Persistence of H\"{o}lder continuity for non-local integro-differential equations

In this paper, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of H\"{o}lder continuity for their solutions. In other words, we prove that a solution stays in $C^\beta$ for all time if its initial data lies in $C^\beta$. This result has an application for a fully non-linear problem, which is used in the field of image processing. The proof is in the spirit of the paper [18] of Kiselev and Nazarov where they established H\"{o}lder continuity of the critical surface quasi-geostrophic (SQG) equation.


Introduction and the main result
Let N ≥ 1 be any dimension. We consider the following evolution equation where K satisfies the weak-( * )-kernel condition, which will be given in Definition 1.2. The above integral is understood in the sense of principal value. More precisely, we denote the integral operator T K t and (T K t ) ǫ for ǫ > 0 corresponding to any given kernel K at time t by x, y)dy and Then, (1) is equivalent to (∂ t w)(t, x) + T K t (w(t, ·))(x) = 0. Related to the above singular integral, there have been many interests recently, not only from the field of analysis, but also from the field of probability (e.g. Caffarelli and Silvestre [6], Schwab [24], Bass and Levin [2], Jacob, Potrykus, and Wu [16], and Chen, Kim, and Kumagai [9]).
Our main concern is to obtain a priori estimate for solutions of (1). The aim is to prove the result [4] of Caffarelli, Chan, and Vasseur with different techniques (a similar result for the stationary case was obtained by Kassmann in [17]). In particular, we prove persistence of Hölder continuity in L ∞ (0, ∞; C β (R N )), which is a new result, by observing the evolution of a dual class of test functions. This class, which appears in the work of Kiselev and Nazarov [18], plays a similar role of the dual space of C β . They obtained, in [18], Hölder regularity for solutions of the critical surface quasi-geostrophic (SQG) equation. It is interesting to compare this method with that of Caffarelli and Vasseur [7]. In [7], the estimate C β ([t, ∞) × R N )) for any t > 0 was proved by using a De Giorgi iteration technique (for other different proofs, we refer to Kiselev, Nazarov, and Volberg [19] and Constantin and Vicol [10]).
We present the definition of the weak-( * )-kernel condition, which is slightly weaker than the above ( * )-kernel condition in Definition 1.1. Definition 1.2. Under the same setting of the parameters in Definition 1.1, we say that K satisfies the weak-( * )-kernel condition on [0, T ] if K satisfies (2) and (3) for the case α < 1. If α ≥ 1, we ask K to hold the following condition (5) as well as (2) and (3).
Remark 1.1. The ( * )-kernel condition in Definition 1.1 implies the weak-( * )-kernel condition in Definition 1.2. Indeed, for the case α < 1, they are exactly same. If α ≥ 1, then the only difference between them is that the ( * )-kernel condition needs (4) while the weak-( * )-kernel condition requires (5). Also, it is easy to verify that (4) implies (5)  are the upper and the lower hemispheres, respectively. Then, thanks to (4), we have Remark 1.2. In the work of [4], the upper bound for k is just Λ while, in this paper, we have Λ · (1 + |x − y| ω ) in (3), which is slightly more general than that of [4]. Some examples with the upper bound (3) can be found in Section 4 of Komatsu [21]. Remark 1.3. The purpose of the condition (5) with (α − 1) < ν is to consider T K t (f )(·) not only as a distribution but also as a locally integrable function. In general, without such an additional cancellation condition, if α ≥ 1, then T K t (f )(x) is not well-defined even for f ∈ C ∞ c . In Lemma 2.2 and Lemma 2.1, it will be shown that as long as the corresponding kernel K satisfies the weak-( * )-kernel condition, the operator T K t is well defined, and T K t (f ) is a locally integrable function for some class of functions f . Remark 1.4. Let the kernel K satisfy the weak-( * )-kernel condition for some α ≥ 1. Then we can combine the two conditions (3) and (5) in order to get an estimate of the integral in (5) for all s ∈ (0, ∞). Indeed, the condition (3) implies that, for s ∈ [s 0 , ∞), Thus, together with the condition (5), we have, for s ∈ (0, ∞), Remark 1.5. We present some typical examples satisfying either the ( * )-kernel condition or the weak-( * )-kernel condition.
