Schauder's estimates for nonlocal equations with singular Lévy measures
aa r X i v : . [ m a t h . P R ] F e b SCHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULARL ´EVY MEASURES
ZIMO HAO, ZHEN WANG AND MINGYAN WUA bstract . In this paper, we establish Schauder’s estimates for the following non-local equa-tions in R d : ∂ t u = L ( α ) κ,σ u + b · ∇ u + f , u (0) = , where α ∈ (1 / ,
2) and b : R + × R d → R is an unbounded local β -order H¨older function in x uniformly in t , and L ( α ) κ,σ is a non-local α -stable-like operator with form: L ( α ) κ,σ u ( t , x ) : = Z R d (cid:16) u ( t , x + σ ( t , x ) z ) − u ( t , x ) − σ ( t , x ) z ( α ) · ∇ u ( t , x ) (cid:17) κ ( t , x , z ) ν ( α ) (d z ) , where z ( α ) = z α ∈ (1 , + z | z | α = , κ : R + × R d → R + is bounded from above and below, σ : R + × R d → R d ⊗ R d is a γ -order H¨older continuous function in x uniformly in t , and ν ( α ) isa singular non-degenerate α -stable L´evy measure.
1. I ntroduction
Let b be a measurable vector-valued function on R d , and a be a measurable symmetricmatrix-valued function on R d . Denote by ∂ i the i -th partial derivative ∂∂ x i . Consider the follow-ing elliptic equation: a i , j ∂ i ∂ j u + b i ∂ i u = f . (1.1)Here and below we use the Einstein summation convention. Suppose that f belongs to C β ,where C β stands for the global H¨older spaces (see Subsection 2.1). Assume that there is aconstant λ > a is strictly elliptic, i.e., ξ i a i , j ξ j > λ | ξ | , ∀ ξ ∈ R d , and the H¨older norms of coe ffi cients are all bounded by another constant Λ >
0, i.e., k a k C β + k b k C β Λ . Then, Schauder’s estimates tell us that there is a constant c = c ( d , β, λ, Λ ) > u ∈ C + β of (1.1), k u k C + β c ( k u k L ∞ + k f k C β ) . It is well-known that Schauder’s estimates play a basic role in constructing the classical so-lution for quasilinear PDEs, and also give an approach to show the well-posenesses of SDEs(see [25], [5], [17], [12], etc.). For heat equations, we can find many ways to prove such an es-timate, such as [16], [19], [20], and so on. A natural question is whether Schauder’s estimates
Date : February 25, 2020.
Keywords:
Schauder estimate, Littlewood-Paley’s decomposition, Heat kernel, Supercritical non-localequation. hold when we replace the local operator a i , j ∂ i ∂ j by some non-local ones? These problems aredrawn great interests recently (see [3], [2], [13], [18], and [15]).In this paper, we consider the following equation: ∂ t u = L ( α ) κ,σ u + b · ∇ u + f , u (0) = , (1.2)where b : R + × R d → R d is measurable, b · ∇ = b i ∂ i , and L ( α ) κ,σ is an α -stable-like operator withform: L ( α ) κ,σ u ( t , x ) : = Z R d (cid:16) u ( t , x + σ ( t , x ) z ) − u ( t , x ) − σ ( t , x ) z ( α ) · ∇ u ( t , x ) (cid:17) κ ( t , x , z ) ν ( α ) (d z ) , (1.3)where z ( α ) = z α ∈ (1 , + z | z | α = with α ∈ (0 , σ : R + × R d → R d ⊗ R d and κ : R + × R d → R are measurable, and ν ( α ) is a non-degenerate α -stable L´evy measure which can be very singular(see Subsection 2.2).Throughout this paper, we make the following assumptions on κ, σ and b : (H βκ ) For some c > β ∈ [0 , t > x , y , z ∈ R d , c − κ ( t , x , z ) c , | κ ( t , x , z ) − κ ( t , y , z ) | c | x − y | β . and in the case of α = Z r | z | R z κ ( t , x , z ) ν ( α ) (d z ) = < r < R < ∞ . (H γσ ) For some c > γ ∈ [0 , t > x , ξ ∈ R d , c − | ξ | | σ ( t , x ) ξ | c | ξ | , k σ ( t , x ) − σ ( t , y ) k c | x − y | γ . (H β b ) For some c > β ∈ [0 , t > x , y ∈ R d with | x − y | | b ( t , | c , | b ( t , x ) − b ( t , y ) | c | x − y | β . Here, k · k denotes the Hilbert-Schmidt norm of a matrix, and | · | denotes the Euclidean norm.Notice that L ( α ) κ,σ u ( t , x ) is meaningful when u ( t , · ) ∈ C γ for some γ > α .From the view point of PDEs, the drift term, instead of the di ff usion term, plays a dominantrole in the supercritical case α ∈ (0 , α ∈ (0 , σ is the identity matrix I , κ ≡
1, and ν ( α ) (d z ) = / | z | d + α d z with α ∈ (0 , L ( α ) κ,σ =∆ α , Silvestre [22] obtained an interior Schauder estimate under some H¨older continuous andbounded drifted coe ffi cents. Moreover, Zhang and Zhao [26] studied Schauder’s estimates forPDE (1.2) with L´evy measure ν ( α ) (d z ) = / | z | d + α d z . In addition, for singular L´evy measures,Chen, Zhang and Zhao [11] showed a Besov-type apriori estimate: for every p > d / ( α + β − c > k u k L ∞ ([0 , T ]; B α + β p , ∞ ) c k f k L ∞ ([0 , T ]; B β p , ∞ ) , where B α + β p , ∞ is the usual Besov space (see Definition 2.6 below), and the drift b ∈ L ∞ ([0 , T ]; B β p , ∞ )with p , ∞ . Recently, under (H β b ) with α ∈ ( ,
1) and α + β >
1, and σ ≡ I , Chaudru de Raynal, CHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULAR L ´EVY MEASURES 3
Menozzi and Priola [6] proved the following Schauder estimate for PDE (1.2): k u k L ∞ ([0 , T ]; C α + β ) c k f k L ∞ ([0 , T ]; C β ) . It was not known untill now if the above Schauder’s estimates hold when α ∈ (0 ,
1) and σ depends on x .In the sequel, use : = as a way of definition. For a Banach space B and T >
0, we denote L ∞ T ( B ) : = L ∞ ([0 , T ]; B ) , L ∞ loc ( B ) : = ∩ T > L ∞ T ( B ) , L ∞ T : = L ∞ ([0 , T ] × R d ) . Additionally, a ∨ b : = max( a , b ), a ∧ b : = min( a , b ). The aim of this paper is to prove thefollowing theorem which gives Schauder estimate for PDE (1.2) and the existence of classicalsolutions (see Definition 3.1 below). Theorem 1.1.
