Local well-posedness of the fifth-order KdV-type equations on the half-line

This paper is a continuation of authors' previous work \cite{CK2018-1}. We extend the argument \cite{CK2018-1} to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the $X^{s,b}$ nonlinear estimates for $b<\frac12$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b>\frac12$ \cite{CGL2010, JH2009}. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line.


Introduction
This paper is a continuation of authors' previous work [5]. In [5], the authors studied the Duhamel boundary forcing operator associated to the fifth-order linear operator, and established the local well-posedness of Kawahara equation posed on the right / left half-line. In this paper, we extend the previous study to the fifth-order KdV-type equations whose nonlinearities are different, in particular, do not satisfy the scaling symmetry. Consider the following fifth-order KdV-type equation: where u(t, x) is real-valued function and F (u) is a nonlinearity. We, here, take F (u) = (1 − ∂ 2 x ) 1 2 ∂ x (u 2 ) or F (u) = ∂ x (u 3 ). When F (u) = (1 − ∂ 2 x ) 1 2 ∂ x (u 2 ), the equation (1.1) was introduced by Tina, Gui and Liu [31] to understand the role of dispersive and nonlinear convection effects in the fifth-order K(m, n, p) equations of the form ∂ t u + β 1 ∂ x (u m ) + β 2 ∂ 3 x (u n ) + β 3 ∂ 5 x (u p ) = 0 when (m, n, p) = (2, 2, 1), in particular, β 1 = 1 and β 2 = β 3 = −1.
When F (u) = ∂ x (u 3 ), the equation (1.1) is well-known as the modified Kawahara equation, which was proposed first by Kawahara [18]. The modified Kawahara equation arises in the theory of shallow water waves, the theory of magneto-acoustic waves in plasmas and propagation of nonlinear water-waves in the long-wavelength region as in the case of KdV equations. This equation is also regarded as a singular perturbation of KdV equation. We refer to [1,15,13] and references therein for more background informations.
1.1. Main analysis. The principal contribution in the paper is to establish the nonlinear estimates for both nonlinearities F (u) = (1 − ∂ 2 x ) 1 2 ∂ x (u 2 ) and F (u) = ∂ x (u 3 ), in particular, for a certain regularity s ∈ R, 0 < b < 1 2 < α < 1 − b. In contrast with the initial value problem (IVP), the standard X s,b space with b > 1/2 is not suitable for the study of IBVP due to the presence of the Duhamel boundary forcing operator. Moreover, the time trace norm of the Duhamel parts restricts the regularity range (see Section 5). A natural countermeasure is to use additional auxiliary spaces: the (timeadapted) Bourgain space Y s,b and the low frequency localized X s,b space, which will be introduced in Section 2.
The followings are the main results established in the paper.
The proof of Theorems 1.1 (a) and 1.2 (a) are based on the Taos [k; Z]-multiplier norm method [28], which has become now very standard to prove the multilinear estimates. The elementary L 2 -block estimates have already been provided by Chen, Li, Miao and Wu [9] for the bilinear case (Chen and Guo [6] corrected the high × high ⇒ high case), and by the second author [21] for the trilinear case (in [21], the L 2 -block estimates for periodic functions in the spatial variable are given, but the proof for non-periodic functions is almost identical). In contrast to [7], a direct proof of trilinear estimates is given, while new bilinear estimates and T T * argument are used to prove the trilinear estimates in [7]. The Strichartz estimate for the linear operator group {e t∂ 5 x } [12] is needed to deal with high × high × high ⇒ high interaction component, since the trilinear L 2 -block estimates (Lemma 3.2) are given for only high × low × low ⇒ high case. The proof of Theorems 1.1 (b) and 1.2 (b) are based on the proof of Lemma 5.10 (b) in [14], but more careful examination of frequency relations is needed (also for the proof of Theorems 1.1 (a) and 1.2 (a)).
1.3. Review on the well-posedness results. Both IVP and IBVP of the fifth-order KdVtype equations have been extensively studied. When , the local wellposedness of (1.1) was first established by Tina, Gui and Liu [31] in H s (R), s ≥ −11/16 by using the Fourier restriction norm method [2]. In addition to the technique Tao's [K; Z] multiplier norm method [28], Chen and Liu [8] improved the local well-posedness in H s (R), s > −5/4 and they also showed the ill-posedness, in the sense of the lack of continuity of the flow map, for s < −5/4. At the endpoint regularity H −5/4 (R), Chen, Guo and Liu [7] proved the local well-posedness by using Besov-type function spaces. This is the optimal result until now as far as authors' know.
When F (u) = ∂ x (u 3 ), the Cauchy problem for (1.1) was studied by Jia and Huo [16] and Chen, Li, Miao and Wu [9], independently. They established the local well-posedness in H s (R), s ≥ −1/4, by using the Fourier restriction norm method. In [9], the authors used bilinear L 2 -block estimates and T T * argument to prove the trilinear estimate, while a direct calculation of trilinear integral operator was performed in [16]. The global well-posedness of (1.1) in H s (R), s > −3/22 was shown by Yan, Li and Yang [33] via the I-method [10].
The IBVP of the modified Kawahara equation posed on the right half-line in the high regularity Sobolev space H s (R + ) ( 1 4 ≤ s < 2) has been studied by Tao and Lu [30]. On the other hand, the IBVP of (1.1) on the half-lines (both right and left), where the nonlinearity is given by both , in the low regularity setting (in particular, negative regularities) is first considered here as far as we know. We end this section with referring to [24,22,25,23,5] and references therein for IBVP results of the fifth-order KdV-type equations posed on the half-line.
1.4. Organization of the paper. The rest of paper is organized as follows: In Section 2, we mainly construct the solution space and observe several basic properties for IBVP of (1.1). In Section 3, we give the proofs of Theorems 1.1 and 1.2. In Section 4, we briefly introduce the Duhamel boundary forcing operator for the fifth-order equations. In Section 5, we establish energy estimates, in particular time trace estimate, with a short time cut-off function. In Sections 6, we prove Theorems 1.3 -1.6.