(II) One may assume that the kernel has the form not of K(t, x, y) but of K(t, x − y) (for more general cases, we refer to Silvestre [25]). Then the natural symmetry we would impose to the kernel is K(t, x − y) = K(t, y − x), which implies (2) directly. This K holds the weak-( * )-kernel condition for any α ∈ (0, 2) once we assume the bounds condition (3). Indeed, for α ≥ 1, the integral in (5) is always zero due to the cancellation from the symmetry (i.e. any ν ∈ (α − 1, 1) with τ = 0 and s 0 = ∞ works).
We concentrate our effort first to prove the part (II) of the above theorem in Section 2, 3, and 4. In fact, we will show the part (II) carefully to ensure that the two constants C and β in the conclusion of the part (II) depend only on the parameters in Definition 1.1. Thus, these two constants C and β depend neither on T nor on any actual norms coming from the smoothness assumption (10). As a result, the part (I), which will be proved in Appendix, follows the part (II) by a limit argument. Unfortunately, if α ≥ 1, then we need the condition (4), which is more restrictive than (5). Remark 1.6. More precisely, the conclusion of the part (I) follows once we regularize the function k in a proper way, which should keep all the parameters. In short, since k may not be bounded due to (3), we make it bounded first. Then take a convolution with a mollifier. This process does not hurt the parameter set essentially if α < 1. However for the case α ≥ 1, the cancellation condition (5) is not preserved during the process. That is the reason we impose the ( * )-kernel condition to the part (I) of Theorem 1.1 instead of the weak-( * )-kernel condition.
As in [4], we show how our result can be applied to a fully non-linear problem. We introduce the following non-linear evolution problem: This equation can be considered as the evolution problem coming from the Euler-Lagrange equation for the variational integral (for more detailed explanation, see [4]). This non-linear problem can be found in Giacomin, Lebowitz, and Presutti [14], or in the field of image processing (e.g. see Gilboa and Osher [15], Lou, Zhang, Osher, and Bertozzi [23]).
We impose the following conditions to the equation (11).
Following the approach of [4], we present the following important consequence of the part (II) of Theorem 1.1. Theorem 1.2. We have two constants β > 0 and C > 0 which depend only on the above parameters, and there exists a global-time weak solution θ of the equation (11) with the following estimates for a.e. t ∈ (0, ∞): and The main idea of the above theorem 1.2 is the following: First, we regularize θ 0 , G, and φ in a proper way so that we obtain a sequence of smooth solutions of (11). Then, we take a derivative (w := D e θ) to the non-linear equation (11), and we freeze some coefficients. As a result of this process, we obtain the linear equation (1) together with the weak-( * )-kernel condition on the K satisfying (10). Thus, we can use the conclusion of the part (II) of Theorem 1.1. Finally, we extract a weak solution by a limit argument. This proof will be given in Appendix.
Now we want to explain the main idea of the part (II) of Theorem 1.1, which is the heart of this paper. As mentioned earlier, our proof follows the spirit of the paper [18]. First, thanks to the duality of the equation (1) from the symmetry condition (2), we can focus only on the evolution of U r , a class of test functions, which is related to the dual space of the Hölder space C β (see Definition 2.3 of U r , Lemma 2.4, and Lemma 2.5). In this paper, we take the same definition of the class U r from the paper [18] while other classes can be found in Dabkowski [12] and Chamorro [8]. In particular, the class introduced in [12] is quite different from that of [18] and it was successfully used to obtain eventual regularity of the super-critical surface quasi-geostrophic (SQG) equation.
Second, we prove the short-time evolution of test functions (Proposition 3.1). In order to obtain it, we need to manage the competition (refer to Remark 3.4) between the L p condition and the concentration condition. The former condition, which can be proved from the lower bound Λ −1 · 1 |z|≤ζ , of the kernel has a regularization effect (Lemma 3.4, Lemma 3.5) as a diffusion term in usual PDEs does. However, the latter condition comes from the upper bound Λ · (1 + |z| ω ) of the kernel and this upper bound plays a similar role as a source term in usual PDEs (Lemma 3.3).
In addition, since the length of the time interval coming from the conclusion of Proposition 3.1 is proportional to r α where r is the parameter of U r , it should be verified that we can repeat the short-time evolution (Proposition 3.1) as many times as we want in order to reach any fixed time (refer to Remark 4.1).
For the case α < 1, the main difficulty is to handle both lower and upper bounds (3) of the kernel: in particular, both the finite size ζ of support of the lower bound and the term (1 + |z| ω ) of the upper bound cause some troubles. In order to cover the case α ≥ 1, we use the cancellation condition (5), which is designed to cancel desirable amount of singularity at x = y of the kernel. Then we can interpret T K t (f ) as locally integrable functions for some class of functions (see Lemma 2.1, Lemma 2.2). This fact will be crucial to prove the concentration condition (Lemma 3.3).