Suppose that α ∈ (1 / , , γ ∈ ( − αα ∨ , , β ∈ ((1 − α ) ∨ , ( α ∧ γ ) , and α + β < N . Under (H βκ ) , (H γσ ) , and (H β b ) , for any f ∈ L loc ( C β ) , there is a unique classicalsolution u in the sense of Definition 3.1 such that for any T > and some constant c = c ( T , c , d , α, β, γ ) > , k u k L ∞ T ( C α + β ) c k f k L ∞ T ( C β ) , k u k L ∞ T T k f k L ∞ T . (1.4)We point out that there are few results of heat kernel estimates for the operator ∂ t − L ( α ) κ,σ when ν ( α ) (d z ) , / | z | d + α d z and σ is not a constant. Hence, it seems to be quite di ffi cult to obtainSchauder’s estimates as stated in Theorem 1.1 by using methods from [6]. A key ingredient inour approach is the use of Littlewood-Paley’s theory. Remark 1.2.
Notice that b ( x ) = x satisfies the condition (H β b ) for any β ∈ [0 , . Hence,Theorem 1.1 covers some unbounded drift cases. Remark 1.3.
In the Theorem 1.1, α is required to be greater than / due to some momentproblems (see Remark 3.10 ). This restriction also appears in [6] . An open problem is to dropthe restriction α > / . If L ( α ) κ,σ = ∆ α/ , i.e. σ ≡ I , κ ≡ , we can drop the restriction α > / ,and then obtain Schauder’s estimates for α ∈ (0 , in our method( see Remark 3.12 ). This paper is organized as follows: In Section 2, we introduce some basic function spaces,and present the estimates of Littlewood-Paley’s types for heat kernels of nonlocal operatorswith constant coe ffi cients (see Lemma 2.12 below). In Section 3, we show the the maximumprinciple Lemma 3.3, and prove Schauder’s estimates Theorem 3.4 for PDE (1.2) by freezingcoe ffi cients along the characterization curve. In Section 4, through the continuity method, weapply the apriori estimate Theorem 3.4 to show the main result Theorem 1.1.Throughout this paper, we shall use the following conventions and notations: • We use A . B to denote A cB for some unimportant constant c > • N : = N ∪ { } , R + : = [0 , ∞ ), and for R >
0, we shall denote B R : = { x ∈ R d : | x | < R } . • If Ω is a domain in R d , for any p ∈ [1 , ∞ ], we denote by L p ( Ω ) the space of p -summablefunction on Ω . Denote L p : = L p ( R d ) with the norm denoted k · k p . • For two operators A , A , we use [ A , A ] : = A A − A A to denote their commuta-tor. ZIMO HAO, ZHEN WANG AND MINGYAN WU
2. P reliminaries
H ¨older spaces and Besov spaces.
We first introduce the H¨older spaces. For h ∈ R d and f : R d → R , the first order di ff erence operator is defined by δ h f ( x ) : = f ( x + h ) − f ( x ) . For 0 < β < N , if Ω is a domain in R d , let C β ( Ω ) be the usual β -order H¨older space on Ω consisting of all functions f : Ω → R with k f k C β ( Ω ) : = k f k L ∞ ( Ω ) + · · · + k∇ [ β ] f k L ∞ ( Ω ) + [ ∇ [ β ] f ] C β − [ β ] ( Ω ) < ∞ , where [ β ] denotes the greatest integer less than β , and ∇ j stands for the j -order gradient, and[ f ] C γ ( Ω ) : = sup x , x + h ∈ Ω h , | δ h f ( x ) | / | h | γ , γ ∈ [0 , . Denote C β : = C β ( R d ) and [ · ] C β : = [ · ] C β ( R d ) . We introduce another notation:[ f ] C γ : = sup < | h | k δ h f k ∞ / | h | γ , γ ∈ [0 , . For any integer n >
1, define C n be the set of n -order continuous di ff erentiable functions on R d with k f k C n : = n X k = k∇ n f k L ∞ ( R d ) < ∞ . If f belongs to C , then it is Lipschtiz. Remark 2.1.
For < s < , note that the set consisting of all functions whose C s -seminormsare finte is bigger than C s -seminorms. For example, let f ( x ) = x, then [ f ] C s < ∞ but [ f ] C s = ∞ . This fact tells us that for some unbounded functions, their C s -seminorms can be finite. The following result is simple but important (see [25, Lemma2.1]).
Lemma 2.2. If < s < , then for any x , y ∈ R d with | x − y | > , | f ( x ) − f ( y ) | f ] C s | x − y | . In the sequel, let χ : R d → R be a smooth function with χ ( x ) = | x | ;0 if | x | > . (2.1)Notice that [ f ] C s < ∞ is the local property of f and [ f ] C s < ∞ is the global property. Thefollowing results are useful and their proofs are straightforward and elementary. Lemma 2.3.
For fixed x ∈ R d , define ˜ f ( x ) : = ( f ( x ) − f ( x )) χ ( x − x ) , ∀ x ∈ R d . Then, for anys ∈ (0 , , there exists a constant c = c ( s , χ ) > such that k ˜ f k C s c [ f ] C s . Lemma 2.4.
