) , R > µ that is supported on the unit ball B (0), with µ satisfying(1.7) Z | y | µ ( dy ) + Z | ξ | [1 + υ ( ξ )] d +3 exp {− ζ ( ξ ) } dξ < ∞ , where υ ( ξ ) = Z χ α ( y ) | y | [( | ξ | | y | ) ∧ µ ( dy ) ,ζ ( ξ ) = Z [1 − cos (2 πξ · y )] µ ( dy ) . In addition, for all ξ ∈ S d − = { ξ ∈ R d : | ξ | = 1 } , there is a constant c > Z | y |≤ | ξ · y | µ ( dy ) ≥ c . (ii) (Symmetry) If α = 1, then(1.8) Z r< | y | 1) if α ∈ (0 , α , α ∈ (1 , 2] if α ∈ (1 , α ∈ (1 , 2] and α ∈ [0 , 1) if α = 1,and Z | y |≤ | y | α ˜ ν R ( dy ) + Z | y | > | y | α ˜ ν R ( dy ) ≤ N . The N > R .(iv) (Scalability) Suppose ς ( r ) := ν ( | y | > r ) , r > r and Z sς ( rs ) ς ( r ) − ds ≤ C for some positive C independent of r .Under A(w,l) , Z νt possesses a smooth density function whose regularityestimates were derived in [9]. Moreover, Z νt is approximately distributed as R Z νw ( R ) t , R > 0. This property gives a uniform description of L´evy measuresthat were considered in [18], [7] and [8]. In [18], ν is assumed to be confinedby two α -stable measures of the same order, namely, Z S d − Z ∞ B ( rw ) drr α Σ ( dw ) ≤ ν ( B ) ≤ Z S d − Z ∞ B ( rw ) drr α Σ ( dw )(1.9)for any Borel measurable set B . They also assumed Σ and Σ are twofinite measures defined on the unit sphere and Σ is nondegenerate. In FANHUI XU this situation, ν satisfies A(w,l) with w ( r ) = l ( r ) = r α , r > 0. Anotherinteresting class of L´evy measures was investigated in [7] and [8], where(1.10) ν ( B ) = Z ∞ Z | w | =1 B ( rw ) a ( r, w ) j ( r ) r d − Σ ( dw ) dr, ∀ B ∈ B (cid:16) R d (cid:17) , Σ ( dw ) is a finite measure on the unit sphere, and j ( r ) = Z ∞ (4 πt ) − d/ exp (cid:18) − r t (cid:19) Λ ( dt ) , r > , with Λ ( dt ) being a measure on (0 , ∞ ) such that R ∞ (1 ∧ t ) Λ ( dt ) < ∞ . Let φ ( r ) = R ∞ (cid:0) − e − rt (cid:1) Λ ( dt ) , r ≥ H. there is a function ρ ( w ) defined on the unit sphere such that ρ ( w ) ≤ a ( r, w ) ≤ , ∀ r > 0, and for all | ξ | = 1, Z S d − | ξ · w | ρ ( w ) ≥ c > . G. (i) There is C > C φ (cid:0) r − (cid:1) r − d ≤ j ( r ) ≤ Cφ (cid:0) r − (cid:1) r − d . (ii) There are 0 < σ ≤ σ < C > < r ≤ RC − (cid:18) Rr (cid:19) σ ≤ φ ( R ) φ ( r ) ≤ C (cid:18) Rr (cid:19) σ . It can be verified that H and G produce L´evy measures of A(w,l) -typewith w ( r ) = j ( r ) − r − d , r > 0, and l ( r ) = (cid:26) Cr σ if r ≤ ,Cr σ if r > C > 0. (See [7, 8, 9, 18] for details and examples.)Write H T = [0 , T ] × R d . In this note, we consider the following parabolicintegro-differential equation: ∂ t u ( t, x ) = L u ( t, x ) − λu ( t, x ) + f ( t, x ) , λ ≥ , (1.11) u (0 , x ) = 0 , ( t, x ) ∈ H T , where L = A + Q or L = G + Q , and for any function ϕ ∈ C ∞ (cid:0) R d (cid:1) , A ϕ ( x ) := Z [ ϕ ( x + y ) − ϕ ( x ) − χ α ( y ) y · ∇ ϕ ( x )] ρ ( t, x, y ) ν ( dy ) , G ϕ ( x ) := Z [ ϕ ( x + G ( x ) y ) − ϕ ( x ) − χ α ( y ) G ( x ) y · ∇ ϕ ( x )] ν ( dy ) , Q ϕ ( x ) := 1 α ∈ (1 , b ( t, x ) · ∇ ϕ ( x ) + p ( t, x ) ϕ ( x ) + Z R d [ ϕ ( x + q ( t, x, y )) − ϕ ( x ) − ∇ ϕ ( x ) · q ( t, x, y ) 1 α ∈ (1 , | y |≤ ] ̺ ( t, x, y ) ν ( dy ) . (1.12) HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 5 We assume for the underlying L´evy measure ν : ˜A ( w, l, γ ). (i) ν satisfies A(w,l) .(ii) There is ε ∈ (0 , 1) such that for any β ′ ∈ (0 , β + ε ), Z l ( t ) β ′ dtt + Z ∞ l ( t ) β ′ dtt + 1 α ∈ [1 , Z l ( t ) β ′ dtt < ∞ . (iii) Set γ ( t ) = inf { s > l ( s ) ≥ t } for t > 0. There exist 0 < δ < min (cid:0) , β (cid:1) and 0 < δ ′ < min (cid:0) , ε (cid:1) for the ε in (ii) such that1 α ∈ (0 , Z ∞ t δ γ ( t ) − dt < ∞ , α =1 (cid:18)Z t δ γ ( t ) − dt + Z ∞ t − δ ′ γ ( t ) − + t δ γ ( t ) − dt (cid:19) < ∞ , α ∈ (1 , (cid:18)Z t − δ γ ( t ) − dt + Z ∞ t δ γ ( t ) − + t − + δ γ ( t ) − dt (cid:19) < ∞ . Suppose the kernel function ρ satisfies H( K, β ) . (i) There is K > ∀ t ∈ [0 , T ], | ρ ( t, x, y ) | ≤ K, ∀ x, y ∈ R d , (1.13) | ρ ( t, x , y ) − ρ ( t, x , y ) | ≤ Kw ( | x − x | ) β , ∀ y ∈ R d . (1.14)(ii) If α = 1, then for ∀ x ∈ R d , ∀ r ∈ (0 , , ∀ t ∈ [0 , T ], Z r< | y | < yρ ( t, x, y ) ν ( dy ) = 0 . We assume for the main part G : G( c , K, β ) . (i) G ( z ) , z ∈ R d is an invertible and uniform continuous d × d -matrix, and G ( z ) = G ( z ′ ) if z = z ′ .(ii) | det G ( z ) | ≥ c , k G ( z ) k ≤ K, ∀ z ∈ R d for some c , K > K , g ( z, z ′ ) ≤ Kw ( | z − z ′ | ) β , ∀ z, z ′ ∈ R d , where ¯ G z,z ′ := k G ( z ) − G ( z ′ ) k and g (cid:0) z, z ′ (cid:1) = w (cid:16) ¯ G − z,z ′ (cid:17) − if α ∈ (0 , ,w (cid:16) ¯ G − z,z ′ (cid:17) − w (cid:0) ¯ G z,z ′ (cid:1) − δ ′ ∨ ¯ G z,z ′ if α = 1 , ¯ G z,z ′ if α ∈ (1 , . For the same w, l, K, β , we assume the lower order part Q satisfies: B( K, β ) . (i) | b ( t, · ) | β + | p ( t, · ) | β + | ̺ ( t, · , y ) | β ≤ K, ∀ y ∈ R d , ∀ t ∈ [0 , T ].(ii) For all α ∈ (0 , , z ′ ∈ R d , ∀ t ∈ [0 , T ], q ( t, · , y ) = 0 if y = 0. Besides,lim ε → sup t,z ′ Z | q ( t,z ′ ,y ) |≤ ε (cid:0) w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) + 1 α =1 (cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ν ( dy ) = 0 . (iii) For all α ∈ (0 , , z ′ ∈ R d , ∀ t ∈ [0 , T ], Z R d α< (cid:0) w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ∧ (cid:1) + 1 α =1 (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ∧ (cid:1) ν ( dy ) ≤ K. FANHUI XU (iv) For all α ∈ (1 , , z ′ ∈ R d , ∀ t ∈ [0 , T ], Z | y |≤ w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ν ( dy ) + Z | y | > (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ∧ (cid:1) ν ( dy ) ≤ K. (v) For all z ′ , h ∈ R d , ∀ t ∈ [0 , T ], α ∈ (1 , Z | y |≤ w ( | q ( t, x + h, y ) | ) β | q ( t, x + h, y ) − q ( t, x, y ) | ν ( dy ) ≤ Kw ( | h | ) β , Z | y |≤ w ( | q ( t, x + h, y ) − q ( t, x, y ) | ) β | q ( t, x, y ) | ν ( dy ) ≤ Kw ( | h | ) β , Z | y | > (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ + h, y (cid:1) − q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ∧ (cid:1) ν ( dy ) ≤ Kw ( | h | ) β . If α ∈ (0 , Z R d (cid:0) w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ + h, y (cid:1) − q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ∧ (cid:1) ν ( dy ) ≤ Kw ( | h | ) β . And if α = 1, Z R d (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ + h, y (cid:1) − q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ∧ (cid:1) ν ( dy ) ≤ Kw ( | h | ) β . The main conclusion of this paper is Theorem 1.1. Let ˜A ( w, l, γ ) , B( K, β ) and H( K, β ) (resp. G( c , K, β ) )hold. If f ( t, x ) ∈ ˜ C β ( H T ) , β ∈ (cid:0) , α (cid:1) , then there is a unique solution u ( t, x ) ∈ ˜ C β ( H T ) to (1.11) with L = A + Q (resp. L = G + Q ). Moreover,there exists a constant C depending on c , c , N , K, β, d, T , µ, ν such that | u | β ≤ C (cid:0) λ − ∧ T (cid:1) | f | β , | u | β ≤ C | f | β . And there is a constant C depending on c , c , N , K, κ, β, d, T, µ, ν such thatfor all ≤ s < t ≤ T , κ ∈ [0 , and κ + β > , | u ( t, · ) − u ( s, · ) | κ + β ≤ C | t − s | − κ | f | β . Due to generality of the measure ν we are considering, the L´evy symbol ψ ν ( ξ ) , ξ ∈ R d of the process Z νt is generally not smooth in ξ . This wasalready an obstacle for applying the standard Fourier multiplier theorem tosolutions of equations with space-independent coefficients, and it continuesto be a difficulty in this work. Thus, probabilistic representations are usedinstead and continuity of the operators are proved in that approach. Thenwe apply continuation of parameters, which was also used in [13] and [14],to show well-posedness of the Cauchy problem. In [14], a parabolic-typeKolmogorov equation with an operator L = A + Q was considered in the HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 7 standard H¨older-Zygmund space, where Q is the lower order part and theprincipal part A u ( t, x ) := Z [ u ( t, x + y ) − u ( t, x ) − χ α ( y ) y · ∇ u ( t, x )] ρ ( t, x, y ) dy | y | d + α . With more flavor of probability, in [13] a stochastic parabolic integro-differentialequation with operators L u ( t, x ) := Z (cid:2) u ( t, x + y ) − u ( t, x ) − α ≥ | y |≤ y · ∇ u ( t, x ) (cid:3) ν ( t, x, dy )+ 1 α =2 a ij ( t, x ) ∂ ij u ( t, x ) + 1 α ≥ ˜ b i ( t, x ) ∂ i u ( t, x ) + l ( t, x ) u ( t, x )was studied in H¨older spaces. A deterministic model with a similar operatorwas addressed in the little H¨older-Zygmund spaces in [12]. Besides, theCauchy problem for a second order linear SPDE was considered in [11] and[16] in standard H¨older