We want to mention a few articles related to the integral operator T K t corresponding a kernel K. For smooth bounded kernels, we may use a theory of pseudo differential operators (e.g. Kumano-go [22], Komatsu [20]), while for measurable kernels, there exists a fundamental solution (see [21]). Also, we refer to [17] and Barlow, Bass, Chen, and Kassmann [1]. Recently, in Dyda and Kassmann [13], assumptions of kernels have been extended in some geometrical sense. If we focus on non-divergence case, we refer to [6].
As mentioned before, the following three sections 2, 3, and 4 are dedicated to the proof of the part (II) of the main theorem 1.1. More precisely, in Section 2, we introduce some definitions and few important lemmas. After that, we present and prove the main proposition 3.1 in Section 3. Finally, the proof of the part (II) of Theorem 1.1 ends in Section 4. At the end of this paper, Appendix contains the proofs of the part (I) of Theorem 1.1 and Theorem 1.2.
Before considering a general ζ ∈ (0, ∞], we will prove first the conclusion of the part (II) of Theorem 1.1 for a fixed ζ = ζ 0 where (14) ζ . This definition of ζ 0 will help us to obtain enough regularization directly so that the proof becomes more straightforward. Once we prove the part (II) of Theorem 1.1 with ζ = ζ 0 , a general proof for any value ζ ∈ (0, ∞] will follow a scaling argument.
On the other hand, for the case ζ < ζ 0 , we define a scaling: for a kernel K with ζ < ζ 0 , then w ǫ is a solution on [0, T /ǫ α ] for the kernel K ǫ with a new ζ = ζ 0 once we pick up ǫ by ǫ = ζ/ζ 0 . Then we can apply the part (II) of Theorem 1.1 for w ǫ and the same result for w follows.
In this paper, we denote Sobolev spaces by W k,p and H k := W k,2 for integers k ≥ 0 and for p ∈ [1, ∞] in the usual way. In addition, the symbol S is used to represent the Schwartz space in R N .
. Moreover the operator can be extended to more general spaces. For example, if f is locally integrable and |f | 1+|x| N +α−ω dx < ∞, then we can define T K t (f ) as an element of S ′ where S ′ is the dual of Schwartz space S (see also Silvestre [26]). We will make use of the following Lemma 2.1, which says that is not only an element of S ′ but also a locally integrable function with a desirable estimate. This fact will be used to obtain the concentration condition (Lemma 3.3) for the evolution of U r , which will be introduced in Definition 2.3.

Lemma 2.1. We have an estimate
Proof.
Moreover, the following properties hold: (I). Duality of T: (II). Mean zero of T: where we used the Taylor expansion of f in the first integral. For (b), we use the upper bound of (3): (1)) · r 2 from the Taylor error estimate where B x (r) is the ball of radius r centered at x: (1)) .

Now we can easily verify that
Note that if f ∈ C 2 (R N ) ∩ L 1 (R N ), then the above argument implies T K t (f ) ∈ L ∞ . Then, the proof of (I) follows the symmetry in x, y of K. Indeed, if f, g ∈ C 2 (R N ) ∩ L 1 (R N ), then we have In addition, for the case f ∈ C 2 (R N ) ∩ L 1 (R N ) with g(x) = |x| γ , then the integral |f (x)T K t (g)(x)|dx is bounded due to the assumption f ∈ L ∞ ∩ L 1 with Lemma 2.1. Indeed, Lemma 2.1 implies that T K t (| · | γ ) is integrable in the unit ball containing the origin and is bounded outside of the ball. Then, together with f ∈ L ∞ ∩ L 1 , we obtain f · T K t (| · | γ ) ∈ L 1 . Thus all equalities of (16) can be justified via a limit argument.
To prove (II), we take g ∈ C ∞ c such that g = 1 in B(1) and supp(g) ⊂ B(2) and define g n by g n (·) := g(·/n). Then, thanks to the property (I) with g n , the conclusion follows by taking a limit n → ∞.
In the following lemma, we present a maximum principle for solutions of (1).
Remark 2.2. Also we assume that the solutions are smooth. However the estimate of the result does not depend on this smoothness.