Assume that γ ∈ (0 , and β ∈ (0 , γ ) . Let φ : R d → R d with [ φ ] C γ < ∞ , and letf ∈ C β/γ . Then, g ( · ) : = f ( · + φ ( · )) ∈ C β with k g k C β (cid:16) + [ φ ] β/γ C γ (cid:17) k f k C β/γ . CHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULAR L ´EVY MEASURES 5
Next we are going to introduce the Besov spaces. Let S ( R d ) be the Schwartz space of allrapidly decreasing functions on R d , and S ′ ( R d ) be the dual space of S ( R d ) called Schwartzgeneralized function (or tempered distribution) space. For any f ∈ S ( R d ), the Fourier trans-form ˆ f and the inverse Fourier transform ˇ f are defined byˆ f ( ξ ) : = (2 π ) − d / Z R d e − i ξ · x f ( x )d x , ξ ∈ R d , ˇ f ( x ) : = (2 π ) − d / Z R d e i ξ · x f ( ξ )d ξ, x ∈ R d . For any f ∈ S ′ ( R d ), the Fourier transform ˆ f and the inverse Fourier transform ˇ f are definedby h ˆ f , ϕ i : = h f , ˆ ϕ i , h ˇ f , ϕ i : = h f , ˇ ϕ i , ∀ ϕ ∈ S ( R d ) . Let φ be a radial C ∞ -function on R d with φ ( ξ ) = | ξ | φ ( ξ ) = | ξ | > . For ξ = ( ξ , · · · , ξ n ) ∈ R d and j ∈ N , define φ j ( ξ ) : = φ (2 − j ξ ) − φ (2 − ( j − ξ ) . It is easy to see that for j ∈ N , φ j ( ξ ) = φ (2 − ( j − ξ ) > φ j ⊂ { ξ ∈ R d | j − | ξ | j + } , k X j = φ j ( ξ ) = φ (2 − k ξ ) → , k → ∞ . In particular, if | j − j ′ | >
2, thensupp φ (2 − j · ) ∩ supp φ (2 − j ′ · ) = ∅ . From now on we shall fix such φ and φ . For j ∈ N , the block operator ∆ j is defined on S ′ ( R d ) by ∆ j f ( x ) : = ( φ j ˆ f )ˇ( x ) = ˇ φ j ∗ f ( x ) = ( j − d Z R d ˇ φ (2 j − y ) f ( x − y )d y . (2.2) Remark 2.5.
For j ∈ N , by definitions it is easy to see that ∆ j = ∆ j e ∆ j , where e ∆ j : = ∆ j − + ∆ j + ∆ j + with ∆ − ≡ , (2.3) and ∆ j is symmetric in the sense that h ∆ j f , g i = h f , ∆ j g i . Here is the definition for the Besov spaces.
Definition 2.6 (Besov spaces) . For any s ∈ R and p , q ∈ [1 , ∞ ] , the Besov space B sp , q is definedas the set of all f ∈ S ′ ( R d ) such that k f k B sp , q : = (cid:16) ∞ X j = jqs k ∆ j f k qp (cid:17) / q < ∞ . When q = ∞ , it is in the following sense k f k B sp , ∞ : = sup j ∈ N js k ∆ j f k p < ∞ . ZIMO HAO, ZHEN WANG AND MINGYAN WU
Recall the Bernstein’s inequality( [1, Lemma 2.1]).
Lemma 2.7 (Bernstein’s inequality) . For any k = , , , · · · , there is a constant c = c ( k , d ) > such that for all j > , k∇ k ∆ j f k ∞ c k j k ∆ j f k ∞ . (2.4) Remark 2.8.
It is well-known that for any < s < N and n ∈ N , k f k B s ∞ , ∞ ≍ k f k C s , k f k B n ∞ , ∞ . k f k C n . The proof can be found in [24] or [1] . We also need the following interpolation inequality [4, Theorem 6.4.5-(3)].
Lemma 2.9.
Let A be a Banach space and β < β be two positive noninteger numbers. If T is a bounded linear operator from C β to A , then there is a constant c = c ( β , β ) > such that kT k B ( C β , A ) c kT k θ B ( C β , A ) kT k − θ B ( C β , A ) , where β = θβ + (1 − θ ) β < N for some θ ∈ [0 , . L´evy measures and heat kernel estimates.
We call a measure ν on R d a L´evy measureif ν ( { } ) = , Z R d (cid:0) ∧ | x | (cid:1) ν (d x ) < + ∞ . In particular, for α ∈ (0 , ν ( α ) is α -stable if it has form ν ( α ) ( A ) = Z ∞ Z S d − A ( r θ ) Σ (d θ ) r + α ! d r , A ∈ B ( R d ) , (2.5)where Σ is a finite measure over the unit sphere S d − (called spherical measure of ν ( α ) ). Noticethat, for any γ > α > γ > Z R d ( | z | γ ∧ | z | γ ) ν ( α ) (d z ) < ∞ . (2.6)One says that an α -stable measure ν ( α ) is non-degenerate if Z S d − | θ · θ | α Σ (d θ ) > θ ∈ S d − . (2.7) Example 2.10 (Standard α -stable measures) . If Σ is the uniform measure on unit sphere S d − ,then ν ( α ) is the standard or strict α -stable L´evy measure and ν ( α ) (d y ) = d y | y | d + α . In this case, [ L ( α )1 , I f ( ξ ) = −| ξ | α ˆ f ( ξ ) , ∀ ξ ∈ R d , where L ( α ) κ,σ is defined by (1.3) and I is the identity matrix. CHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULAR L ´EVY MEASURES 7
Example 2.11 (Cylindrical α -stable measures) . If Σ = P dk = δ e k , where δ e k is the Dirac measureat the e k = (0 , · · · , , k th , , · · · , , then ν ( α ) (d x ) = n X k = δ (d x ) · · · δ (d x k − ) d x k | x k | + α δ (d x k + ) · · · δ (d x d ) , where δ is the Dirac measure at the zero. Such measure is called the cylindrical L´evy measure.Moreover, [ L ( α )1 , I f ( ξ ) = − d X i = | ξ i | α ˆ f ( ξ ) , ∀ ξ = ( ξ , .., ξ d ) ∈ R d . Notice that | ξ | α is not smooth at origin, and P di = | ξ i | α is not smooth on all axes ∪ di = { ξ i = } . Inother words, P di = | ξ i | α is more singular than | ξ | α . Fix α ∈ (0 , ν ( α ) be a non-degenerate α -stable measure, κ : R + × R d → R + and σ : R + → R d ⊗ R d be measurable functions satisfying the following assumptions: c − κ ( t , z ) c , c − | ξ | | σ ( t ) ξ | c | ξ | , ∀ ( t , z , ξ ) ∈ R + × R d × R d , (2.8)for some constant c > α = Z r | z | R z κ ( t , z ) ν ( α ) (d z ) = < r < R < ∞ . (2.9)Let N (d t , d z ) be the Possion random measure with intensity measure κ ( t , z ) ν ( α ) (d z )d t . For 0 s t , define L κ s , t : = Z ts Z R d z ˜ N (d r , d z ) + Z ts Z R d ( z − z ( α ) ) κ ( r , z ) ν ( α ) (d z )d r , (2.10)where ˜ N (d r , d z ) : = N (d r , d z ) − κ ( r , z ) ν ( α ) (d z )d r is the compensated Poisson random measureand z ( α ) : = z α ∈ (1 , + z | z | α = . Precisely, L κ s , t = R ts R R d zN (d r , d z ) , if α ∈ (0 , R ts R | z | z ˜ N (d r , d z ) + R t R | z | > zN (d r , d z ) , if α = R ts R R d z ˜ N (d r , d z ) , if α ∈ (1 , . Next, we consider the following process: X κ,σ s , t : = Z ts σ ( r )d L κ s , r , < s < t < ∞ . (2.11)By the same argument as in [7], we have the crucial lemma in this paper ZIMO HAO, ZHEN WANG AND MINGYAN WU
Lemma 2.12.