classes.The outline of this note is as follows.In section 2, notation is introduced. Definitions of function spaces andresults on norm equivalence from [15] are briefly mentioned at the conve-nience of readers. In section 3, we show continuity of the operators by usingprobability representations. In section 4, we derive some a priori estimatesand prove the main theorem by applying continuation of parameters. Otherauxiliary results are collected in the Appendix section.2. Notation and Function Spaces Basic Notation. We use N for the set of nonnegative integers, N + for N \{ } , and ℜ for the real part of a complex-valued quantity.For a function u = u ( t, x ) on H T = [0 , T ] × R d , ∂ t u := ∂u/∂t , ∂ i u := ∂u/∂x i , ∂ ij u := ∂ u/∂x i x j . The gradient of u with respect to x is denotedby ∇ u , and D | γ | u := ∂ | γ | u/∂x γ . . . ∂x γ d d , where γ = ( γ , . . . , γ d ) ∈ N d is amulti-index.As usual, C ∞ b (cid:0) R d (cid:1) denotes the set of infinitely differentiable functions on R d whose derivative of arbitrary order is finite, S (cid:0) R d (cid:1) is the space of rapidlydecreasing functions on R d and S ′ (cid:0) R d (cid:1) denotes the space of continuousfunctionals on S (cid:0) R d (cid:1) . It is well-known that Fourier transform is a bijectionon S ′ (cid:0) R d (cid:1) . We adopt the normalized definition for Fourier and its inversetransforms in this note, i.e., F ϕ ( ξ ) = ˆ ϕ ( ξ ) := Z e − i πx · ξ ϕ ( x ) dx, F − ϕ ( x ) = ˇ ϕ ( x ) := Z e i πx · ξ ϕ ( ξ ) dξ, ϕ ∈ S (cid:16) R d (cid:17) . For any L´evy measure ν we may symmetrize it as below.¯ ν ( dy ) := 12 ( ν ( dy ) + ν ( − dy )) . (2.1) FANHUI XU As a convention, C is a positive constant that represents different values invarious contexts. Explicit dependence on certain quantities may be indicatedwhen necessary.2.2. Function Spaces of Generalized Smoothness. Our primary func-tion spaces of generalized smoothness in this note are ˜ C β (cid:0) R d (cid:1) , β ∈ (0 , /α )endowed with the norm | u | β = sup t,x | u ( t, x ) | + sup t,x,h =0 | u ( t, x + h ) − u ( t, x ) | w ( | h | ) β := | u | + [ u ] β < ∞ and ˜ C β (cid:0) R d (cid:1) , β ∈ (0 , /α ) with the norm | u | β := | u | + | L µ u | + [ L µ u ] β < ∞ , where µ is a reference measure satisfying A(w,l) for the same w and l as ν ,and L µ is the associated operator defined as (1.3).By [15, Proposition 1], these generalized H¨older norms are equivalent tothe norm of generalized Besov spaces ˜ C β ∞ , ∞ (cid:0) R d (cid:1) : | u | β, ∞ = sup j ∈ N w (cid:0) N − j (cid:1) − β | u ∗ ϕ j | < ∞ , β ∈ (0 , ∞ ) . Given the choice of [15], in above definition ϕ j ∈ S (cid:0) R d (cid:1) for any j ∈ N and P ∞ j =0 F ϕ j = 1. Moreover, when j ≥ F ϕ j = φ (cid:0) N − j · (cid:1) for some N such that l (cid:0) N − (cid:1) < < l ( N ) and for some φ ∈ C ∞ (cid:0) R d (cid:1) so that supp ( φ ) = { ξ : N − ≤ | ξ | ≤ N } .Set κ ∈ [0 , 1] and β > 0. Denote the L´evy symbol associated with L µ by ψ µ ( ξ ) = Z h e i πξ · y − − i πχ α ( y ) ξ · y i µ ( dy ) , ξ ∈ R d , and denote ψ µ,κ = ψ µ if κ = 1 , − ( −ℜ ψ µ ) κ if κ ∈ (0 , , κ = 0 . Then the auxiliary space C µ,κ,β (cid:0) R d (cid:1) is a class of functions whose norm | u | µ,κ,β := | u | + | L µ,κ u | β, ∞ < ∞ , where L µ,κ u := F − [ ψ µ,κ F u ] , u ∈ S ′ (cid:16) R d (cid:17) . (2.2)Set ( I − L µ ) κ u = (cid:26) ( I − L µ ) u if κ = 1 , F − [(1 − ℜ ψ µ ) κ F u ] if κ ∈ [0 , . Another auxiliary space ˜ C µ,κ,β (cid:0) R d (cid:1) is introduced as the collection of func-tions whose norm k u k µ,κ,β := | ( I − L µ ) κ u | β, ∞ < ∞ . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 9 Lemmas 1-3 below comprise a list of probabilistic representations thatwere derived in [15] and will be intensively used in next section. Lemma 1. [15, Lemma 8] Let ν be a L´evy measure satisfying (iii) in A(w,l) and L ˜ ν R ,κ be defined as (2.2) . Then for any ϕ ( x ) ∈ C ∞ b (cid:0) R d (cid:1) , L ˜ ν R ,κ ϕ ( x ) = C Z ∞ t − − κ E h ϕ (cid:16) x + Z ˜ ν R t (cid:17) − ϕ ( x ) i dt, κ ∈ (0 , , where C − = R ∞ t − κ − (cid:0) − e − t (cid:1) dt and ˜ ν R ( dy ) = 12 (˜ ν R ( dy ) + ˜ ν R ( − dy )) , R > . Besides, L ˜ ν R ,κ ϕ ∈ C ∞ b (cid:0) R d (cid:1) . If furthermore ϕ ( x ) ∈ S (cid:0) R d (cid:1) , then (cid:12)(cid:12) L ˜ ν R ,κ ϕ (cid:12)(cid:12) L Let a > and ν be a L´evy measure satisfying(iii) in A(w,l) . Then aI − L ν defines a bijection on C ∞ b (cid:0) R d (cid:1) . Moreover,for all C ∞ b (cid:0) R d (cid:1) functions ϕ , the following representations hold. ϕ ( x ) = Z ∞ e − at E ( aI − L ν ) ϕ ( x + Z νt ) dt, ( aI − L ν ) − ϕ ( x ) = Z ∞ e − at E ϕ ( x + Z νt ) dt, x ∈ R d . Lemma 3. [15, Lemma 10] Let a > and κ ∈ (0 , . Suppose ν is aL´evy measure satisfying (iii) in A(w,l) . Then ( aI − L ν ) κ is a bijection on C ∞ b (cid:0) R d (cid:1) . Moreover, for all C ∞ b (cid:0) R d (cid:1) functions ϕ , ( aI − L ν ) κ ϕ ( x ) = C Z ∞ t − κ − (cid:2) ϕ ( x ) − e − at E ϕ (cid:0) x + Z ¯ νt (cid:1)(cid:3) dt, (2.3) ( aI − L ν ) − κ ϕ ( x ) = C ′ Z ∞ t κ − e − at E ϕ (cid:0) x + Z ¯ νt (cid:1) dt, (2.4) where C − = R ∞ t − κ − (cid:0) − e − t (cid:1) dt , C ′− = R ∞ t κ − e − t dt and Z ¯ νt is theL´evy process associated with ¯ ν . Remark: Lemmas 1 and 3 imply that L µ,κ , ( aI − L ν ) κ , ( aI − L ν ) − κ , κ ∈ (0 , 1] are closed operations in C ∞ b (cid:0) R d (cid:1) . Therefore, they may be all ex-tended to κ ∈ (1 , 2) through composition of operators. It was shown in [15,Corollary 2] that (2.4) also holds for κ ∈ (1 , Lemma 4. [15, Lemma 6 and Proposition 6] Let β > and κ ∈ [0 , .Suppose ν is a L´evy measure satisfying A(w,l) . Then (2.2) is well-definedfor all κ and all u ∈ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) , L ν,κ u ( x ) = lim n →∞ L ν,κ u n ( x ) , x ∈ R d , (2.5) and this convergence is uniform with respect to x . Moreover, | L ν,κ u | ≤ | L ν,κ u | β, ∞ ≤ C | u | κ + β, ∞ for some C > independent of u . Based on Lemmas 1-4, norm equivalence were established. Proposition 1. [15, Theorems 3.2 and 3.3] Let ν be a L´evy measure satis-fying A(w,l) , β > , κ ∈ (0 , . Then norms | u | ν,κ,β , k u k ν,κ,β and | u | κ + β, ∞ are mutually equivalent. Continuity of the Operator In this section, we study respectively operators that have a kernel de-pending on the spatial variable x and operators that have space-dependentcoefficients. The first lemma explains the relation between generalized reg-ularity and the ordinary smoothness. Lemma 5. Let β, δ ∈ (0 , ∞ ) , σ ∈ [0 , and k be a positive integer so that Z l ( t ) β t − k − dt < ∞ . a) Any function u ∈ ˜ C β ∞ , ∞ (cid:0) R d (cid:1) is k -times continuously differentiable andthere is C depending only on N, β so that for any multi-index | γ | ≤ k andany σ ∈ [0 , with | γ | + σ ≤ k , | ∂ σ D γ u | ≤ C | u | β, ∞ Z l ( t ) β t −| γ |− σ − dt. Moreover, ∂ σ D γ u = D γ ∂ σ u = ∞ X j =0 ( ∂ σ D γ u ) ∗ ϕ j converges uniformly.b) Any function u ∈ ˜ C β + δ ∞ , ∞ (cid:0) R d (cid:1) is k -times continuously differentiable andthere is C depending only on N, β so that for any multi-index | γ | ≤ k andany σ ∈ [0 , with | γ | + σ ≤ k , | ∂ σ D γ u | δ, ∞ ≤ C | u | β + δ, ∞ Z l ( t ) β t −| γ |− σ − dt. Proof. Recall properties of the convolution functions ϕ j , j ∈ N . If we write˜ ϕ j = ϕ j − + ϕ j + ϕ j +1 , j ≥ , ˜ ϕ = ˇ φ + ϕ + ϕ , ˜ ϕ = ϕ + ϕ , then, F ˜ ϕ j ( ξ ) = ˆ˜ ϕ j ( ξ ) = F ˜ ϕ (cid:0) N − j ξ (cid:1) , ξ ∈ R d , j ≥ , where F ˜ ϕ ( ξ ) = φ ( N ξ ) + φ ( ξ ) + φ (cid:0) N − ξ (cid:1) . Note that φ is necessarily 0 on the boundary of its support. Then, ϕ j = ϕ j ∗ ˜ ϕ j , j ≥ , ˜ ϕ j ( x ) = N jd ˜ ϕ (cid:0) N j x (cid:1) , j ≥ . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 11 And then, u = ∞ X j =0 u ∗ ϕ j = ∞ X j =0 ˜ ϕ j ∗ u ∗ ϕ j . a) We only show cases in which | γ | = 1. The proof for other higher ordersis an application of induction on γ . Denote( ∂ σ D γ ˜ ϕ ) j ( x ) = N jd ( ∂ σ D γ ˜ ϕ ) (cid:0) N j x (cid:1) , x ∈ R d , j ≥ . Then ∞ X j =1 ∂ σ D γ ( ˜ ϕ j ∗ u ∗ ϕ j ) = ∞ X j =1 N ( | γ | + σ ) j ( ∂ σ D γ ˜ ϕ ) j ∗ u ∗ ϕ j . Since ∞ X j =0 w (cid:0) N − j (cid:1) β ( N − j ) | γ | + σ ≤ w ( N ) β Z ∞ l ( N − x ) β ( N − x ) | γ | + σ dx ≤ C Z l ( t ) β t | γ | + σ dtt < ∞ , we have ∞ X j =0 | ∂ σ D γ ( ˜ ϕ j ∗ u ∗ ϕ j ) | ≤ C ∞ X j =0 w (cid:0) N − j (cid:1) β N ( | γ | + σ ) j w (cid:0) N − j (cid:1) − β | u ∗ ϕ j | ≤ C | u | β, ∞ Z l ( t ) β t −| γ |− σ − dt < ∞ . Therefore, P ∞ j =0 ∂ σ D γ ( ˜ ϕ j ∗ u ∗ ϕ j ) ∈ C (cid:0) R d (cid:1) converges uniformly and there-fore it converges in the weak topology of S ′ ( R ). By continuity of the Fouriertransform, D γ ∂ σ u = ∂ σ D γ u = ∞ X j =0 ∂ σ D γ ( ˜ ϕ j ∗ u ∗ ϕ j ) = ∞ X j =0 ( ∂ σ D γ u ) ∗ ϕ j . Moreover, | ∂ σ D γ u | ≤ ∞ X j =0 | ∂ σ D γ ( ˜ ϕ j ∗ u ∗ ϕ j ) | ≤ C | u | β, ∞ Z l ( t ) β t −| γ |− σ − dt. b) From a), w (cid:0) N − j (cid:1) − δ | ( ∂ σ D γ u ) ∗ ϕ j | ≤ C ∞ X j =0 w (cid:0) N − j (cid:1) β N ( | γ | + σ ) j w (cid:0) N − j (cid:1) − β − δ | u ∗ ϕ j | ≤ C | u | β + δ, ∞ Z l ( t ) β t −| γ |− σ − dt, ∀ j ∈ N . And the conclusion follows. (cid:3) Remark: As a conclusion of Lemma 5 and ˜A ( w, l, γ ) (iii), if u ∈ ˜ C β ∞ , ∞ (cid:0) R d (cid:1) , β > α ∈ [1 , u has classical first-order derivatives. Lemma 6. Let κ ∈ (0 , and µ be the reference measure. Then for anyfunction ϕ ∈ C ∞ b (cid:0) R d (cid:1) , ( aI − L µ ) κ ϕ → − L µ,κ ϕ, κ ∈ (0 , , ( aI − L µ ) κ ϕ → L µ,κ ϕ, κ ∈ (1 , . uniformly as a → + .Proof. Apparently, ( aI − L µ ) ϕ ( x ) → − L µ ϕ ( x ) uniformly as a → 0. Usethe representation (2.3) for κ ∈ (0 , aI − L µ ) κ ϕ ( x ) = C ( κ ) Z ∞ t − κ − e − at (cid:2) ϕ ( x ) − E ϕ (cid:0) x + Z ¯ µt (cid:1)(cid:3) dt + a κ ϕ ( x ) , where C ( κ ) − = R ∞ t − κ − (cid:0) − e − t (cid:1) dt . Note (2.3), then | ( aI − L µ ) κ ϕ ( x ) + L µ,κ ϕ ( x ) |≤ C ( κ ) Z ∞ t − κ − (cid:12)(cid:12) e − at − (cid:12)(cid:12) (cid:12)(cid:12) ϕ ( x ) − E ϕ (cid:0) x + Z ¯ µt (cid:1)(cid:12)(cid:12) dt + a κ | ϕ ( x ) |≤ C ( κ ) | ϕ | (cid:20) a κ Z a t − κ − (cid:0) − e − t (cid:1) dt + Z ∞ t − κ − (cid:0) − e − at (cid:1) dt + a κ (cid:21) → a → + , ∀ x ∈ R d . To be precise, for any ε > 0, there is δ > | ( aI − L µ ) κ ϕ + L µ,κ ϕ | < ε | ϕ | whenever 0 < a < δ . Besides,( aI − L µ ) κ ϕ − L µ, κ ϕ = [( aI − L µ ) κ + L µ,κ ] ◦ [( aI − L µ ) κ + L µ,κ ] ◦ ϕ − aI − L µ ) κ ◦ L µ,κ ϕ − L µ, κ ϕ. By arguments above, when 0 < a < δ , | [( aI − L µ ) κ + L µ,κ ] ◦ [( aI − L µ ) κ + L µ,κ ] ◦ ϕ | ≤ ε | ϕ | , and − aI − L µ ) κ ◦ L µ,κ ϕ − L µ, κ ϕ → a → + . Therefore, ( aI − L µ ) κ ϕ → L µ, κ ϕ uniformly as a → + . (cid:3) The following derivation is needed in next two lemmas. Given (1.8), ψ µ ( ξ ) = w ( R ) − ψ ˜ µ R ( Rξ ) , ξ ∈ R d , ∀ R ∈ R + . Using the L´evy-Khintchine formula, we obtain p ( t, z ) = R − d p R (cid:16) w ( R ) − t, R − z (cid:17) , z ∈ R d , ∀ R ∈ R + , HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 13 where p ( t, z ) , z ∈ R d denotes the density function of Z µt if κ = 1 and that of Z ¯ µt if κ ∈ (0 , ∪ (1 , p R ( t, z ) , z ∈ R d denotes the density of Z Rt := Z ˜ µ R t if κ = 1 and Z ˜¯ µ R t otherwise. Existence of p R ( t, z ) is guaranteed by Lemma18 in Appendix. Lemma 7. Let κ ∈ (0 , , β ∈ (0 , ∞ ) and µ be the reference measure.Assume Z ∞ t κ − γ ( t ) − dt < ∞ . (3.1) Then for any function ϕ ∈ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) and any R > , ϕ ( x + y ) − ϕ ( x ) = C ( κ ) w ( R ) κ Z ∞ t κ − Z L µ,κ ϕ ( x + Rz ) · (cid:2) p R (cid:0) t, z − R − y (cid:1) − p R ( t, z ) (cid:3) dzdt, (3.2) where p R ( t, x ) , x ∈ R d follows the definition above. In particular, | ϕ ( x + y ) − ϕ ( x ) | ≤ Cw ( | y | ) κ | L µ,κ ϕ | , ∀ x, y ∈ R d . (3.3) Proof. We first assume ϕ ∈ C ∞ b (cid:0) R d (cid:1) ∩ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) . By (2.4), ϕ ( x + y ) − ϕ ( x )= C Z ∞ t κ − e − at E [( aI − L µ ) κ ϕ ( x + y + Z t ) − ( aI − L µ ) κ ϕ ( x + Z t )] dt = C Z ∞ t κ − e − at Z ( aI − L µ ) κ ϕ ( x + z ) [ p ( t, z − y ) − p ( t, z )] dzdt, where Z t = Z µt if κ = 1 and Z t = Z ¯ µt otherwise, and p ( t, x ) denotes theprobability density function of Z t . Recall that Lemma 18 claims Z |∇ p ( t, z ) | dz < C ′ γ ( t ) − . Let a → κ ∈ (0 , ϕ ( x + y ) − ϕ ( x )= C ( κ ) Z ∞ t κ − Z L µ,κ ϕ ( x + z ) [ p ( t, z − y ) − p ( t, z )] dzdt = C ( κ ) w ( R ) κ Z ∞ t κ − Z L µ,κ ϕ ( x + Rz ) (cid:2) p R (cid:0) t, z − R − y (cid:1) − p R ( t, z ) (cid:3) dzdt. Now consider ϕ ∈ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) . By [15, Proposition 5] and Lemma 4, thereis a sequence of functions v n ∈ C ∞ b (cid:0) R d (cid:1) ∩ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) such thatlim n →∞ | L µ,κ v n − L µ,κ ϕ | = 0 , ∀ κ ∈ [0 , . Moreover, v n ( x + y ) − v n ( x ) = C ( κ ) w ( R ) κ Z ∞ t κ − Z L µ,κ v n ( x + Rz ) · (cid:2) p R (cid:0) t, z − R − y (cid:1) − p R ( t, z ) (cid:3) dzdt. Pass the limit on both sides. Then (3.2) holds for ϕ ∈ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) . If y = 0,by setting R = | y | , we obtain (3.3) immediately. (cid:3) Denote ∇ α u ( x ; y ) = u ( x + y ) − u ( x ) − χ α ( y ) y · ∇ u ( x ) . Lemma 8. Let κ ∈ (0 , , β ∈ (0 , ∞ ) and µ be the reference measure.Assume α ∈ [1 , Z t κ − γ ( t ) − dt < ∞ , (3.4) Z ∞ α ∈ (0 , t κ − γ ( t ) − + 1 α ∈ [1 , t κ − γ ( t ) − dt < ∞ . (3.5) Then for all α ∈ (0 , ∪ (1 , and any function ϕ ∈ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) , ∇ α ϕ ( x ; y ) = C ( κ ) Z ∞ t κ − Z L µ,κ ϕ ( x + z ) ∇ α p ( t, z ; − y ) dzdt = C ( κ ) w ( R ) κ Z ∞ t κ − Z L µ,κ ϕ ( x + Rz ) ∇ α p R (cid:0) t, z ; − R − y (cid:1) dzdt, ∀ R > . (3.6) Moreover, |∇ α ϕ ( x ; y ) | ≤ Cw ( | y | ) κ | L µ,κ ϕ | , ∀ x, y ∈ R d . (3.7) If α = 1 , then (3.7) hold for | y | ≤ and all ϕ ∈ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) .Proof. Let a > 0. Similarly as in Lemma 7, we first consider ϕ ∈ C ∞ b (cid:0) R d (cid:1) ∩ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) . By (2.4), ∇ α ϕ ( x ; y ) = C Z ∞ t κ − e − at Z ( aI − L µ ) κ ϕ ( x + z ) ∇ α p ( t, z ; − y ) dzdt, where C = (cid:0)R ∞ t κ − e − t dt (cid:1) − . Let a → , (3.5).Then for all κ ∈ (0 , ∇ α ϕ ( x ; y ) = C Z ∞ t κ − Z L µ,κ ϕ ( x + z ) ∇ α p ( t, z ; − y ) dzdt = Cw ( R ) κ Z ∞ t κ − Z L µ,κ ϕ ( x + Rz ) ∇ α p R (cid:0) t, z ; − R − y (cid:1) dzdt. For general ϕ ∈ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) . By [15, Proposition 5] and Lemma 4, thereis a sequence of functions v n ∈ C ∞ b (cid:0) R d (cid:1) ∩ ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) such thatlim n →∞ | L µ,κ v n − L µ,κ ϕ | = 0 , ∀ κ ∈ [0 , , HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 15 and ∇ α v n ( x ; y ) = Cw ( R ) κ Z ∞ t κ − Z L µ,κ v n ( x + Rz ) ∇ α p R (cid:0) t, z ; − R − y (cid:1) dzdt. Passing the limit on both sides, we obtain (3.6) for all functions in ˜ C κ + β ∞ , ∞ (cid:0) R d (cid:1) .Setting R = | y | , y = 0, we have ∇ α ϕ ( x ; y ) = C ( κ ) w ( | y | ) κ Z ∞ t κ − Z L µ,κ ϕ ( x + | y | z ) ∇ α p | y | (cid:16) t, z ; − | y | − y (cid:17) dzdt. (3.8)(3.7) then follows from (3.8) , (3.4) and (3.5). (cid:3) Now we are ready to prove the stronger continuity of the operator. Choose η ( x ) ∈ C ∞ (cid:0) R d (cid:1) such that 0 ≤ η ( x ) ≤ , ∀ x ∈ R d , supp ( η ) ⊆ { x : | x | ≤ } ,and η ( x ) ≡ B (0). η m,z ( x ) := η ( m ( x − z )) , m ≥ Operators with Space-Dependent Kernels. Let ν be a L´evy mea-sure satisfying ˜A ( w, l, γ ). We now consider ρ ( t, x, y ) ν ( dy ), where ρ ( t, x, y )satisfies H( K, β ) . Obviously, ρ ( t, x, y ) ν ( dy ) is a L´evy measure for eachfixed x ∈ R d and t ∈ [0 , T ]. Denote L t,z u ( x )(3.9) = Z [ u ( x + y ) − u ( t, x ) − χ α ( y ) y · ∇ u ( x )] ρ ( t, z, y ) ν ( dy ) , h u, η m,z i t,z (3.10) = Z [ u ( x + y ) − u ( x )] [ η m,z ( x + y ) − η m,z ( x )] ρ ( t, z, y ) ν ( dy ) . Lemma 9. Let ν be a L´evy measure satisfying ˜A ( w, l, γ ) and ρ be a boundedmeasurable function. β ∈ (0 , /α ) . u ∈ ˜ C β ∞ , ∞ (cid:0) R d (cid:1) . Then there is β ′ ∈ (0 , β ) such that | L t,z u | ≤ C sup t,z,y | ρ ( t, z, y ) | | u | β ′ , ∞ , [ L t,z u ] β ≤ C sup t,z,y | ρ ( t, z, y ) | | u | β, ∞ . where C does not depend on t, z or u .Proof. Recall parameters introduced in A(w,l) . Clearly, | L t,z u | ≤ sup t,z,y | ρ ( t, z, y ) | Z | y |≤ |∇ α u ( x ; y ) | ν ( dy ) + Z | y | > |∇ α u ( x ; y ) | ν ( dy ) ! . Choose κ ∈ (0 , 1) sufficiently small so that lim r →∞ w ( r ) κ /r α = 0.According to [15, Lemma 1], such a κ must exist. Then by (3.3) and ˜A ( w, l, γ )(iii), for all α ∈ (0 , Z | y | > |∇ α u ( x ; y ) | ν ( dy ) ≤ C | L µ,κ u | Z | y | > w ( | y | ) κ ν ( dy ) ≤ C ( d, κ, α ) | L µ,κ u | , ∀ x ∈ R d . If α ∈ (1 , A(w,l) . Z | y | > |∇ α u ( x ; y ) | ν ( dy ) ≤ C ( d, α ) (cid:16) | u | + | u | , ∞ (cid:17) , ∀ x ∈ R d . It follows from [15, Proposition 4] and Lemma 4 that for all α ∈ (0 , 2) andany