Proof. To prove (I), we multiply η ′ (w) to the equation (1) To prove (II), for p ≥ 2, we use (17) with putting η(·) = | · | p and taking an integral in x variable in order to use (II) of Lemma 2.2. For p < 2, we need to regularize η(·) = | · | p first. Now we adopt the notion of the class U r of test functions following the paper [18]. Let A ≥ 1 be a constant which will be chosen later. Definition 2.3. We say that a measurable function ϕ(·) on R N lies in U r for some r ∈ (0, ∞) if ϕ satisfies the following four conditions: In addition, we say that ϕ lies in aU r for some a > 0 when (1/a)ϕ ∈ U r . We call x 0 a center of ϕ.
The following lemma connects between C β space and U r , which tells us that r −β U r plays a similar role of the dual space of C β .
for any w ∈ C β (R N ), for any 0 < r < ∞, and for any ϕ ∈ U r .
(II) Conversely, we have a constant C such that if a bounded function w satisfies sup ϕ∈S∩Ur,0<r≤1 Proof. For the part (I), let x 0 be a center of ϕ. Then, from the mean zero property, For the part (II), we recall Littlewood-Paley projections ∆ j , which is defined by for |ξ| ≤ 1 and η = 0 for |ξ| ≥ 2. We use the characterization of C β in terms of Littlewood-Paley projections (see Stein [27]). Indeed, if a bounded function w in R N satisfies sup j=1,2,3,··· 2 βj ∆ j (w) L ∞ (R N ) < ∞ then w lies in C β (R N ) and it has the estimate where C 1 depends only on β, N and the choice of Ψ. In order to show (19), it is enough to find 0 < a < ∞ such that Ψ 2 −j ∈ aU 2 −j for all j ≥ 1 because ∆ j (w)(x) = R 3 w(y)Ψ 2 −j (x − y)dy and U r is translation invariant. It is clear that Ψ is a Schwartz function from the fact η ∈ C ∞ 0 . Thus we can take Thus (19) follows with C = C 1 · max{1, a}.
We define the backward kernel K (T ) corresponding to any finite time T < T and to the kernel K by (20) K (T ) (s, x, y) = K(T − s, x, y).
Then it is easy to see T K t = T K (T ) T −t and they share the weak-( * )-kernel condition with the same parameter set.
Lemma 2.5. Let w, ϕ ∈ L ∞ (0, T ; (L 1 ∩ L ∞ )(R N )) be two smooth solutions of (1) with T < ∞ for each smooth initial data w 0 , ϕ 0 ∈ (L 1 ∩ L ∞ )(R N ) and for each associated kernels K and K (T ) , respectively. In addition, we assume ϕ 0 ∈ U r ∩ S for some r ∈ (0, 1]. Then, we have Then, we use Lemma 2.2 and the fact As a result, we conclude that R 3 w(t, x)ϕ(T − t, x)dx is constant in t. Then put t := 0 and t := T .

The main proposition and its proof
We are ready to present the main proposition about the evolution of test functions in a short time interval, whose length is proportional to r α . Roughly speaking, if ϕ 0 ∈ U r , then there exist z = z(r, s) and β such that ϕ(s) ∈ r z β U z for s ∈ [0, δr α ].
Proof. Let ϕ 0 ∈ U r ∩ S for some 0 < r ≤ 1. Then there exists a weak solution ϕ corresponding to the initial data ϕ 0 (this can be proved by following [21]. Or refer to the approximation scheme in [4]). Moreover this solution is smooth, and it lies in L ∞ t (H b x ) for every integer b ≥ 0 due to the smoothness assumption (10) of k (it can be proved by using a standard energy argument).
First we state the following elementary inequalities without proof.
In order to obtain (21), we need to verify the mean zero, the concentration, the L ∞ , and the L 1 conditions. First the mean-zero condition is easily verified in STEP 1. Second, we derive some estimates for remained three other conditions in STEP 2-4. Then, in STEP 5, we combine all the estimates we obtained in STEP 2-4 to finish the proof. Without loss of generality, we can assume that a center of ϕ 0 is the origin (i.e. x 0 = 0). STEP 1. Mean zero-condition. From (II) of Lemma 2.2, we have, for any t ∈ (0, T ), ·))(x)dx = 0.

STEP 2. Concentration-condition.
Lemma 3.3. There exists a constant C conc > 0 such that, for any s ∈ (0, T ), we have where C conc does not depend on A as long as A ≥ 1.
Remark 3.1. This lemma says that test functions lose their concentration with certain rate as time goes on. In Step 5, it will be shown that the rate can be absorbed into the regularization effect from the L 1 and the L ∞ conditions. Proof.