Let α ∈ (0 , . Under (2.8) and (2.9) , the random variable X κ,σ s , t defined by (2.11) has a smooth density p κ,σ s , t . Furthermore, for any T > , β ∈ [0 , α ) , and n ∈ N , there isa constant c = c ( c , α, ν ( α ) , β, T , d ) such that for any s , t ∈ [0 , T ] and j ∈ N , Z t Z R d | x | β | ∆ j p κ,σ s , t ( x ) | d x d s c − ( α + β ) j . (2.12)3. S chauder ’ s estimates for nonlocal equations In this section, we show Schauder’s estimates for nonlocal equations: ∂ t u = L ( α ) κ,σ u + b · ∇ u + f , u (0) = , (3.1)where b : R + × R d → R d is a measurable function and L ( α ) κ,σ is defined by (1.3) with α ∈ (0 , L ( α ) κ,σ u ( t , x ) : = Z R d (cid:16) u ( t , x + σ ( t , x ) z ) − u ( t , x ) − σ ( t , x ) z ( α ) · ∇ u ( t , x ) (cid:17) κ ( t , x , z ) ν ( α ) (d z ) . Throughout this section, we assume that κ, σ and b satisfy, respectively, conditions (H βκ ) , (H γσ ) and (H β b ) . Definition 3.1 (Classical solutions) . We call a bounded continuous function u defined on R + × R d a classical solution of PDE (3.1) if for some ε ∈ (0 , ,u ∈ (cid:16) ∩ M > C ( R + ; C ( α ∨ + ε ( B M )) (cid:17) ∩ L ∞ loc ( C ( α ∨ + ε ) and for all ( t , x ) ∈ [0 , ∞ ) × R d ,u ( t , x ) = Z t (cid:16) L ( α ) κ,σ u + b · ∇ u + f (cid:17) ( s , x )d s . (3.2) Remark 3.2.
Note that under the conditions (H κ ) and (H σ ) , L ( α ) κ,σ u ( t , x ) and b · ∇ u ( t , x ) ispointwisely well defined for any u ∈ L ∞ loc ( C γ ) with γ > α ∨ . Hence, the classical solution iswell-defined. We have the following maximum principle for classical solutions.
Lemma 3.3 (Maximum principle) . Assume that σ ( t , x ) and κ ( t , x , z ) > are bounded measur-able functions. Let b ( t , x ) be a measurable function and bounded in R + for fixed x ∈ R d . Then,for any T > and classical solution u of PDE (3.1) in the sense of Definition 3.1, it holds that k u k L ∞ T T k f k L ∞ T . Proof.
Define¯ u ( t , x ) : = − u ( t , x ) + Z t k f ( s , · ) k ∞ d s and u ( t , x ) : = − u ( t , x ) − Z t k f ( s , · ) k ∞ d s . By (3.2), it is easy to see that for Lebesgue almost all t > ∂ t ¯ u − L ( α ) κ,σ ¯ u − b · ∇ ¯ u > ∂ t u − L ( α ) κ,σ u − b · ∇ u , where lim t → ¯ u ( t , x ) = lim t → u ( t , x ) =
0. Notice that the form of ν ( α ) does not a ff ect the resultof [8, Theorem 6.1]. Thus, by [8, Theorem 6.1] and Lemma 2.2, we have u ( t , x ) ¯ u ( t , x ) CHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULAR L ´EVY MEASURES 9 which implies that | u ( t , x ) | Z t k f ( s , · ) k ∞ d s T k f k L ∞ T . The desired estimate is proved. (cid:3)
Our goal of this section is to prove the following Schauder’s apriori estimates.
Theorem 3.4 (Schauder’s estimates) . Let α ∈ (1 / , , γ ∈ ( − αα ∨ , and β ∈ ((1 − α ) ∨ , ( α ∧ γ ) with α + β < N . Under the conditions (H βκ ) , (H γσ ) , and (H β b ) , for any T > andf ∈ L ∞ T ( C β ) , there is a constant c = c ( T , c , d , α, β, γ ) > such that for any classical solutionu of PDE (3.1) , k u k L ∞ T ( C α + β ) c k f k L ∞ T ( C β ) . (3.3)To prove this theorem, we use the perturbation argument by freezing coe ffi cients along thecharacterization curve as showed in [17]. We need the following well-known fact from ODE,whose proof can be found in [17, Lemma 6.5]. Lemma 3.5.
Let b : R + × R d → R d be a measurable vector field. Suppose that for each t > ,x b ( t , x ) is continuous and there is a constant c > such that for all ( t , x ) ∈ R + × R d , | b ( t , x ) | c (1 + | x | ) . Then, for each x ∈ R d , there is a global solution θ t to the following ODE: ˙ θ t = b ( t , θ t ) , θ = x . Moreover, if we denote by S x : = { θ · : θ = x } the set of all solutions with starting point x, thenfor each T > , ∪ x ∈ R d ∪ θ · ∈ S x { θ T } = R d . Bounded drift case.
In this subsection, assuming b ∈ L ∞ loc ( C β ), we prove the followingapriori estimate. Theorem 3.6.