Preliminaries
Let R + = (0, ∞). For positive real numbers x, y ∈ R + , we mean x y by x ≤ Cy for some C > 0. Also, x ∼ y means x y and y x. Similarly, a and ∼ a can be defined, where the implicit constants depend on a.
For a cut-off function ψ given by we fix the time localized function ψ T (t) = ψ(t/T ), 0 < T < 1.
2.1. Riemann-Liouville fractional integral. A brief summary of the Riemann-Liouville fractional integral operator is, here, given, see [11,14] for more details. Let t + be a function defined by and t − can be defined by t − = (−t) + . Let α be a complex number. For Re α > 0, the tempered distribution is defined as a locally integrable function by It is straightforward to obtain 2) for all k ∈ N. The expression (2.2) facilitates to extend the definition of where (τ − i0) −α is the distributional limit. When α / ∈ Z, (2.3) can be rewritten by Together with (2.3) and (2.4), we see For f ∈ C ∞ 0 (R + ), we define the Riemann-Liouville fractional integral operator by for Re α > 0. The well-know properties are We end this subsection with introducing some lemmas associated to the Riemann-Liouville fractional integral operator I α f without proofs. Lemma 2.1 (Lemma 2.1 in [14]). If f ∈ C ∞ 0 (R + ), then I α f ∈ C ∞ 0 (R + ), for all α ∈ C. Lemma 2.2 (Lemma 5.3 in [14]). If 0 ≤ Re α < ∞ and s ∈ R, then  [14]). If 0 ≤ Re α < ∞, s ∈ R and µ ∈ C ∞ 0 (R), then .

Moreover, we have
We refer to [5] for the details. We end this subsection with introducing some lemmas associated to B(x) without proofs.

2.3.
Sobolev spaces on the half line and solution spaces. Let s ≥ 0. We say f ∈ . For s ∈ R, we say f ∈ H s 0 (R + ) if there exists F ∈ H s (R) such that F is the extension of f on R and F (x) = 0 for x < 0. In this case, we set f H s 0 (R + ) = F H s (R) . For s < 0, we define H s (R + ) as the dual space of H −s 0 (R + ). We also set C ∞ 0 (R + ) = {f ∈ C ∞ (R); suppf ⊂ [0, ∞)}, and define C ∞ 0,c (R + ) as the subset of C ∞ 0 (R + ), whose members have a compact support on (0, ∞). We remark that C ∞ 0,c (R + ) is dense in H s 0 (R + ) for all s ∈ R. We end this subsection with stating elementary properties of the Sobolev spaces. We refer to [17,11,5] for the proofs. Lemma 2.6 (Lemma 2.1 in [5]). For − 1 2 < s < 1 2 and f ∈ H s (R), we have , where the constant c depends only on s and ψ.
Lemma 2.8 (Proposition 2.4 in [11]). If 1 2 < s < 3 2 the following statements are valid: . Lemma 2.9 (Proposition 2.5. in [11]). Let f ∈ H s 0 (R + ). For the cut-off function ψ defined in . We define the Fourier transform of f with respect to both spatial and time variables by and denote by f or F(f ). We use F x and F t (or without distinction of variables) to denote the Fourier transform with respect to space and time variable respectively.
For s, b ∈ R, the classical Bourgain spaces X s,b [2] associated to (1.1) is defined as the completion of S ′ (R 2 ) under the norm where · = (1 + | · | 2 ) 1/2 . As already mentioned in Section 1, the modifications of X s,b spaces are needed for our analysis due to the Duhamel boundary forcing operator and the time trace estimates. The modulation exponent b of the standard X s,b space is forced to be taken in the range (0, 1 2 ) from the X s,b estimation of the Duhamel boundary forcing terms (see Lemma 5.3 (c)). On the other hand, very low frequency interactions in the nonlinear estimates compel the exponent b to be bigger than 1/2. To balance these inter-contradiction conditions, we define the low frequency localized X s,b -type space D α as the completion of S ′ (R 2 ) under the norm where 1 A is the characteristic functions on a set A. Besides, the time trace estimate of the Duhamel parts holds only for the positive regularity. To meet the negative regularity in the nonlinear estimate (see Lemma 5.2 (b)), it is necessary to define the (time-adapted) Bourgain space Y s,b associated to (1.1) as the completion of S ′ (R 2 ) under the norm We make a Littlewood-Paley decomposition. Let Z + = Z ∩ [0, ∞). For k ∈ Z + , we set Let η 0 : R → [0, 1] denote a smooth bump function supported in [−2, 2] and equal to 1 in [−1, 1]. For k ∈ Z + , we define χ 0 (ξ) = η 0 (ξ), and χ k (ξ) = η 0 (ξ/2 k ) − η 0 (ξ/2 k−1 ), k ≥ 1, on the support I k . Let P k denote the L 2 operators defined by P k v(ξ) = χ k (ξ) v(ξ). For the modulation decomposition, we use the multiplier η j , but the same as η j (τ − ξ 5 ) = χ j (τ − ξ 5 ). For k, j ∈ Z + , let The Littlewood-Paley theory allows that and We define the solution space denoted by Z s,b,α ℓ under the norm 1 :