Lemma 3.4. There exist two constants δ L ∞ > 0 and C L ∞ > 0 such that, for any s ∈ [0, min{δ L ∞ r α , T }], we have Remark 3.2. This lemma is proved by using the lower bound of (3), which gives us some regularization effect. We follow a similar argument of Theorem 4.1 in the paper Córdoba and Córdoba [11], which showed a L ∞ decay for smooth solutions of the 2D surface QG equation.
Proof. First, we define M (t) := ϕ(t, ·) L ∞ . We claim that there exist δ 1 > 0 and . To prove the above claim (25), first pick any t ∈ (0, T ) such that r N for all t * < t from Lemma 2.3. It can be easily proved that there exists a point x t such that |ϕ(t, x t )| = M (t). Indeed, because our kernel lies in C ∞ t,x,y [0, T ] × R N × R N ) with ϕ 0 ∈ S, we can show ϕ ∈ L ∞ t H d for every integer d ≥ 0 by standard energy estimates. In particular, ϕ(t, ·) ∈ H b for some integer b > (N/2) for every time. Then, ϕ(t, ·) vanishes at the infinity thanks to a Fourier transform argument. Since ϕ(t, ·) is continuous, there exists a maximum (or minimum) point.
Then, for almost every time t ∈ (0, T ), there exist a pointx t such that |ϕ(t,x t )| = M (t) with the following inequality: x t ) = −M (t) (this can be proved by following the argument of [11]).
We assume the first case ϕ(t,x t ) = M (t) > 0 (the other one can be dealt in similar fashion). Then We used the fact ϕ(t,x t ) − ϕ(t, y) ≥ 0 with the lower bound of the kernel (3).
Let R be any number between 0 and ζ, which will be chosen soon. We separate the ball B R (x t ) into two disjoint regions Ω 1 and Ω 2 by the following way: x t ) implies y ∈ Ω 1 . Otherwise, y ∈ Ω 2 . Then we have the following upper bound of measure of Ω 2 : As a result, from R < ζ, we have ). (14). Coming back to (26), we have Solving this differential inequality, we obtain For any p > 0, it is easy to see (1 + x) −p ≤ (1 − 1 2 px) for 0 ≤ x ≤ C p . Thus, we have Thanks to (25), the whole case (24) can be achieved easily by taking δ L ∞ := 1 2 δ 1 and C L ∞ : Lemma 3.5. There exist two constants δ L 1 > 0 and C L 1 > 0 such that, for any s ∈ [0, min{δ L 1 r α , T }], we have where C L 1 does not depend on A as long as A ≥ 1.
Remark 3.3. In this time, we obtain L 1 decay by using the lower bound of the kernel (3). In general, without the mean zero property, we do not expect L 1 decay (refer to [11]). However, with the mean zero property, we can manage certain amount of cancellation of the L 1 -norm. This idea comes from the argument in [18] where L 1 decay for mean-zero solutions for the 2D-SQG equation in a periodic setting was obtained.
In order to use the lower bound of the kernel (3), we need to restrict the above integral on a subset of {|x−y| ≤ ζ}. For this purpose, we define the subsetsD s + ,D s − andS s byD s ± = D s ± ∩ B(ar) andS s = S s ∩ B(ar). Then, if x, y ∈ B(ar), then |x − y| ≤ 2ar ≤ 2a ≤ ζ from (29). Thus, from the lower bound of the kernel (3), we have where, for the last equality, the estimate (30) was used.

STEP 5.
Combining all conditions. Now we are ready to finish the proof of the main proposition 3.1. In STEP 2-4, we proved that Note that the constants C L 1 , C L ∞ , and C conc are independent of A as long as A ≥ 1 while δ L 1 and δ L ∞ depend on A. We define δ 3 := min{δ L 1 , δ L ∞ } so that the above three estimates (31), (32), and (33) hold at the same time for all s ∈ [0, min{δ 3 r α , T }]. Without loss of generality, we can assume Recall that we are looking for β > 0 and z(r, s) such that ϕ(s) ∈ r z β U z . Thus, from Definition 2.3 of U r and from the above three estimates (31), (32), and (33), we need the followings: Remark 3.4. (34) is equivalent to The power of A in (35) is strictly less than that of A in (36). This fact is crucial because we can choose A large enough to hold (35) and (36) at the same time. Then we can make β small enough to hold (37), too. We will now give all the details.