Let α ∈ (1 / , , γ ∈ ( − αα ∨ , and β ∈ ((1 − α ) ∨ , ( α ∧ γ ) with α + β < N .Under the conditions (H βκ ) , (H γσ ) , and b ∈ L ∞ loc ( C β ) , for any T > and f ∈ L ∞ T ( C β ) , there is aconstant c = c ( T , k b k L ∞ T ( C β ) , d , α, β, γ ) > such that for any classical solution u of PDE (3.1) , k u k L ∞ T ( C α + β ) c k f k L ∞ T ( C β ) . (3.4)Fix x ∈ R d . Let θ t solve the following ODE in R d :˙ θ t = − b ( t , θ t ) , θ = x . Define ˜ u ( t , x ) : = u ( t , x + θ t ) , ˜ f ( t , x ) : = f ( t , x + θ t ) , ˜ σ ( t , x ) : = σ ( t , x + θ t ) , ˜ κ ( t , x , z ) : = κ ( t , x + θ t , z ) , ˜ σ ( t ) : = ˜ σ ( t , , ˜ κ ( t , z ) : = ˜ κ ( t , , z ) , and ˜ b ( t , x ) : = b ( t , x + θ t ) − b ( t , θ t ) . It is easy to see that ˜ u satisfies the following equation: ∂ t ˜ u = L ( α )˜ κ , ˜ σ u + ˜ b · ∇ ˜ u + (cid:16) L ( α )˜ κ, ˜ σ − L ( α )˜ κ , ˜ σ (cid:17) ˜ u + ˜ f , where (see (1.3)) L ( α )˜ κ , ˜ σ u ( t , x ) : = Z R d Ξ ( α ) u ( t , x ; ˜ σ z )˜ κ ( t , z ) ν ( α ) (d z )with Ξ ( α ) u ( t , x ; ˜ σ z ) : = u ( t , x + ˜ σ z ) − u ( t , x ) − ˜ σ z ( α ) · ∇ u ( t , x ) . Under (H βκ ) , (H γσ ) and b ∈ L ∞ loc ( C β ), we have | ˜ b ( t , x ) | + | ˜ κ ( t , x , z ) − ˜ κ ( t , z ) | . | x | β and | ˜ σ ( t , x ) − ˜ σ ( t ) | . | x | γ . (3.5)Taking κ ( t , z ) = ˜ κ ( t , z ) and σ ( t ) = ˜ σ ( t ) in (2.11) and obesrving that (2.8) and (2.9) are stillvalid in this case, we have a smooth density p s , t for X ˜ κ , ˜ σ s , t . Define P s , t f ( s , x ) : = E f ( s , x + X ˜ κ , ˜ σ s , t ) = Z R d f ( s , x + y ) p s , t ( y )d y , ∀ x ∈ R d . Then, by Duhamel’s formula [7, Lemma 3.1] we have˜ u ( t , x ) = Z t P s , t (cid:16) L ( α )˜ κ, ˜ σ − L ( α )˜ κ , ˜ σ (cid:17) ˜ u ( s , x )d s + Z t P s , t (˜ b · ∇ ˜ u )( s , x )d s + Z t P s , t ˜ f ( s , x )d s . (3.6)Below, without loss of generality, we drop the tilde over u , κ, κ , σ, σ , b and f .We prepare the folowing lemmas which are analogues of [17, Lemma 6.6, 6.8, 6.9]. Lemma 3.7.
Let α ∈ (0 , , γ ∈ (0 , and β ∈ (0 , ( α ∧ γ ) . Under conditions (H βκ ) and (H γσ ) ,for any T > , there is a constant c > and ε ∈ (0 , β ) such that for all j ∈ N , t ∈ [0 , T ] andu ∈ L ∞ T ( C α + ε ) , Z t | ∆ j P s , t (cid:16) L ( α ) κ,σ − L ( α ) κ ,σ (cid:17) u | ( s , s c − ( α + β ) j k u k L ∞ T ( C α + ε ) . (3.7) Proof.
First of all, by defnitions, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) L ( α ) κ,σ − L ( α ) κ ,σ (cid:17) u ( s , x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) ( L ( α ) κ,σ − L ( α ) κ ,σ ) u ( s , x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ( L ( α ) κ ,σ − L ( α ) κ ,σ ) u ( s , x ) (cid:12)(cid:12)(cid:12) : = J + J . For simplicity of notation, we drop the time variable t and the superscript α of ν ( α ) . For any ε >
0, by (2.6) and (3.5), we obtain that for u ∈ C α + ε , J = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R d Ξ ( α ) u ( x ; σ z ) · ( κ ( x , z ) − κ ( z )) ν (d z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . | x | β Z R d (cid:12)(cid:12)(cid:12) Ξ ( α ) u ( x ; σ z ) (cid:12)(cid:12)(cid:12) ν (d z ) . | x | β k u k C α + ε , where provided | Ξ ( α ) f ( x ; z ) | k f k C α + ε ( | z | α + ε ∧ Z t | ∆ j P s , t J | ( s , s . k u k C α + ε Z t Z R d | x | β | ∆ j p s , t ( x ) | d x d s . − j ( α + β ) k u k C α + ε . Next, we estimate J for α ∈ (0 , α ∈ (1 , α = CHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULAR L ´EVY MEASURES 11 (1)
Case: α ∈ (0 , ε ∈ (0 , β ) such that β < ( α − ε ) γ and α + ε <
1, by (3.5), wehave J k κ k ∞ (cid:16) Z | z | + Z | z | > (cid:17) | u ( x + σ ( x ) z ) − u ( x + σ (0) z ) | ν ( α ) (d z ) . k u k C α + ε (cid:16) [ σ ] α + ε C γ | x | ( α + ε ) γ Z | z | | z | α + ε ν (d z ) (cid:17) + k u k C α − ε (cid:16) [ σ ] α − ε C γ | x | ( α − ε ) γ Z | z | > | z | α − ε ν (d z ) (cid:17) . Hence, by (2.12), we get that for all u ∈ C α + ε , Z t | ∆ j P s , t J | ( s , s . k u k C α + ε Z t Z R d ( | x | ( α + ε ) γ + | x | ( α − ε ) γ ) | ∆ j p s , t ( x ) | d x d s . − j α (2 − j ( α + ε ) γ + − j ( α − ε ) γ ) k u k C α + ε . − j ( α + β ) k u k C α + ε . (2) Case: α ∈ (1 , ε ∈ (0 , β ) such that α + ε <
2, by(3.5) , we have J . (cid:16) Z | z | + Z | z | > (cid:17) | ( σ ( x ) − σ ) z | Z |∇ u ( x + r σ ( x ) z + (1 − r ) σ z ) − ∇ u ( x ) | d r ν (d z ) . [ σ ] C γ | x | γ (cid:16) k∇ u k C α + ε − k σ k α + ε − ∞ Z | z | | z | α + ε ν (d z ) + k∇ u k ∞ Z | z | > | z | ν (d z ) (cid:17) . Therefore, by (2.12) and β < γ , we obtain that for all u ∈ C α + ε , Z t | ∆ j P s , t J | ( s , s . k u k C α + ε Z t Z R d | x | γ | ∆ j p s , t ( x ) | d x d s . − j ( α + β ) k u k C α + ε . (3) Case: α =
1. As above proofs, we decompose the integral on R d into two parts, the smalljump part and the large jump part. The small jump part, that is the integral on { z ∈ R d | | z | } ,is same as the case of α ∈ (1 , { z ∈ R d | | z | > } issame as the case of α ∈ (0 , Z t | ∆ j P s , t J | ( s , s . − j ( α + β ) k u k C α + ε . Combining J with J , we complete the proof. (cid:3) Lemma 3.8.