β ′ ∈ (0 , β ), Z | y | > |∇ α u ( x ; y ) | ν ( dy ) ≤ C (cid:0) d, κ, α, β ′ (cid:1) | u | β ′ , ∞ , ∀ x ∈ R d . On the other hand, suggested by ˜A ( w, l, γ )(iii), we apply (3.7) by setting β ′ ∈ (0 , δ ) if α = 1 and β ′ = δ if α = 1. Then Z | y |≤ |∇ α u ( x ; y ) | ν ( dy ) ≤ C (cid:0) d, β ′ , α (cid:1) (cid:12)(cid:12)(cid:12) L µ, β ′ u (cid:12)(cid:12)(cid:12) Z | y |≤ w ( | y | ) β ′ ν ( dy ) , ∀ x ∈ R d . By Lemma 17 (c) and Lemma 4, Z | y |≤ |∇ α u ( x ; y ) | ν ( dy ) ≤ C (cid:0) d, β ′ , α (cid:1) | u | β + β ′ ) / , ∞ , ∀ x ∈ R d . Now consider | L z u ( x ) − L z u ( x ) | . Set a = | x − x | . Then, | L t,z u ( x ) − L t,z u ( x ) |≤ sup t,z,y | ρ ( t, z, y ) | Z | y |≤ a |∇ α ( u ( x ; y ) − u ( x ; y )) | ν ( dy )+ sup t,z,y | ρ ( t, z, y ) | Z | y | >a |∇ α ( u ( x ; y ) − u ( x ; y )) | ν ( dy ) . Denote ς ( r ) = ν ( | y | > r ) and take β ′ ∈ (0 , δ ) if α = 1 and β ′ = δ if α = 1.Apply (3.8), [15, Proposition 1], Lemmas 4 and 17(a). Z | y |≤ a |∇ α ( u ( x ; y ) − u ( x ; y )) | ν ( dy ) ≤ C sup z (cid:12)(cid:12)(cid:12) L µ, β ′ u ( x + z ) − L µ, β ′ u ( x + z ) (cid:12)(cid:12)(cid:12) Z | y |≤ a w ( | y | ) β ′ ν ( dy ) ≤ − C h L µ, β ′ u i β − β ′ w ( a ) β − β ′ Z a ς ( r ) − − β ′ dς ( r ) ≤ C | u | β, ∞ w ( a ) β − β ′ ς ( r ) − β ′ | a ≤ C | u | β, ∞ w ( a ) β . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 17 Recall ˜A ( w, l, γ ) and set κ = β + min ( δ, ε ) / α = 1 and κ = β +( δ ′ + ε ) / α = 1. Apply (3.8), [15, Proposition 1] and Proposition 1. Z | y | >a |∇ α ( u ( x ; y ) − u ( x ; y )) | ν ( dy ) ≤ C sup z (cid:12)(cid:12)(cid:12) L µ, β − κ u ( x + z ) − L µ, β − κ u ( x + z ) (cid:12)(cid:12)(cid:12) Z | y | >a w ( | y | ) β − κ ν ( dy ) ≤ C | u | β, ∞ w ( a ) κ Z | y | >a w ( | y | ) β − κ ν ( dy ) . Similarly as above, Z | y | >a |∇ α ( u ( x ; y ) − u ( x ; y )) | ν ( dy ) ≤ − C | u | β, ∞ w ( a ) κ Z ∞ a ς ( r ) − − β + κ dς ( r ) ≤ − C | u | β, ∞ w ( a ) κ ς ( r ) κ − β | ∞ a ≤ C | u | β, ∞ w ( a ) β . As a conclusion, [ L t,z u ] β ≤ C sup t,z,y | ρ ( t, z, y ) | | u | β, ∞ . (cid:3) Corollary 1. Let ν be a L´evy measure satisfying ˜A ( w, l, γ ) and ρ satisfy H( K, β ) , β ∈ (0 , /α ) . Then for any u ∈ ˜ C β ∞ , ∞ (cid:0) R d (cid:1) , |A u | β ≤ C (cid:18) sup t,z,y | ρ ( t, z, y ) | | u | β, ∞ + sup t,y | ρ ( t, · , y ) | β | u | β ′ , ∞ (cid:19) , where β ′ ∈ (0 , β ) and C does not depend on u .Proof. Obviously, |A u | ≤ sup t,z | L t,z u | ≤ C sup t,z,y | ρ ( t, z, y ) | | u | β ′ , ∞ forsome β ′ ∈ (0 , β ). Meanwhile, | L t,x + y u ( x + y ) − L t,x u ( x ) |≤ | L t,x + y u ( x + y ) − L t,x u ( x + y ) | + | L t,x u ( x + y ) − L t,x u ( x ) |≤ C sup t,y [ ρ ( t, · , y )] β | u | β ′ , ∞ w ( | y | ) β + C sup t,z,y | ρ ( t, z, y ) | | u | β, ∞ w ( | y | ) β . Namely, [ A u ] β ≤ C (cid:16) sup t,y [ ρ ( t, · , y )] β | u | β ′ , ∞ + sup t,z,y | ρ ( t, z, y ) | | u | β, ∞ (cid:17) . (cid:3) Lemma 10. Let ν be a L´evy measure satisfying ˜A ( w, l, γ ) and ρ satisfy H( K, β ) . β ∈ (0 , /α ) . Then for any u ∈ ˜ C β ∞ , ∞ (cid:0) R d (cid:1) and any ε ∈ (0 , , sup t,z |h u, η m,z i t,z | β, ∞ ≤ Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , (3.11) where C ε depends on ε but is independent of u .Proof. Direct computation shows that for κ ∈ (0 , L µ,κ η m,z ( x ) = w (cid:0) m − (cid:1) − κ L ˜ µ m − ,κ η ( m ( x − z )) . By ˜A ( w, l, γ ), there is κ ∈ (1 / , 1) such that R ∞ t κ − γ ( t ) − dt < ∞ .Apply (3.3) with such a κ . |h u, η m,z i t,z | ≤ C Z | u ( x + y ) − u ( x ) | | η m,z ( x + y ) − η m,z ( x ) | ν ( dy ) ≤ Cw (cid:0) m − (cid:1) − κ | L µ,κ u | (cid:12)(cid:12) L ˜ µ m − ,κ η (cid:12)(cid:12) Z | y |≤ w ( | y | ) κ ν ( dy ) + C | u | . According to Lemmas 17, 4 and [15, Proposition 4], |h u, η m,z i t,z | ≤ Cw (cid:0) m − (cid:1) − κ ( | L µ,κ u | + | u | ) ≤ Cl ( m ) κ (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . For the difference estimate, let us set a = | x − x | and denote |h u, η m,z i t,z ( x ) − h u, η m,z i t,z ( x ) |≤ | Z | y |≤ [ u ( x + y ) − u ( x ) − u ( x + y ) + u ( x )] · [ η m,z ( x + y ) − η m,z ( x )] ρ ( t, z, y ) ν ( dy ) | + | Z | y |≤ [ u ( x + y ) − u ( x )] [ η m,z ( x + y ) − η m,z ( x ) − η m,z ( x + y ) + η m,z ( x )] ρ ( t, z, y ) ν ( dy ) | + | Z | y | > { [ u ( x + y ) − u ( x )] [ η m,z ( x + y ) − η m,z ( x )] − [ u ( x + y ) − u ( x )] [ η m,z ( x + y ) − η m,z ( x )] } ρ ( t, z, y ) ν ( dy ) | := I + I + I . Similarly, we use (3.3). Then for some κ ∈ (1 / , I ≤ Cw (cid:0) m − (cid:1) − κ sup z | L µ,κ u ( x + z ) − L µ,κ u ( x + z ) | (cid:12)(cid:12) L ˜ µ m − ,κ η (cid:12)(cid:12) ≤ Cw (cid:0) m − (cid:1) − κ w ( a ) β | L µ,κ u | β, ∞ ≤ Cl ( m ) κ w ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . For the same κ , I ≤ Cw (cid:0) m − (cid:1) − κ sup z (cid:12)(cid:12) L ˜ µ m − ,κ η ( m ( x − z )) − L ˜ µ m − ,κ η ( m ( x − z )) (cid:12)(cid:12) | L µ,κ u | ≤ Cw (cid:0) m − (cid:1) − κ l ( m ) β w ( a ) β | u | κ + β, ∞ ≤ Cl ( m ) β w ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 19 Besides, I ≤ C Z | y | > | [ u ( x + y ) − u ( x )] [ η m,z ( x + y ) − η m,z ( x ) − η m,z ( x + y ) + η m,z ( x )] | ν ( dy )+ C Z | y | > | [ u ( x + y ) − u ( x + y ) − u ( x ) + u ( x )][ η m,z ( x + y ) − η m,z ( x )] | ν ( dy ):= I + I , where I ≤ C | u | Z | y | > | η m,z ( x + y ) − η m,z ( x + y ) − η m,z ( x )+ η m,z ( x ) | ν ( dy ) ≤ C | u | l ( m ) β w ( a ) β , and obviously, I ≤ Cw ( a ) β | u | β, ∞ ≤ Cw ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . Summarizing, for any z ∈ R d ,[ h u, η m,z i t,z ] β ≤ Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . It follows immediately from [15, Proposition 1] thatsup t,z |h u, η m,z i t,z | β, ∞ ≤ C sup t,z |h u, η m,z i t,z | β ≤ Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . (cid:3) Operators with Space-Dependent Coefficients. In this section,we study the operator G ϕ ( x ) := Z [ ϕ ( x + G ( x ) y ) − ϕ ( x ) − χ α ( y ) G ( x ) y · ∇ ϕ ( x )] ν ( dy ) . We define the norm of an d × d -invertible matrix function G ( x ) , x ∈ R d to be its operator norm, i.e., | G ( x ) | := sup y ∈ R d , | y | =1 | G ( x ) y | , and k G k := sup x ∈ R d | G ( x ) | . If all entries of G are constants, then G is viewed as a constant functionand definitions above apply. Note k G k being finite implies finiteness of eachentry. If furthermore | det G ( z ) | ≥ c for some c > 0, then (cid:13)(cid:13) G − (cid:13)(cid:13) is alsofinite. Lemma 11. Let G be an invertible d × d -matrix and β > . f ∈ ˜ C β ∞ , ∞ (cid:0) R d (cid:1) . g ( x ) := f ( Gx ) , x ∈ R d . Then, | g | β, ∞ ≤ C | f | β, ∞ (3.12) for some C only depending on (cid:13)(cid:13) G − (cid:13)(cid:13) and k G k .Proof. Consider the mapping T : R d → R d such that T ( x ) = Gx . Then T − ( x ) = G − x . Clearly, both T and T − are continuous and k T k − = k G k − ≤ (cid:13)(cid:13) T − (cid:13)(cid:13) = (cid:13)(cid:13) G − (cid:13)(cid:13) . For any j = 0, | g ∗ ϕ j | = sup x (cid:12)(cid:12)(cid:12)(cid:12)Z f ( Gy ) ϕ j ( x − y ) dy (cid:12)(cid:12)(cid:12)(cid:12) = sup x | det G | (cid:12)(cid:12)(cid:12)(cid:12)Z f ( y ) ϕ j (cid:0) G − x − G − y (cid:1) dy (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12) F − [ φ j ( Gξ ) F f ] (cid:12)(cid:12) . Note φ j ( G · ) is supported on { ξ : N j − ≤ | Gξ | ≤ N j +1 } ⊂ { ξ : N j − k G k − ≤| ξ | ≤ N j +1 (cid:13)(cid:13) G − (cid:13)(cid:13) } . Denote n ( j ) = min { i : N j +1 (cid:13)(cid:13) G − (cid:13)(cid:13) ≤ N i } ∨ ,m ( j ) = max { i : N i ≤ N j − k G k − } ∨ . Then n ( j ) = n (1) + j − m ( j ) = m (1) + j − 1, and that n ( j ) − m ( j ) ≤ n (1) − m (1) + 1 which is independent of j . Moreover, φ j ( Gξ ) = φ j ( Gξ ) n ( j ) X i = m ( j ) φ i ( ξ ) . Therefore, | g ∗ ϕ j | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F − φ j ( Gξ ) n ( j ) X i = m ( j ) φ i ( ξ ) F f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( n (1) − m (1) + 1) sup j | f ∗ ϕ j | (cid:12)(cid:12) F − [ φ j ( Gξ )] (cid:12)(cid:12) L ( R d ) ≤ Cw (cid:0) N − j (cid:1) β | f | β, ∞ . Similarly, if j = 0, | g ∗ ϕ | = (cid:12)(cid:12) F − [ c ϕ ( Gξ ) F f ] (cid:12)(cid:12) , and supp ( c ϕ ( Gξ )) = { ξ : | Gξ | ≤ N } ⊂ { ξ : | ξ | ≤ N (cid:13)(cid:13) G − (cid:13)(cid:13) } . Denote k = min { i : N (cid:13)(cid:13) G − (cid:13)(cid:13) ≤ N i } ∨ 1. Then, c ϕ ( Gξ ) = c ϕ ( Gξ ) c ϕ ( ξ ) + k X i =1 φ i ( ξ ) ! . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 21 Therefore, | g ∗ ϕ j | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F − "c ϕ ( Gξ ) c ϕ ( ξ ) + k X i =1 φ i ( ξ ) ! F f ≤ C sup j | f ∗ ϕ j | (cid:12)(cid:12) F − [ c ϕ ( Gξ )] (cid:12)(cid:12) L ( R d ) ≤ Cw (cid:0) N − j (cid:1) β | f | β, ∞ . Summarizing, | g | β, ∞ ≤ C | f | β, ∞ . (cid:3) Proposition 2. Let ν be a L´evy measure satisfying A(w,l) and G be aninvertible d × d -matrix. For any function f ∈ C b (cid:0) R d (cid:1) , Lf ( x ) := Z [ f ( x + Gy ) − f ( x ) − χ α ( y ) Gy · ∇ f ( x )] ν ( dy ) . Then for β ∈ (0 , /α ) , there exists C depending on (cid:13)(cid:13) G − (cid:13)(cid:13) and k G k suchthat | Lf | β, ∞ ≤ C | f | β, ∞ . Proof. If g ( x ) := f ( Gx ). Then Lf ( Gx ) = L ν g ( x ). By previous continuityand equivalence results and Lemma 11, | Lf | β, ∞ ≤ C | Lf ( G · ) | β, ∞ = | L ν g | β, ∞ ≤ C | g | β, ∞ ≤ C | f | β, ∞ . (cid:3) Let us denote ∇ α,z u ( x ; y ) = u ( x + G ( z ) y ) − u ( x ) − χ α ( y ) G ( z ) y · ∇ u ( x ) ,L z u ( x ) = Z ∇ α,z u ( x ; y ) ν ( dy ) , (3.13) ¯ G z,z ′ := (cid:13)(cid:13) G ( z ) − G (cid:0) z ′ (cid:1)(cid:13)(cid:13) , and g (cid:0) z, z ′ (cid:1) = w (cid:16) ¯ G − z,z ′ (cid:17) − if α ∈ (0 , ,w (cid:16) ¯ G − z,z ′ (cid:17) − w (cid:0) ¯ G z,z ′ (cid:1) − δ ′ ∨ ¯ G z,z ′ if α = 1 , ¯ G z,z ′ if α ∈ (1 , . Lemma 12. Let β ∈ (0 , /α ) , ν be a L´evy measure satisfying ˜A ( w, l, γ ) ,and G ( z ) , ∀ z ∈ R d satisfy G( c , K, β ) . If u ∈ ˜ C β (cid:0) R d (cid:1) , then, | L z u − L z ′ u | ≤ Cg (cid:0) z, z ′ (cid:1) | u | β ′ , ∞ , [ L z u − L z ′ u ] β ≤ C (cid:0) ¯ G z,z ′ (cid:1) σ | u | β, ∞ for some β ′ ∈ (0 , β ) , σ ∈ (0 , . C is independent z, z ′ and u . Proof. Write for simplicity G = G ( z ), G ′ = G ( z ′ ) and ¯ G = ¯ G z,z ′ . Use (3.6). ∇ α,z u ( x ; y ) − ∇ α,z ′ u ( x ; y )= Cw ( R ) κ Z ∞ t κ − Z L µ,κ u ( x + Rϑ ) · h ∇ α,z p R (cid:0) t, ϑ ; − R − Gy (cid:1) − ∇ α,z ′ p R (cid:0) t, ϑ ; − R − G ′ y (cid:1)i dϑdt := D ( κ, R )with κ and R to be determined according to our needs. If α ∈ (0 , ˜A ( w, l, γ ), we may split the integral as follows. | L z u ( x ) − L z ′ u ( x ) |≤ Z ¯ G | y |≤ (cid:12)(cid:12) D (cid:0) κ, ¯ G | y | (cid:1)(cid:12)(cid:12) ν ( dy ) + Z ¯ G | y | > (cid:12)(cid:12) u ( x + Gy ) − u (cid:0) x + G ′ y (cid:1)(cid:12)(cid:12) ν ( dy ):= I + I for some κ ∈ (1 , δ ). By Lemmas 17 and 18 in Appendix, I ≤ C | L µ,κ u | Z ¯ G | y |≤ Z ∞ t κ − (cid:16) ∧ γ ( t ) − ¯ G − | y | − (cid:12)(cid:12) Gy − G ′ y (cid:12)(cid:12)(cid:17) dtw (cid:0) ¯ G | y | (cid:1) κ ν ( dy ) ≤ Cw (cid:0) ¯ G − (cid:1) − | L µ,κ u | Z | y |≤ w ( | y | ) κ ˜ ν ¯ G − ( dy ) ≤ Cw (cid:0) ¯ G − (cid:1) − | L µ,κ u | . Besides, I ≤ Cw (cid:0) ¯ G − (cid:1) − Z | y | > | u | ˜ ν ¯ G − ( dy ) ≤ Cw (cid:0) ¯ G − (cid:1) − | u | . Therefore when α ∈ (0 , β ′ ∈ (0 , β ) such that | L z u − L z ′ u | ≤ Cw (cid:0) ¯ G − (cid:1) − | u | β ′ , ∞ . (3.14)If α = 1, we write instead | L z u ( x ) − L z ′ u ( x ) |≤ Z | y |≤ | D (1 + δ, | y | ) | ν ( dy ) + Z ¯ G | y |≤ , | y | > (cid:12)(cid:12) u ( x + Gy ) − u (cid:0) x + G ′ y (cid:1)(cid:12)(cid:12) ν ( dy )+ Z ¯ G | y | > , | y | > (cid:12)(cid:12) u ( x + Gy ) − u (cid:0) x + G ′ y (cid:1)(cid:12)(cid:12) ν ( dy ) := I + I + I , where I ≤ C (cid:12)(cid:12)(cid:12) L µ, δ u (cid:12)(cid:12)(cid:12) Z | y |≤ w ( | y | ) δ Z ∞ t κ − ¯ G (cid:16) γ ( t ) − ∧ γ ( t ) − (cid:17) dtν ( dy ) ≤ C ¯ G (cid:12)(cid:12)(cid:12) L µ, δ u (cid:12)(cid:12)(cid:12) . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 23 Meanwhile, similarly as I , I , we have w (cid:0) ¯ G − (cid:1) ( I + I ) ≤ C (cid:12)(cid:12)(cid:12) L µ, − δ ′ u (cid:12)(cid:12)(cid:12) Z | y |≤ , | y | > ¯ G w ( | y | ) − δ ′ ˜ ν ¯ G − ( dy ) + Z | y | > | u | ˜ ν ¯ G − ( dy ) ! ≤ C (cid:18) δ ′ (cid:12)(cid:12)(cid:12) L µ, − δ ′ u (cid:12)(cid:12)(cid:12) + 1 δ ′ (cid:12)(cid:12)(cid:12) L µ, − δ ′ u (cid:12)(cid:12)(cid:12) w (cid:0) ¯ G (cid:1) − δ ′ + | u | (cid:19) . Thus for α = 1, there is β ′ ∈ (0 , β ) so that | L z u − L z ′ u | ≤ Cδ ′ (cid:16) ¯ G ∨ w (cid:0) ¯ G − (cid:1) − w (cid:0) ¯ G (cid:1) − δ ′ (cid:17) | u | β ′ , ∞ . (3.15)Next, we discuss the case α ∈ (1 , | L z u ( x ) − L z ′ u ( x ) |≤ Z | y |≤ | D ( κ, | y | ) | ν ( dy ) + Z | y | > (cid:12)(cid:12)(cid:12) ∇ α,z u ( x ; y ) − ∇ α,z ′ u ( x ; y ) (cid:12)(cid:12)(cid:12) ν ( dy ):= I + I . Then as how we estimated I , we have I ≤ C ¯ G | L µ,κ u | for some κ ∈ (1 , δ ). Clearly, I ≤ C ¯ G |∇ u | . Thus for α ∈ (1 , β ′ ∈ (0 , β )so that | L z u − L z ′ u | ≤ C ¯ G | u | β ′ , ∞ . (3.16)We now estimate the difference. Without loss of generality, we set | x − x | = a ∈ (0 , ∇ α,z u ( x ; y ) − ∇ α,z ′ u ( x ; y ) − ∇ α,z u ( x ; y ) + ∇ α,z ′ u ( x ; y )= Cw ( R ) κ Z ∞ t κ − Z [ L µ,κ u ( x + Rϑ ) − L µ,κ u ( x + Rϑ )] · h ∇ α,z p R (cid:0) t, ϑ ; − R − y (cid:1) − ∇ α,z ′ p R (cid:0) t, ϑ ; − R − y (cid:1)i dϑdt := e D ( κ, R )with κ and R to be determined. Then, | L z u ( x ) − L z ′ u ( x ) − L z u ( x ) + L z ′ u ( x ) |≤ Z | y |≤ a (cid:12)(cid:12)(cid:12) e D ( κ, | y | ) (cid:12)(cid:12)(cid:12) ν ( dy ) + Z | y | >a (cid:12)(cid:12)(cid:12) e D (cid:0) κ ′ , | y | (cid:1)(cid:12)(cid:12)(cid:12) ν ( dy ):= I + I . Denote ς ( r ) = ν ( | y | > r ). By Lemma 18 and [15, Lemma 1], for all α ∈ (0 , β ′ ∈ (0 , β ) and σ ∈ (0 , 1) such that I ≤ − C h L µ, β ′ u i β − β ′ ¯ G σ w ( a ) β − β ′ Z a ς ( r ) − − β ′ dς ( r ) ≤ C | u | β, ∞ ¯ G σ w ( a ) β − β ′ ς ( r ) − β ′ | a ≤ C | u | β, ∞ ¯ G σ w ( a ) β . Recall ˜A ( w, l, γ ). Using the symmetry assumption for α = 1 and non-degeneracy of G , we can set κ = β + min ( δ, ε ) / α = 1 and κ = β +( δ ′ + ε ) / α = 1, there is β ′ ∈ (0 , β ) and σ ∈ (0 , 1) such that I ≤ C h L µ, β − κ u i κ ¯ G σ w ( a ) κ Z | y | >a w ( | y | ) β − κ ν ( dy ) ≤ − C | u | β, ∞ ¯ G σ w ( a ) κ Z ∞ a ς ( r ) − − β + κ dς ( r ) ≤ C | u | β, ∞ ¯ G σ w ( a ) κ ς ( a ) κ − β ≤ C | u | β, ∞ ¯ G σ w ( a ) β . This ends the proof. (cid:3) Corollary 2. Let β ∈ (0 , /α ) , ν be a L´evy measure satisfying ˜A ( w, l, γ ) ,and G ( z ) , ∀ z ∈ R d satisfy G( c , K, β ) . If u ∈ ˜ C β (cid:0) R d (cid:1) , then, |G u − L z u | β ≤ C (cid:16)(cid:0) ¯ G x,z (cid:1) σ | u | β, ∞ + | u | β ′ , ∞ (cid:17) for some β ′ ∈ (0 , β ) . C is independent of x, z and u .Proof. First by Lemma 12, |G u − L z u | ≤ sup z,z ′ | L z ′ u − L z u | ≤ C | u | β ′ , ∞ for some β ′ ∈ (0 , β ). In the meantime, | L x + y u ( x + y ) − L z u ( x + y ) − L x u ( x ) + L z u ( x ) |≤ | L x u ( x + y ) − L z u ( x + y ) − L x u ( x ) + L z u ( x ) | + | L x + y u ( x + y ) − L x u ( x + y ) |≤ C (cid:0) ¯ G x,z (cid:1) σ | u | β, ∞ w ( | y | ) β + C | u | β ′ , ∞ w ( | y | ) β . Namely, [ G u − L z u ] β ≤ C (cid:16)(cid:0) ¯ G x,z (cid:1) σ | u | β, ∞ + | u | β ′ , ∞ (cid:17) . (cid:3) For η m,z introduced in previous section, we denote h u, η m,z i z (3.17)= Z [ u ( x + G ( z ) y ) − u ( x )] [ η m,z ( x + G ( z ) y ) − η m,z ( x )] ν ( dy ) . Lemma 13. Let ν be a L´evy measure satisfying ˜A ( w, l, γ ) and k G ( z ) k ≤ K, ∀ z ∈ R d for some K > . β ∈ (0 , /α ) . Then for any u ∈ ˜ C β ∞ , ∞ (cid:0) R d (cid:1) and any ε ∈ (0 , , sup z |h u, η m,z i z | β, ∞ ≤ Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , (3.18) where C ε depends on ε but is independent of u . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 25 Proof. We proceed in the same manner as in Lemma 10. First, since k G ( z ) k is uniformly bounded, there is κ ∈ (1 / , 1) such that |h u, η m,z i z | ≤ C Z | u ( x + G ( z ) y ) − u ( x ) | | η m,z ( x + G ( z ) y ) − η m,z ( x ) | ν ( dy ) ≤ Cw (cid:0) m − (cid:1) − κ | L µ,κ u | (cid:12)(cid:12) L ˜ µ m − ,κ η (cid:12)(cid:12) Z | y |≤ w ( | y | ) κ ν ( dy ) + C | u | ≤ Cl ( m ) κ (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . For the difference, again, let us set a = | x − x | ∈ (0 , 1) and estimate |h u, η m,z i z ( x ) − h u, η m,z i z ( x ) |≤ | Z | y |≤ [ u ( x + G ( z ) y ) − u ( x ) − u ( x + G ( z ) y ) + u ( x )] · [ η m,z ( x + G ( z ) y ) − η m,z ( x )] ν ( dy ) | + | Z | y |≤ [ u ( x + G ( z ) y ) − u ( x )] [ η m,z ( x + G ( z ) y ) − η m,z ( x ) − η m,z ( x + G ( z ) y ) + η m,z ( x )] ν ( dy ) | + | Z | y | > [ u ( x + G ( z ) y ) − u ( x )] [ η m,z ( x + G ( z ) y ) − η m,z ( x ) − η m,z ( x + G ( z ) y ) + η m,z ( x )] ν ( dy ) | + | Z | y | > [ u ( x + G ( z ) y ) − u ( x ) − u ( x + G ( z ) y ) + u ( x )] · [ η m,z ( x + G ( z ) y ) − η m,z ( x )] ν ( dy ) | := I + I + I + I . Then (3.3) implies that I , I ≤ Cl ( m ) β w ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , where C depends on K . Meanwhile, I ≤ C | u | Z | y | > | η m,z ( x + G ( z ) y ) − η m,z ( x + G ( z ) y ) − η m,z ( x )+ η m,z ( x ) | ν ( dy ) ≤ C | u | l ( m ) β w ( a ) β , and obviously, I ≤ Cw ( a ) β | u | β, ∞ ≤ Cw ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . Summarizing,sup z [ h u, η m,z i z ] β ≤ Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , and thus,sup z |h u, η m,z i z | β, ∞ ≤ Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . (cid:3) Lower Order Operators. For any function u ∈ C b ( H T ), denote Q t,z,z ′ u ( t, x ):= 1 α ∈ (1 , b ( t, z ) · ∇ u ( t, x ) + p ( t, z ) u ( t, x ) + Z R d [ u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) − u ( t, x ) − ∇ u ( t, x ) · q (cid:0) t, z ′ , y (cid:1) α ∈ (1 , | y |≤ ] ̺ ( t, z, y ) ν ( dy ):= 1 α ∈ (1 , b ( t, z ) · ∇ u ( t, x ) + p ( t, z ) u ( t, x ) + ˜ Q t,z,z ′ u ( t, x ) . Lemma 14. Let B( K, β ) hold. β ∈ (0 , /α ) . Then for any u ∈ ˜ C β ∞ , ∞ ( H T ) and any ε ∈ (0 , , there exists β ′ ∈ (0 , β ) , sup t,z,z ′ (cid:12)(cid:12)(cid:12) ˜ Q t,z,z ′ u ( t, · ) (cid:12)(cid:12)(cid:12) ≤ C sup t,z,y | ρ ( t, z, y ) | | u | β ′ , ∞ , (3.19) sup t,z,z ′ h ˜ Q t,z,z ′ u ( t, · ) i β ≤ sup t,z,y | ρ ( t, z, y ) | (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , (3.20) where C, C ε are independent of u .Proof. We split the integral. (cid:12)(cid:12)(cid:12) ˜ Q t,z,z ′ u ( t, x ) (cid:12)(cid:12)(cid:12) ≤ C sup t,z,y | ρ ( t, z, y ) | Z | y |≤ α ∈ (1 , (cid:12)(cid:12) ∇ α u (cid:0) t, x ; q (cid:0) t, z ′ , y (cid:1)(cid:1)(cid:12)(cid:12) ν ( dy )+ C sup t,z,y | ρ ( t, z, y ) | Z R d α ∈ (0 , (cid:12)(cid:12) u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) − u ( t, x ) (cid:12)(cid:12) ν ( dy )+ C sup t,z,y | ρ ( t, z, y ) | Z | y | > α ∈ (1 , (cid:12)(cid:12) u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) − u ( t, x ) (cid:12)(cid:12) ν ( dy ):= C sup t,z,y | ρ ( t, z, y ) | ( I + I + I ) . Use assumptions ˜A ( w, l, γ ) and B( K, β ) . By Lemma 8, I ≤ C | L µ u | Z | y |≤ w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ν ( dy ) ≤ C | L µ u | . Take β ′ ∈ (0 , δ ). Then by Lemma 7, I ≤ C α ∈ (0 , Z R d (cid:16)(cid:12)(cid:12)(cid:12) L µ, β ′ u (cid:12)(cid:12)(cid:12) w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) β ′ ∧ | u | (cid:17) ν ( dy )+ C α =1 Z R d (cid:0) |∇ u | (cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ∧ | u | (cid:1) ν ( dy ) ≤ C (cid:16) α ∈ (0 , (cid:12)(cid:12)(cid:12) L µ, β ′ u (cid:12)(cid:12)(cid:12) + 1 α =1 |∇ u | + | u | (cid:17) . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 27 Clearly, I ≤ C ( |∇ u | + | u | ). Summarizing, there exists β ′ ∈ (0 , β ) so thatsup t,z,z ′ (cid:12)(cid:12)(cid:12) ˜ Q t,z,z ′ u ( t, x ) (cid:12)(cid:12)(cid:12) ≤ C sup t,z,y | ρ ( t, z, y ) | | u | β ′ , ∞ . Meanwhile,sup t,z,y | ρ ( t, z, y ) | − (cid:12)(cid:12)(cid:12) ˜ Q t,z,z ′ u ( t, x ) − ˜ Q t,z,z ′ u ( t, x ) (cid:12)(cid:12)(cid:12) ≤ C Z | y |≤ α ∈ (1 , (cid:12)(cid:12) ∇ α u (cid:0) t, x ; q (cid:0) t, z ′ , y (cid:1)(cid:1) − ∇ α u (cid:0) t, x ; q (cid:0) t, z ′ , y (cid:1)(cid:1)(cid:12)(cid:12) ν ( dy )+ C Z | y | > α ∈ (1 , | u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) − u ( t, x ) − u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) + u ( t, x ) | ν ( dy )+ C Z R d α ∈ (0 , | u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) − u ( t, x ) − u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) + u ( t, x ) | ν ( dy ):= C ( I + I + I ) . Set | x − x | = a . Then for any ǫ ∈ (0 , I = Z | q ( t,z ′ ,y ) |≤ ǫ, | y |≤ (cid:12)(cid:12) ∇ α u (cid:0) t, x ; q (cid:0) t, z ′ , y (cid:1)(cid:1) − ∇ α u (cid:0) t, x ; q (cid:0) t, z ′ , y (cid:1)(cid:1)(cid:12)(cid:12) ν ( dy )+ Z | q ( t,z ′ ,y ) | >ǫ, | y |≤ (cid:12)(cid:12) ∇ α u (cid:0) t, x ; q (cid:0) t, z ′ , y (cid:1)(cid:1) − ∇ α u (cid:0) t, x ; q (cid:0) t, z ′ , y (cid:1)(cid:1)(cid:12)(cid:12) ν ( dy ):= I + I . By B( K, β ) , for any ε ∈ (0 , ǫ ∈ (0 , 1) such that I = w ( a ) β [ L µ u ( t, · )] β Z | q ( t,z ′ ,y ) |≤ ǫ, | y |≤ w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ν ( dy ) ≤ Cw ( a ) β | u ( t, · ) | β, ∞ Z | q ( t,z ′ ,y ) |≤ ǫ w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ν ( dy ) ≤ ε w ( a ) β | u ( t, · ) | β, ∞ . There also exists κ ∈ (0 , 1) so that I = w ( a ) β [ L µ,κ u ( t, · )] β Z | q ( t,z ′ ,y ) | >ǫ, | y |≤ w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) κ ν ( dy ) ≤ Cl (cid:0) ǫ − (cid:1) − κ w ( a ) β | u ( t, · ) | κ + β, ∞ Z | y |≤ w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ν ( dy ) ≤ Cl (cid:0) ǫ − (cid:1) − κ w ( a ) β | u ( t, · ) | κ + β, ∞ . Applying [15, Proposition 4], we can always attain I ≤ w ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , which concludes I ≤ w ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) .In the mean time, by B( K, β ) , Lemma 5 and [15, Proposition 4], I ≤ α ∈ (1 , w ( a ) β Z | y | > [ ∇ u ( t, · )] β (cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ∧ [ u ( t, · )] β ν ( dy ) ≤ C α ∈ (1 , w ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . Besides, for any ǫ ∈ (0 , I ≤ Z | q ( t,z ′ ,y ) |≤ ǫ α ∈ (0 , | u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) − u ( t, x ) − u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) + u ( t, x ) | ν ( dy )+ Z | q ( t,z ′ ,y ) | >ǫ α ∈ (0 , | u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) − u ( t, x ) − u (cid:0) t, x + q (cid:0) t, z ′ , y (cid:1)(cid:1) + u ( t, x ) | ν ( dy ):= I + I . We first discuss the case α ∈ (0 , I ≤ C [ L µ u ( t, · )] β w ( a ) β Z | q ( t,z ′ ,y ) |≤ ǫ α ∈ (0 , w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ν ( dy ) ≤ Cw ( a ) β | u ( t, · ) | β, ∞ Z | q ( t,z ′ ,y ) |≤ ǫ α ∈ (0 , w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ν ( dy ) . On the other hand, I ≤ Cw ( a ) β Z | q ( t,z ′ ,y ) | >ǫ h L µ, u ( t, · ) i β w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ∧ [ u ] β ν ( dy ) ≤ Cw ( a ) β l (cid:0) ǫ − (cid:1) | u ( t, · ) | + β, ∞ Z | q ( t,z ′ ,y ) | >ǫ (cid:0) w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ∧ (cid:1) ν ( dy ) . As what we did for I , by choosing an appropriate ǫ , we have I ≤ w ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , If α = 1, then I ≤ C [ ∇ u ( t, · )] β ′ w ( a ) β ′ Z | q ( t,z ′ ,y ) |≤ ǫ (cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ν ( dy ) ≤ Cw ( a ) β ′ | u ( t, · ) | β, ∞ Z | q ( t,z ′ ,y ) |≤ ǫ (cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ν ( dy )for all β ′ ∈ (0 , β ). Examining the proof of Lemma 5, we find that thisconstant C is uniformly bounded under ˜A ( w, l, γ )(ii) for all β ′ ∈ (0 , β ).Thus, I ≤ Cw ( a ) β | u ( t, · ) | β, ∞ Z | q ( t,z ′ ,y ) |≤ ǫ (cid:12)(cid:12) q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ν ( dy ) . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 29 Estimate I in the same way as above. By choosing an appropriate ǫ , wearrive at I ≤ w ( a ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . As a summary, for all α ∈ (0 , 2) and any ε ∈ (0 , C ε thatis independent of u so thatsup t,z,z ′ h ˜ Q t,z,z ′ u ( t, · ) i β ≤ sup t,z,y | ̺ ( t, z, y ) | (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . (cid:3) Lemma 15. Let B( K, β ) hold. β ∈ (0 , /α ) . Then for any u ∈ ˜ C β ∞ , ∞ ( H T ) and any ε ∈ (0 , , there exists C ε independent of u such that |Q u ( t, · ) | β, ∞ ≤ ε | u ( t, · ) | β, ∞ + C ε | u ( t, · ) | . (3.21) Proof. Note that Q u ( t, x ) = Q t,x,x u ( t, x ). By Lemmas 14 and 5 |Q u ( t, · ) | ≤ C | u ( t, · ) | β ′ , ∞ for some β ′ ∈ (0 , β ). Meanwhile, for any x, h ∈ R d , | Q t,x + h,x + h u ( t, x + h ) − Q t,x,x u ( t, x ) |≤ α ∈ (1 , | b ( t, x + h ) ∇ u ( t, x + h ) − b ( t, x ) ∇ u ( t, x ) | + | p ( t, x + h ) u ( t, x + h ) − p ( t, x ) u ( t, x ) | + (cid:12)(cid:12)(cid:12) ˜ Q t,x + h,x + h u ( t, x + h ) − ˜ Q t,x,x u ( t, x ) (cid:12)(cid:12)(cid:12) . Obviously, 1 α ∈ (1 , | b ( t, x + h ) ∇ u ( t, x + h ) − b ( t, x ) ∇ u ( t, x ) | + | p ( t, x + h ) u ( t, x + h ) − p ( t, x ) u ( t, x ) |≤ Cw ( | h | ) β (cid:16) | u ( t, · ) | β + 1 α ∈ (1 , |∇ u ( t, · ) | β (cid:17) . In the mean time, (cid:12)(cid:12)(cid:12) ˜ Q t,x + h,x + h u ( t, x + h ) − ˜ Q t,x,x u ( t, x ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ˜ Q t,x + h,x + h u ( t, x + h ) − ˜ Q t,x,x + h u ( t, x + h ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ Q t,x,x + h u ( t, x + h ) − ˜ Q t,x,x u ( t, x + h ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˜ Q t,x,x u ( t, x + h ) − ˜ Q t,x,x u ( t, x ) (cid:12)(cid:12)(cid:12) . By Lemma 14, (cid:12)(cid:12)(cid:12) ˜ Q t,x + h,x + h u ( t, x + h ) − ˜ Q t,x,x + h u ( t, x + h ) (cid:12)(cid:12)(cid:12) ≤ C sup t,y [ ̺ ( t, · , y )] β w ( | h | ) β | u | β ′ , ∞ for some β ′ ∈ (0 , β ), and (cid:12)(cid:12)(cid:12) ˜ Q t,x,x u ( t, x + h ) − ˜ Q t,x,x u ( t, x ) (cid:12)(cid:12)(cid:12) ≤ C sup t,z,y | ̺ ( t, z, y ) | w ( | h | ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) . Besides, (cid:12)(cid:12)(cid:12) ˜ Q t,x,x + h u ( t, x + h ) − ˜ Q t,x,x u ( t, x + h ) (cid:12)(cid:12)(cid:12) ≤ C Z | y |≤ α ∈ (1 , |∇ α u ( t, x + h ; q ( t, x + h, y )) − ∇ α u ( t, x + h ; q ( t, x, y )) | ν ( dy )+ C Z | y | > α ∈ (1 , | u ( t, x + h + q ( t, x + h, y )) − u ( t, x + h + q ( t, x, y )) | ν ( dy )+ C Z R d α ∈ (0 , | u ( t, x + h + q ( t, x + h, y )) − u ( t, x + h + q ( t, x, y )) | ν ( dy ):= C ( I + I + I ) . Similarly as what we did in Lemma 14, I ≤ C Z | y |≤ Z |∇ u ( t, x + h + θq ( t, x + h, y )) − ∇ u ( t, x + h ) | dθ | q ( t, x + h, y ) − q ( t, x, y ) | ν ( dy )+ C Z | y |≤ Z |∇ u ( t, x + h + θq ( t, x + h, y )) −∇ u ( t, x + h + θq ( t, x, y )) | dθ | q ( t, x, y ) | ν ( dy ) ≤ C [ ∇ u ( t, · )] β Z | y |≤ w ( | q ( t, x + h, y ) | ) β | q ( t, x + h, y ) − q ( t, x, y ) | ν ( dy )+ C [ ∇ u ( t, · )] β Z | y |≤ w ( | q ( t, x + h, y ) − q ( t, x, y ) | ) β | q ( t, x, y ) | ν ( dy ) ≤ C | u ( t, · ) | κ + β, ∞ w ( | h | ) β for some κ ∈ (0 , I ≤ C Z | y | > |∇ u ( t, · ) | (cid:12)(cid:12) q (cid:0) t, z ′ + h, y (cid:1) − q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ∧ | u | ν ( dy ) ≤ C | u ( t, · ) | β ′ , ∞ w ( | h | ) β for some β ′ ∈ (0 , β ). And I ≤ C Z R d α ∈ (0 , (cid:0) | L µ u ( t, · ) | w (cid:0)(cid:12)(cid:12) q (cid:0) t, z ′ + h, y (cid:1) − q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12)(cid:1) ∧ | u | (cid:1) ν ( dy )+ C Z R d α =1 (cid:0) |∇ u ( t, · ) | (cid:12)(cid:12) q (cid:0) t, z ′ + h, y (cid:1) − q (cid:0) t, z ′ , y (cid:1)(cid:12)(cid:12) ∧ | u | (cid:1) ν ( dy ) ≤ C α ∈ (0 , | u ( t, · ) | β ′ , ∞ w ( | h | ) β HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 31 for some β ′ ∈ (0 , β ).Summarizing, we obtain (3.21). (cid:3) Proof of the Main Result Auxiliary Results. In this section, we state or prove well-posednessfor integro-differential equations with space-independent operators. Theorem 4.1. [15, Theorem 1.1] Let β ∈ (0 , ∞ ) , λ ≥ and ν be a L´evymeasure satisfying A(w,l) . If f ( t, x ) ∈ ˜ C β ∞ , ∞ ( H T ) . Then there is a uniquesolution u ∈ ( t, x ) ∈ ˜ C β ∞ , ∞ ( H T ) to ∂ t u ( t, x ) = Lu ( t, x ) − λu ( t, x ) + f ( t, x ) , λ ≥ , (4.1) u (0 , x ) = 0 , ( t, x ) ∈ H T , where for any function ϕ ∈ C b (cid:0) R d (cid:1) , Lϕ ( x ) := Z [ ϕ ( x + y ) − ϕ ( x ) − χ α ( y ) y · ∇ ϕ ( x )] ν ( dy ) . (4.2) Moreover, there exists a constant C depending on κ, β, d, T, µ, ν such that | u | β, ∞ ≤ C (cid:0) λ − ∧ T (cid:1) | f | β, ∞ , (4.3) | u | β, ∞ ≤ C | f | β, ∞ (4.4) And there is a constant C depending on κ, β, d, T, µ, ν such that for all ≤ s < t ≤ T , κ ∈ [0 , , (4.5) | u ( t, · ) − u ( s, · ) | κ + β, ∞ ≤ C | t − s | − κ | f | β, ∞ . Theorem 4.2. Let ν be a L´evy measure, α ∈ (0 , , β ∈ (0 , , λ ≥ . G isan invertible d × d -matrix. Assume that f ( t, x ) ∈ ˜ C β ∞ , ∞ ( H T ) . Then thereis a unique solution u ∈ ( t, x ) ∈ ˜ C β ∞ , ∞ ( H T ) to ∂ t u ( t, x ) = Lu ( t, x ) − λu ( t, x ) + f ( t, x ) , λ ≥ , (4.6) u (0 , x ) = 0 , ( t, x ) ∈ H T , where for any function ϕ ∈ C b (cid:0) R d (cid:1) , Lϕ ( x ) := Z [ ϕ ( x + Gy ) − ϕ ( x ) − χ α ( y ) Gy · ∇ ϕ ( x )] ν ( dy ) . Moreover, there exists a constant C depending on κ, β, d, T, µ, ν, (cid:13)(cid:13) G − (cid:13)(cid:13) , k G k such that | u | β, ∞ ≤ C (cid:0) λ − ∧ T (cid:1) | f | β, ∞ , (4.7) | u | β, ∞ ≤ C | f | β, ∞ (4.8) And there is a constant C depending on κ, β, d, T, µ, ν, (cid:13)(cid:13) G − (cid:13)(cid:13) , k G k such thatfor all ≤ s < t ≤ T , κ ∈ [0 , , (4.9) | u ( t, · ) − u ( s, · ) | κ + β, ∞ ≤ C | t − s | − κ | f | β, ∞ . Proof. We first assume f ( t, x ) ∈ C ∞ b ( H T ) ∩ ˜ C β ∞ , ∞ ( H T ). Existence. Denote F ( r, Z νr ) = e − λ ( r − s ) f ( s, x + GZ νr − GZ νs ) , s ≤ r ≤ t, and apply the Itˆo formula to F ( r, Z νr ) on [ s, t ]. e − λ ( t − s ) f ( s, x + GZ νt − GZ νs ) − f ( s, x )= − λ Z ts F ( r, Z νr ) dr + Z ts Z χ α ( y ) y · ∇ F (cid:0) r, Z νr − (cid:1) ˜ J ( dr, dy )+ Z ts Z (cid:2) F (cid:0) r, Z νr − + y (cid:1) − F (cid:0) r, Z νr − (cid:1) − χ α ( y ) y · ∇ F (cid:0) r, Z νr − (cid:1)(cid:3) J ( dr, dy ) . Take expectation for both sides. e − λ ( t − s ) E f ( s, x + GZ νt − GZ νs ) − f ( s, x )= − λ Z ts e − λ ( r − s ) E f ( s, x + GZ νr − GZ νs ) dr + Z ts Le − λ ( r − s ) E f (cid:0) s, x + GZ νr − − GZ νs (cid:1) dr. Integrate both sides over [0 , t ] with respect to s and obtain Z t e − λ ( t − s ) E f ( s, x + GZ νt − GZ νs ) ds − Z t f ( s, x ) ds = − λ Z t Z r e − λ ( r − s ) E f ( s, x + GZ νr − GZ πs ) dsdr + Z t L Z r e − λ ( r − s ) E f (cid:0) s, x + GZ νr − − GZ νs (cid:1) dsdr, which shows u ( t, x ) = R t e − λ ( t − s ) E f (cid:0) s, x + GZ νt − s (cid:1) ds solves (4.6) in the in-tegral sense. As a result of the dominated convergence theorem and Fubini’stheorem, u ∈ C ∞ b ( H T ). And by the equation, u is continuously differen-tiable in t . Uniqueness. Suppose there are two solutions u , u solving the equation,then u := u − u solves ∂ t u ( t, x ) = Lu ( t, x ) − λu ( t, x ) , (4.10) u (0 , x ) = 0 . Apply the Itˆo formula to v ( t − s, Z νs ) := e − λs u ( t − s, x + GZ νs ), 0 ≤ s ≤ t, over [0 , t ] and take expectation for both sides of the resulting identity,then u ( t, x ) = − E Z t e − λs (cid:2) ( − ∂ t u − λu + Lu ) ◦ (cid:0) t − s, x + GZ νs − (cid:1)(cid:3) ds = 0 . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 33 Estimates for Smooth Inputs. Denote g ( t, x ) = f ( t, Gx ) , x ∈ R d .Then by Lemma 11, for any β ∈ (0 , | u ( t, · ) | β, ∞ ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t e − λ ( t − s ) E f (cid:0) s, G · + GZ νt − s (cid:1) ds (cid:12)(cid:12)(cid:12)(cid:12) β, ∞ = (cid:12)(cid:12)(cid:12)(cid:12)Z t e − λ ( t − s ) E g (cid:0) s, · + Z νt − s (cid:1) ds (cid:12)(cid:12)(cid:12)(cid:12) β, ∞ ≤ C (cid:0) λ − ∧ T (cid:1) | g ( t, · ) | β, ∞ ≤ C (cid:0) λ − ∧ T (cid:1) | f ( t, · ) | β, ∞ . Similarly, we can prove (4.8) , (4.9). Estimates for H¨older Inputs. This part of proof is identical tosection 5 of [15]. (cid:3) Proof of Theorem 1.1. We aim at providing a unifying proof forboth L = A + Q and L = G + Q . Before that, we claim Lemma 16. Let β ∈ (0 , /α ) and f, g ∈ ˜ C β (cid:0) R d (cid:1) . Then | f g | β ≤ | f | | g | + | f | [ g ] β + | g | [ f ] β , | f | = sup z | η m,z f | , ∀ k ∈ N + , and for some positive constant C that does not depend on m , | f | β ≤ Cl ( m ) β | f | + sup z | η m,z f | β , sup z | η m,z f | β ≤ Cl ( m ) β | f | + | f | β . Proof. Proof for the first two is identical to that for the standard H¨oldernorm. By monotonicity of the scaling factor, | f | β ≤ C sup | x − y | > m l (cid:18) | x − y | (cid:19) β | f | + sup | x − y |≤ m | f ( x ) − f ( y ) | w ( | x − y | ) β ≤ Cl ( m ) β | f | + sup z | η m,z ( x ) f ( x ) − η m,z ( y ) f ( y ) | w ( | x − y | ) β ≤ Cl ( m ) β | f | + sup z | η m,z f | β . Meanwhile,sup z | η m,z f | β ≤ | f | | η | + | f | sup z [ η m,z ] β + | η | [ f ] β ≤ Cl ( m ) β | f | + | f | β . (4.11) (cid:3) Without introducing much confusion, we will use L z to represent (3.9) and(3.13) at the same time, and h u, η m,z i z for both (3.10) and (3.17). We willalso use |·| β and |·| β, ∞ interchangeably, which is justified by ˜A ( w, l, γ )(ii). Estimates and Uniqueness. Let u ∈ ˜ C β ∞ , ∞ ( H T ) be a solution to(1.11), either L = A + Q or L = G + Q . Obviously, ∂ t ( η m,z u ) = η m,z ( L z u ) − λ ( η m,z u ) + η m,z f + η m,z [( L − L z ) u ] , where by elementary derivation, η m,z ( L z u ) = L z ( η m,z u ) − u ( L z η m,z ) − h u, η m,z i z , (4.12)therefore, η m,z u solves ∂ t ( η m,z u ) = L z ( η m,z u ) − λ ( η m,z u ) − u ( L z η m,z ) + η m,z f + η m,z [( L − L z ) u ] − h u, η m,z i z . (4.13)As an application of Theorems 4.1 and 4.2, we have | η m,z u | β, ∞ ≤ C (cid:0) | u ( L z η m,z ) | β, ∞ + | η m,z f | β, ∞ (4.14) + | η m,z [( L − L z ) u ] | β, ∞ + |h u, η m,z i z | β, ∞ (cid:1) , | η m,z u | β, ∞ ≤ C (cid:0) λ − ∧ T (cid:1) (cid:0) | u ( L z η m,z ) | β, ∞ + | η m,z f | β, ∞ + | η m,z [( L − L z ) u ] | β, ∞ + |h u, η m,z i z | β, ∞ (cid:1) for some C independent of λ . Clearly, | L z η m,z | β, ∞ ≤ C | η m,z | β, ∞ ≤ Cl ( m ) β . Then by Lemma 16 and [15, Proposition 4], | u ( L z η m,z ) | β, ∞ ≤ C | u ( L z η m,z ) | β ≤ | u | β | L z η m,z | + | u | | L z η m,z | β ≤ Cl ( m ) β | u | β, ∞ ≤ Cl ( m ) β (cid:16) ε | u | β + C ε | u | (cid:17) . Apply Lemma 16 again. | η m,z [( L − L z ) u ] | β, ∞ ≤ C (cid:18) l ( m ) β (cid:12)(cid:12)(cid:12)(cid:16) e L − L z (cid:17) u (cid:12)(cid:12)(cid:12) + l ( m ) β |Q u | + (cid:12)(cid:12)(cid:12)(cid:16) e L − L z (cid:17) u (cid:12)(cid:12)(cid:12) β, ∞ + |Q u | β, ∞ (cid:19) , where e L is either A or G . Then by Lemmas 9, 12, 15 and Corollaries 1,2, | η m,z [( L − L z ) u ] | β, ∞ ≤ Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) + CF ( m, x, z ) | u | β, ∞ , where F ( m, x, z ) := sup t,y, | x − z |≤ /m | ρ ( t, x, y ) − ρ ( t, z, y ) | if L = A + Q , and F ( m, x, z ) := sup | x − z |≤ /m k G ( x ) − G ( z ) k σ if L = G + Q . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 35 Combining Lemmas 10, 13, we obtain | u ( L z η m,z ) | β, ∞ + | η m,z f | β, ∞ + | η m,z [( L − L z ) u ] | β, ∞ + |h u, η m,z i z | β, ∞ ≤ l ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) + C (cid:16) l ( m ) β | f | + | f | β, ∞ (cid:17) + CF ( m, x, z ) | u | β, ∞ . (4.15)An immediate conclusion of this estimate is | η m,z u | β, ∞ ≤ C (cid:0) λ − ∧ T (cid:1) (cid:16) F ( m, x, z ) | u | β, ∞ + l ( m ) β | f | + | f | β, ∞ (cid:17) + C (cid:0) λ − ∧ T (cid:1) l ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , (4.16)where C does not depend on λ, m . Thus, | u | ≤ C sup z | η m,z u | β ≤ C (cid:0) λ − ∧ T (cid:1) l ( m ) β (cid:16) | u | β, ∞ + | f | β, ∞ (cid:17) , (4.17)Combining (4.14) , (4.15) , (4.17), we then have | η m,z u | β, ∞ ≤ εl ( m ) β | u | β, ∞ + C ε (cid:0) λ − ∧ T (cid:1) l ( m ) β (cid:16) | u | β, ∞ + | f | β, ∞ (cid:17) + Cl ( m ) β | f | β, ∞ + CF ( m, x, z ) | u | β, ∞ . (4.18)On the other hand, by Lemma 16, | L µ u | β, ∞ ≤ C | L µ u | β ≤ Cl ( m ) β | L µ u | + C sup z | η m,z L µ u | β ≤ Cl ( m ) β | L µ u | + C sup z | η m,z L µ u | β, ∞ . Let ρ ( z, y ) = 1 , ν ( dy ) = µ ( dy ) in Lemma 9 and utilize (4.12).sup z | η m,z L µ u | β, ∞ ≤ sup z | η m,z u | β, ∞ + sup z | u ( L µ η m,z ) | β, ∞ + sup z |h u, η m,z i z | β, ∞ ≤ sup z | η m,z u | β, ∞ + Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) , (4.19) where C does not depend on λ, m . Combining (4.17) , (4.18), we obtain | u | β, ∞ ≤ C (cid:16) | u | + | L µ u | β, ∞ (cid:17) ≤ C (cid:0) λ − ∧ T (cid:1) l ( m ) β (cid:16) | u | β, ∞ + | f | β, ∞ (cid:17) + C sup z | η m,z u | β, ∞ + Cl ( m ) β (cid:16) ε | u | β, ∞ + C ε | u | (cid:17) ≤ εl ( m ) β | u | β, ∞ + C ε (cid:0) λ − ∧ T (cid:1) l ( m ) β (cid:16) | u | β, ∞ + | f | β, ∞ (cid:17) + C ε l ( m ) β | f | β, ∞ + CF ( m, x, z ) | u | β, ∞ . In the inequality above, we first set m sufficiently large so that CF ( m, x, z ) | u | β, ∞ ≤ | u | β, ∞ . For such an m , we then select ε such that εl ( m ) β < / 4. At last, wechoose λ large enough so that for such m, ε , C ε (cid:0) λ − ∧ T (cid:1) l ( m ) β < / m, ε, λ , | u | β, ∞ ≤ C ( λ ) | f | β, ∞ .We need λ to be sufficiently large though, say λ ≥ λ . To completelyrelax this constraint, let us consider v ( t, x ) := e ( λ − λ ) t u ( t, x ) , λ > 0, where u solves (1.11). Then v is a solution to ∂ t v ( t, x ) = L v ( t, x ) − λ v ( t, x ) + e ( λ − λ ) t f ( t, x ) , λ ≥ ,v (0 , x ) = 0 , ( t, x ) ∈ H T , and | v | β, ∞ = (cid:12)(cid:12)(cid:12) e ( λ − λ ) t u (cid:12)(cid:12)(cid:12) β, ∞ ≤ C λ (cid:12)(cid:12)(cid:12) e ( λ − λ ) t f (cid:12)(cid:12)(cid:12) β, ∞ . Namely, | u | β, ∞ ≤ C λ | f | β, ∞ . Note C λ is uniform with respect to λ .Now we can conclude from (4.16), (4.17) and Lemma 16 that | u | β, ∞ ≤ Cl ( m ) β | u | + C sup z | η m,z u | β, ∞ ≤ C (cid:0) λ − ∧ T (cid:1) | f | β, ∞ , where C does not depend on λ, u, f .Again, according to Theorems 4.1, 4.2 and (4.15) , (4.13), there is a con-stant C depending on κ, β, d, T, µ, ν such that for all 0 ≤ s < t ≤ T , κ ∈ [0 , | η m,z u ( t, · ) − η m,z u ( s, · ) | β, ∞ ≤ C ( t − s ) − κ (cid:0) | u ( L z η m,z ) | β, ∞ + | η m,z f | β, ∞ + | η m,z [( L − L z ) u ] | β, ∞ + |h u, η m,z i z | β, ∞ (cid:1) ≤ Cl ( m ) β ( t − s ) − κ | f | β, ∞ . HE CAUCHY PROBLEM IN GENERALIZED H ¨OLDER SPACES 37 Apply Lemma 16 and repeat derivation (4.19) for the difference function, | u ( t, · ) − u ( s, · ) | β, ∞ ≤ C (cid:16) | u ( t, · ) − u ( s, · ) | + | L µ u ( t, · ) − L µ u ( s, · ) | β, ∞ (cid:17) ≤ C sup z | η m,z u ( t, · ) − η m,z u ( s, · ) | + Cl ( m ) β sup z | η m,z L µ u ( t, · ) − η m,z L µ u ( s, · ) | β, ∞ ≤ (cid:16) Cl ( m ) β + C ε l ( m ) β (cid:17) | η m,z u ( t, · ) − η m,z u ( s, · ) | β, ∞ + εl ( m ) β | u ( t, · ) − u ( s, · ) | β, ∞ . Choose ε such that εl ( m ) β < / 2. Then we arrive at | u ( t, · ) − u ( s, · ) | β, ∞ ≤ (cid:16) Cl ( m ) β + C ε l ( m ) β (cid:17) | η m,z u ( t, · ) − η m,z u ( s, · ) | β, ∞ ≤ C ( t − s ) − κ | f | β, ∞ . Uniqueness of the solution is a direct consequence of these estimates. Existence. Let V ( H T ) be the linear space that for any v ∈ V ( H T ), thereexists a unique f ∈ ˜ C β ∞ , ∞ ( H T ) such that v ( t, x ) = R t f ( s, x ) ds . Equip V ( H T ) with norm | v | V := | f | β, ∞ . Let U ( H T ) be the linear space that forany u ∈ U ( H T ), there is g ∈ ˜ C β ∞ , ∞ ( H T ) such that u ( t, x ) = R t g ( s, x ) ds .Endow U ( H T ) with norm | u | U := | u | β, ∞ . Then V ( H T ) is a normed linearspace and U ( H T ) is a Banach space. Define for θ ∈ [0 , T θ u ( t, x ) = θ (cid:18) u ( t, x ) − Z t ( L u ( s, x ) − λu ( s, x )) ds (cid:19) + (1 − θ ) (cid:18) u ( t, x ) − Z t ( L ν u ( s, x ) − λu ( s, x )) ds (cid:19) := u ( t, x ) − Z t [ L θ u ( s, x ) − λu ( s, x )] ds, where L θ = θ L + (1 − θ ) L ν . Take u ∈ U ( H T ). Then u ( t, x ) := R t g ( s, x ) ds for some g ∈ ˜ C β ∞ , ∞ ( H T ). Clearly, for any θ ∈ [0 , u solves u ( t, x ) = Z t [ L θ u ( s, x ) − λu ( s, x ) + (cid:16) g ( s, x ) − L θ u ( s, x ) + λu ( s, x ) (cid:17) ] ds. Therefore, T θ u ( t, x ) = Z t [ g ( s, x ) − L θ u ( s, x ) + λu ( s, x )] ds, where by Lemma 15, Proposition 2 and Corollary 1, |T θ u | V = | g − L θ u + λu | β, ∞ ≤ C | u | β, ∞ < ∞ . Then, T θ [ U ( H T )] ⊂ V ( H T ). Meanwhile, by estimates we derived above,there is C independent of u, θ such that | u | U = | u | β, ∞ ≤ C | g − L θ u + λu | β, ∞ ≤ C |T θ u | V . Theorem 4.1 says T maps U onto V . By Theorem 5.2 in [4], so does T .5. Appendix Lemma 17. [15, Lemma 2] Let ν be a L´evy measure and w be the scalingfunction which ν satisfies A(w,l) for. Then,a) there are constants C , C > such that C ς ( r ) ≤ w ( r ) − ≤ C ς ( r ) , ∀ r > . (5.1) b) R | y |≤ w ( | y | ) ν ( dy ) = + ∞ .c) For any ε > , R | y |≤ w ( | y | ) ε ν ( dy ) < ∞ .d) For any ε > , R | y |≤ | y | ε w ( | y | ) ν ( dy ) < ∞ . Lemma 18. [9, Lemma 5] Let ν be a L´evy measure satisfying A(w,l) . Z ˜ ν R t is the L´evy process associated to ˜ ν R , R > . For each t, R , Z ˜ ν R t has a boundedand continuous density function p R ( t, x ) , t ∈ (0 , ∞ ) , x ∈ R d . And p R ( t, x ) has bounded and continuous derivatives up to order . Meanwhile, for anymulti-index | ϑ | ≤ , Z (cid:12)(cid:12)(cid:12) ∂ ϑ p R ( t, x ) (cid:12)(cid:12)(cid:12) dx ≤ Cγ ( t ) −| ϑ | , sup x ∈ R d (cid:12)(cid:12)(cid:12) ∂ ϑ p R ( t, x ) (cid:12)(cid:12)(cid:12) ≤ Cγ ( t ) − d −| ϑ | , where C > is independent of t, R . For any β ∈ (0 , such that | ϑ | + β < , Z (cid:12)(cid:12)(cid:12) ∂ β ∂ ϑ p R ( t, x ) (cid:12)(cid:12)(cid:12) dx ≤ Cγ ( t ) −| ϑ |− β . For any a > , there is a constant C > independent of t, R , so that Z | x | >a (cid:12)(cid:12)(cid:12) ∂ ϑ p R ( t, x ) (cid:12)(cid:12)(cid:12) dx ≤ C (cid:16) γ ( t ) −| ϑ | + tγ ( t ) −| ϑ | (cid:17) . Acknowledgments I would like to thank Prof. Remigijus Mikuleviˇcius for useful discussions. References [1] Bergn, J. and L¨ofstr¨om, J., Interpolation Spaces. 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