We take any A ≥ 1 large enough to satisfy the inequality: In addition, we take any β ∈ (0, γ/2] so small that the following inequality holds: Finally, we define a constant L by and a function z(r, s) by z(r, s) := r(1 + L s r α ). For the Concentration-condition, from (II) of Lemma 3.2, we have where the last inequality holds as long as for s ≤ (1/L)r α . We define δ 4 := min{δ 3 , (1/L)}.
On the other hand, from 0 < β ≤ γ/2 < 1/2, we observe the followings: For the L ∞ -condition, from (38) and from (I) and (IV) of Lemma 3.2, we have For the L 1 -condition, from (39) and (I) and (III) of Lemma 3.2, we have Together with the mean zero property of ϕ in STEP 1, we proved for any s ∈ [0, min{δ 5 r α , T }] with r ∈ (0, 1] and for any ϕ 0 ∈ U r , we have the evolution estimate which proves (21).
It remains to prove (22). Let r = 1 (i.e. ϕ 0 ∈ U 1 ). Note that Lemma 3.3 holds for all time s ∈ [0, T ) and L p norm is decreasing all time s ∈ [0, T ) and for any 1 ≤ p ≤ ∞ from (II) of Lemma 2.3. Thus we have ϕ(s) ∈ (1 + Ls)U 1 for all s ∈ [0, T ). This is the end of the proof of Proposition 3.1.

Proof of the part (II) of Theorem 1.1
Proof of the part (II) of Theorem 1.1. Let t be any time between 0 and T . Thanks to (II) of Lemma 2.4 and (II) of Lemma 2.3, the only thing we need to do is to find an estimate on r −β R 3 w(t, x)ϕ 0 (x)dx for ϕ 0 ∈ S ∩ U r with 0 < r ≤ 1. From Proposition 3.1, we have a smooth solution ϕ on [0, t] correspoding to the initial data ϕ 0 with the kernel K (t) , which is defined by K (t) (s) := K(t − s) (see the definition (20)). From Lemma 2.5, we want a control on r −β Lemma 2.3, we get ϕ(t) L 1 (R N ) ≤ r β as long as t ≥ t k . Therefore, we have a control for any t ∈ (0, T ): Thus, from (40), we have where C does not depend on t. Now we can combine (42) with (45) to get Similarly, we can prove ϕ(t) L ∞ (R N ) ≤ C · r β · max{1, 1 t (N +β)/α }. As a result, from (40), we have Proof of the part (I) of Theorem 1.1. In this subsection, we suppose that the kernel K satisfies not the weak-( * )-kernel condition in Definition 1.2 but the ( * )-kernel condition in Definition 1.1 (we recall that the latter condition implies the former one). Note that the kernel K does not need to satisfy (10) any more. Thus, we first construct a family of kernels K ǫ keeping all the parameters of the ( * )-kernel condition uniformly in ǫ > 0, and satisfying (10). Then we use the conclusion of the part (II) of Theorem 1.1.

Proof of Theorem 1.2.
Proof of Theorem 1.2. For convenience, we define a function g by g(x) = G(x) · |x| N +α . In addition to all the assumptions of Theorem 1.2, we assume further (48) θ 0 ∈ C ∞ (R N ), g ∈ C ∞ (R N ), and φ ∈ C ∞ (R).
Then there exists a weak solution θ of (11) in global time and it is smooth. Indeed, for existence issue, we refer to Benilan and Brezis [3] or the appendix in the paper [4]. Smoothness follows a difference quotient argument.
We will show that the conclusions of Theorem 1.2 hold for this smooth solution θ. Moreover, it will be clear that the constants C and β depend only on the parameters in the hypotheses of Theorem 1.2 and they are independent of the actual norms coming from the above additional assumption (48). Thus the conclusions of Theorem 1.2 without (48) follows by a limit argument.
By putting K(t, x, y) := φ ′′ (θ(t, y) − θ(t, x))G(y − x), this function w(= D e θ) solves the linear equation (1). Moreover, it is easy to see that this new kernel K satisfies (2), (3), and (10) directly (a rigorous proof can be completed by using the difference quotient argument, which is contained in [4]). Then, Theorem 1.2 for the case α < 1 follows once we apply the part (II) of Theorem 1.1 to w.
We use the assumption φ ′′ ∈ C ν : where the proof of non-increasing of ∇θ(t) L ∞ x is in the part (II) of Lemma 2.3. By putting τ := C · M · √ Λ with s 0 := 1, we get the condition (5). Then, we apply the part (II) of Theorem 1.1 to w.