Let α ∈ (1 / , and β ∈ ((1 − α ) ∨ , α ∧ . Under the condition b ∈ L ∞ loc ( C β ) ,for T > and ε ∈ (0 , α + β − , there is a constant c > such that for all j ∈ N , t ∈ [0 , T ] andu ∈ L ∞ T ( C α + β − ε ) , Z t | ∆ j P s , t ( b · ∇ u ) | ( s , s c − ( α + β ) j k u k L ∞ T ( C α + β − ε ) . (3.8) Proof.
By the (3.5) and (2.12), we have Z t | ∆ j P s , t ( b · ∇ u ) | ( s , s = Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R d ∆ j p s , t ( x ) · ( b · ∇ u )( s , x )d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d s . Z t k∇ u ( s ) k ∞ Z R d | x | β | ∆ j p s , t ( x ) | d x d s . − ( α + β ) j k u k L ∞ T ( C α + β − ε ) , where we used the fact that 1 < α + β − ε . (cid:3) Lemma 3.9.
Let α ∈ (0 , and β ∈ R + . For any T > , there is a constant c > such that forall j ∈ N , t ∈ [0 , T ] and f ∈ L ∞ T ( C β ) , Z t | ∆ j P s , t f | ( s , s c − ( α + β ) j k f k L ∞ T ( C β ) . (3.9) Proof.
By (2.3), (2.12) and Remark 2.8, we have Z t | ∆ j P s , t f | ( s , s = Z t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R d e ∆ j p s , t ( x ) · ∆ j f ( s , x )d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d s . Z t k ∆ j f ( s ) k ∞ Z R d | e ∆ j p s , t ( x ) | d x d s . − ( α + β ) j k f k L ∞ T ( C β ) . Thus, we get (3.9). (cid:3)
Now we are in a position to give
Proof of Theorem 3.6.
By (3.6), Lemma 3.7, Lemma 3.8 and Lemma 3.9, we have | ∆ j u ( t , θ t ) | = | ∆ j ˜ u ( t , | . − ( α + β ) j (cid:16) k u k L ∞ T ( C α + β − ε ) + k f k L ∞ T ( C β ) (cid:17) , for some ε ∈ (0 , β ∧ ( α + β − x and Lemma 3.5, we obatin k ∆ j u ( t ) k ∞ . − ( α + β ) j (cid:16) k u k L ∞ T ( C α + β − ε ) + k f k L ∞ T ( C β ) (cid:17) , (3.10)which in turn implies (3.4) by the interpolation inequality k u k L ∞ T ( C α + β − ε ) ε k u k L ∞ T ( C α + β ) + c ε k u k L ∞ T for any ε ∈ (0 ,
1) and some constant c ε , the maximum principle Lemma 3.3, and Remark 2.8.The proof is completed. (cid:3) Remark 3.10.
The restriction of α ∈ (1 / , is only used in Lemma 3.8, which is caused bythe moment problem due to − α < α . Since we consider classical solutions, α + β must belarger than so that ∇ u is meaningful. In addition, we shall assume β < α due to the momentestimate (see Lemma 2.12). The critical case α + β = is a technical problem, and we haveno ideas to fix it. Unbounded drift case.
In this subsection, we use a cuto ff technique depending on char-acterization curve making unbounded drift bounded to prove Theorem 3.4. We first establisha commutator estimate. Lemma 3.11.
Let α ∈ (0 , , γ ∈ (0 , and β ∈ (0 , ( α ∧ γ ) . Under conditions (H βκ ) and (H γσ ) ,for any T > , there is a constant c > such that for any u ∈ L ∞ T ( C α ) , (cid:13)(cid:13)(cid:13) [ χ, L ( α ) κ,σ ] u (cid:13)(cid:13)(cid:13) L ∞ T ( C β ) c k u k L ∞ T ( C α ) . The definition of the notation [ · , · ] can be found at the end of introduction and χ is definedby (2.1). Proof.
Rewrite [ χ, L ( α ) κ,σ ] u ( t , x ) = Z R d Σ ( t , x , z ) κ ( t , x , z ) ν ( α ) (d z ) , (3.11)where Σ ( t , x , z ) : = δ σ ( t , x ) z χ ( x ) u ( t , x + σ ( t , x ) z ) − u ( t , x ) σ ( t , x ) z ( α ) · ∇ χ ( x ) CHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULAR L ´EVY MEASURES 13 with z ( α ) : = z α ∈ (1 , + z | z | α = and the definition of the notation δ h f is defined in the beginningof Subsection 2.1. For simplicity of notation, we drop the time variable t and the superscript α of ν ( α ) . As the proof of Lemma 3.7, we split the integral (3.11) over areas { z ∈ R d | | z | } and { z ∈ R d | | z | > } , that is [ χ, L ( α ) κ,σ ] u ( t , x ) = J + J with J ( x ) : = Z | z | Σ ( x , z ) κ ( x , z ) ν (d z ) and J ( x ) : = Z | z | > Σ ( x , z ) κ ( x , z ) ν (d z ) , where J and J are called the small jump part and the large jump part, respectively. Since z ( α ) has di ff erent forms in cases α < α > α =
1, we estimate J , J for these casesseparately. Here, the key to estimate k · k C β norms of those integrals is the following fact k Z g ( · , z )d z k C β Z k g ( · , z ) k C β d z . (1) Case: α ∈ (0 , (H γσ ) , and the fact k f g k C β k f k C β k g k C β , we derive that k Σ ( · , z ) k C β . ( | z | ∧ | z | β/γ ) k u k C α , (3.12)where we used Σ ( x , z ) = ( z · Z ∇ χ ( x + s σ ( x ) z )d s (cid:17) u ( x + σ ( x ) z ) (3.13)for | z |
1. Therefore, by (H βκ ) and (2.6) with 0 < β/γ < α <
1, we obtain k J k C β + k J k C β . k u k C α Z R d | z | ∧ | z | β/γ ν (d z ) . k u k C α , which in turn gives the desired result.(2) Case: α ∈ (1 , J , rewrite Σ ( x , z ) = ( z · Z ∇ χ ( x + s σ ( x ) z )d s (cid:17) u ( x + σ ( x ) z ) − u ( x ) σ ( x ) z · ∇ χ ( x ) . (3.14)Noticing 0 < β/γ < < α , by Lemma 2.4, (H βκ ) , and (H γσ ) , we have k J k C β . k u k C α Z | z | > ( | z | β/γ + | z | ) ν ( α ) (d z ) . k u k C α . For J , we need the interpolation inequality Lemma 2.9. Let T z u ( x ) : = u ( x + σ ( x ) z ) − u ( x ) . By Lemma 2.4, we have, kT z u k C β = k z · Z ∇ u ( · + r σ ( · ) z )d r k C β . k u k C + β/γ | z | . We also have kT z u k C β k u ( · + σ ( · ) z ) k C β + k u k C β . k u k C β/γ . Choose some θ ∈ ( α − , α − β/γ ) such that ϑ : = θ (1 + βγ ) + (1 − θ ) βγ < α . Then, byLemma 2.9, we obtain that kT z u k C β . | z | θ k u k C ϑ | z | θ k u k C α . (3.15) Since Σ ( x , z ) = (cid:16) σ ( x ) z · Z δ r σ ( x ) z ∇ χ ( x )d r (cid:17) u ( x + σ ( x ) z ) + σ ( x ) z · T z u ( x ) , by (3.15) and Lemma 2.4, we obtain that k J k C β . Z | z | k ( χ ( · + σ ( · ) z ) − χ ) u ( · + σ ( · ) z ) − u σ z · ∇ χ k C β ν ( α ) (d z ) . k u k C α Z | z | ( | z | + | z | + θ ) ν ( α ) (d z ) . k u k C α . (3) Case: α =
1. Observe that for the case α = J is same as the case of α ∈ (1 ,
2) and J issame as the case of α ∈ (0 , (cid:3) Now we are in a position to give
Proof of Theorem 3.4.
Fix x ∈ R d . Let θ t , ˜ u , ˜ f , ˜ σ , ˜ κ , ˜ σ , ˜ κ and ˜ b be the same ones inSubsection 3.1. See that ˜ κ and ˜ σ still satisfy (H βκ ) and (H γσ ) respectively. The only di ff erencewe shall note is that b < L ∞ loc ( C β ) here. We use the cuto ff technique to fix this problem below.By (3.1), it is easy to see that ˜ u satisfies the following equation: ∂ t ˜ u = L ( α )˜ κ, ˜ σ ˜ u + ˜ b · ∇ ˜ u + ˜ f . (3.16)Observe that ∂ t ( χ ˜ u ) = L ( α )˜ κ, ˜ σ ( χ ˜ u ) + ( χ ˜ b ) · ∇ ( χ ˜ u ) + χ ˜ f + [ χ, L ( α )˜ κ, ˜ σ ] ˜ u − ( χ ˜ b ) ˜ u · ∇ χ, (3.17)where χ is defined by (2.1). Moreover, by Lemma 2.3 and (H β b ) , we have χ ˜ b ( t , · ) ∈ C β and k ( χ ˜ b )( t , · ) k C β . [ b ( t , · )] C β c . (3.18)Thus, concluding form Theorem 3.6, (3.18), and Lemma 3.11, for any t T , we obtain that k χ ˜ u ( t , · ) k C α + β . k χ ˜ f k L ∞ T ( C β ) + k [ χ, L ( α )˜ κ, ˜ σ ] ˜ u k L ∞ T ( C β ) + k ( χ ˜ b ) ˜ u · ∇ χ k L ∞ T ( C β ) . k f k L ∞ T ( C β ) + k u k L ∞ T ( C α ) + k u k L ∞ T ( C β ) . Noticing that, for any k ∈ N , ∇ k ( χ ˜ u )( t , x ) = ∇ k ˜ u ( t , x ) , ∀| x | / , t ∈ R + , and for any t > k u ( t , · ) k C α + β ( B ( θ t , / k ( χ ˜ u )( t , · ) k C α + β with B ( θ t , / = { x ∈ R d | | x − θ t | / } . Therefore, by Lemma 2.2, Lemma 3.5, and takingsupremum of x , we get k u k L ∞ T ( C α + β ) . k f k L ∞ T ( C β ) + k u k L ∞ T ( C α ) + k u k L ∞ T ( C β ) . Furthermore, by interpolations and the maximum principle Lemma 3.3, we have k u k L ∞ T ( C α + β ) . k f k L ∞ T ( C β ) . The proof is completed. (cid:3)
CHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULAR L ´EVY MEASURES 15
Remark 3.12.
Observe that, by Lemma 2.2, | ˜ b ( t , x ) | [ b ( t , · )] C β ( | x | β {| x | } + | x | {| x | > } ) . Hence, if we have the following estimate Z t Z R d | x ||∇ ∆ j p s , t ( x ) | d x d s . − j α , (3.19) then we get Lemma 3.8 under (H β b ) with β ∈ ((1 − α ) ∨ , γ ) . Furthermore, we get Theorem 1.1directly without using cuto ff techiques as showned in Subsection 3.1. Fortunately, if α ∈ (1 , or L ( α ) κ,σ = ∆ α/ with α ∈ (0 , , (3.19) is true.
4. P roof of T heorem Lemma 4.1.
Let α ∈ (0 , and β ∈ ((1 − α ) ∨ , ∧ α ) with α + β < N . Under conditions (H βκ ) and (H σ ) , there is a constant c > such that for any T > and u ∈ L ∞ loc ( C α + β ) , k L ( α ) κ,σ u k L ∞ T ( C β ) c k u k L ∞ T ( C α + β ) . Proof.
For the simplicity, we only consider the case 0 < α <
1, and drop the time variable t and the superscript α of ν ( α ) and L ( α ) κ,σ . In this case, by (1.3), L κ,σ u ( x ) = Z R d ( u ( x + σ ( x ) z ) − u ( t , x )) κ ( x , z ) ν (d z ) . (4.1)For any fixed x , h ∈ R d , define κ ( x , z ) : ≡ κ ( x , z ) , σ ( x ) : ≡ σ ( x ) . Notice that L κ,σ u ( x + h ) − L κ,σ u ( x ) = L κ,σ u ( x + h ) − L κ ,σ u ( x + h ) + L κ ,σ u ( x + h ) − L κ ,σ u ( x + h ) + L κ ,σ u ( x + h ) − L κ,σ u ( x ): = I + I + I . For I , under (H βκ ) and (H σ ) , by (2.6), we have | I | . k κ k C β | h | β Z R d h ( k σ k L ∞ k u k C | z | ) ∧ k u k L ∞ i ν (d z ) . k u k C α + β | h | β . For I , under (H σ ) and 0 < β < α <
1, we have | u ( x + h + σ ( x + h ) z ) − u ( x + h + σ ( x ) z ) | | h | β h ( k u k C | z | ) ∧ ( k u k C β | z | β ) i , Hence, by (2.6) and (H βκ ) , we get | I | . k u k C | h | β Z R d ( | z | ∧ | z | β ) ν (d z ) . k u k C α + β | h | β . For I , we use the block operator ∆ j . Define L u ( x ) : = Z R d (cid:16) u ( x + x + σ ( x ) z ) − u ( x + x ) (cid:17) κ ( x ) ν (d z ) . Note that | f ( x + h ) − f ( x ) | ( k∇ f k L ∞ | z | ) ∧ (2 k f k ∞ ). For any j >
0, by Bernstein’s inequality(2.4) and ν (d λ z ) = λ α ν (d z ) with λ >
0, we have | ∆ j L u ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z R d (cid:16) ∆ j u ( x + x + σ ( x ) z ) − ∆ j u ( x + x ) (cid:17) κ ( x ) ν (d z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Z R d ( | σ ( x ) z |k∇ ∆ j u k L ∞ ) ∧ k ∆ j u k L ∞ k κ k L ∞ ν (d z ) . k ∆ j u k L ∞ Z R d ( | j z | ∧ ν (d z ) . α j k ∆ j u k L ∞ , which implies that k L u k C β . k u k α + β . Thus, we have | I | = | L u ( h ) − L u (0) | . | h | β k u k α + β , which completes the proof. (cid:3) Now, we are in a position to give
Proof of Theorem 1.1. ( Step 1 ) Suppose that b is bounded and σ is Lipschitz in this step. Consider the followingcontinuity equation: ∂ t u = λ L ( α )1 ,σ u + (1 − λ ) L ( α ) κ,σ u + b · ∇ u + f , u (0) = . (4.2)When λ =
1, by the same argument as in [7, Section 5] and Theorem 3.4, there is aunique classical solution for Eq.(4.2). Using the continuity method, by Lemma 4.1 andTheorem 3.4, we get a classical solution u for Eq.(4.2) with λ = Step 2 ) For any n ∈ N and ( t , x ) ∈ R + × R d , let b n ( t , x ) : = b ( t , x ) ∧ n and σ n ( t , x ) : = σ ( t ) ∗ ρ n ( x )where ρ n is the usual modifier. By Step 1, there is a classical solution u n of PDE (1.2)with b = b n and σ = σ n , i.e., ∂ t u n = L ( α ) κ,σ n u n + b n · ∇ u n + f , u n (0) = . (4.3)Noting that [ b n ( t )] C β [ b ( t )] C β and k σ n ( t ) k C γ k σ ( t ) k C γ , by Theorem 3.4, there is aconstant c such that for all n ∈ N , k u n k L ∞ T ( C α + β ) c k f k L ∞ T ( C β ) . (4.4)Moreover, by Lemma 4.1, we havesup n k L ( α ) κ,σ n u n k L ∞ T . sup n k u n k L ∞ T ( C α + β ) c k f k L ∞ T ( C β ) . (4.5)Thus, combining (4.3) with the above inequality, for any M > s , t ∈ [0 , T ], k u n ( t ) − u n ( s ) k L ∞ ( B M ) . | t − s | (1 + k b k L ∞ T ( B M ) ) k f k L ∞ T ( C β ) . | t − s | (cid:16) + sup t ∈ R + | b ( t , | M (cid:17) k f k L ∞ T ( C β ) . CHAUDER’S ESTIMATES FOR NONLOCAL EQUATIONS WITH SINGULAR L ´EVY MEASURES 17
Therefore, by Ascolli-Arzela’s theorem, there is a function u ∈ ∩ M > C ( R + ; C ( α ∨ + ε ( B M ))and a subsequence { n k } k > such that for all t ∈ [0 , T ] and M > j → + ∞ sup | x | M |∇ m u n j ( t , x ) − ∇ m u ( t , x ) | = , m = , . (4.6)For the convenience, we drop the subscript k of n k . By definitions and (4.4), we have k u k L ∞ T ( C α + β ) = sup t ∈ [0 , T ] sup j > ( α + β ) j k ∆ j u ( t ) k L ∞ = sup t ∈ [0 , T ] sup j > ( α + β ) j k lim n →∞ ∆ j u n ( t ) k L ∞ sup t ∈ [0 , T ] sup j > ( α + β ) j sup n k ∆ j u n ( t ) k L ∞ sup n k u n k L ∞ T ( C α + β ) c k f k L ∞ T ( C β ) . Notice that u n ( t , x ) = Z t (cid:16) L ( α ) κ,σ n u n ( s , x ) + b n ( s , x ) · ∇ u n ( s , x ) + f ( s , x ) (cid:17) d s (4.7)with sup n k L ( α ) κ,σ n u n k L ∞ T . k f k L ∞ T ( C β ) andsup t ∈ R + , n ∈ N | b n ( t , x ) | sup t ∈ R + | b ( t , x ) | sup t ∈ R + | b ( t , | + | x | h b ( t ) i C β < + ∞ , ∀ x ∈ R d . Letting n → ∞ in (4.7), by (4.6) and the dominated convergence theorem we have u ( t , x ) = Z t (cid:16) L ( α ) κ,σ u ( s , x ) + b ( s , x ) · ∇ u ( s , x ) + f ( s , x ) (cid:17) d s . Hence, the function u is a classical solution of PDE (3.1) in the sense of Definition 3.1.The proof is finished. (cid:3) Acknowledgements.
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