Nonlinear estimates
In this section, we are going to establish nonlinear estimates, in particular, the control of 3.1. L 2 -block estimates. Let a 1 , a 2 , a 3 ∈ R. The quantities a max ≥ a med ≥ a min can be conveniently defined to be the maximum, median and minimum values of a 1 , a 2 , a 3 respectively. Similarly, to be the maximum, sub-maximum, third-maximum and minimum values of b 1 , b 2 , b 3 , b 4 respectively. For ξ 1 , ξ 2 ∈ R, let denote the (quadratic) resonance function by which plays an crucial role in the bilinear X s,b -type estimates. Let f, g, h ∈ L 2 (R 2 ) be compactly supported functions. We define a quantity by The change of variables in the integration yields and We give the bilinear L 2 -block estimates for the quadratic nonlinearity . See [9,6] for the proof.
We first prove Theorem 1.1.
Proof. Let By the duality argument, (3.11) is equivalent to . We divide the frequency regions of integration into several regions associated to the relation of frequencies to prove (3.15).
Case II-a k 1 = 0. Without loss of generality, we may assume that j 2 ≤ j 3 . By (3.14) and Lemma 3.1 (b), (3.16) on this case is dominated by whenever we choose 1/4 < b < 1/2 and for all α > 1/2. We use the Cauchy-Schwarz inequality to obtain Remark 3.1. The proof of Proposition 3.1 is indeed analogous to the proof of Proposition 5.1 in [5]. However, the high-low interaction component with very low frequency ( is slightly worse than the same one of ∂ x (u 2 ) in some sense, since the high-low bilinear local smoothing effect exactly cancels two derivatives in high frequency (two derivative gains). As a consequence of this observation, the argument used in the proof of Proposition 5.1 in [5] causes a logarithmic divergence in k 3 -summation, and thus more delicate computation, here, is required as above compared with Case II-a in the proof of Proposition 5.1 in [5].
The low ×low ⇒low interaction component can be directly controlled by the Cauchy-Schwarz inequality, since the low frequency localized space D α with α > 1/2 allows the L 2 integrability with respect to τ -variables.
Therefore, the proof of (3.11) is completed.
Remark 3.2. In view of the proof of Proposition 5.1 in [5], one can see that the regularity threshold appears in the high × high ⇒ low interaction component, which is the well-known worst component of quadratic nonlinearity (for semi-linear "dispersive" equations), while the regularity threshold −5/4, here, occurs in the high × high ⇒ high interaction case. It is because the high × high ⇒ low interaction component of (1 We state the elementary integral estimates without proof. (3.25) The proof of (3.25) is almost identical to the proof of (3.23) and (3.24), hence we omit the detail.
We now consider (3.31) on the case when |ξ| ≥ 1. We use (3.23) in addition to (3.3) so that the left-hand side of (3.31) is bounded by The support property (|τ − ξ 5 | |H|) and (3.26) implies and hence (3.32) can be controlled by we can reduce (3.33) by By (3.24), (3.34) is bounded by .26) and (3.27) with |ξ| ≥ 1 and s ≤ 0, we obtain In this case, it suffices to show from the Cauchy-Schwarz inequality that the left-hand side of (3.35) is bounded by (3.36) When |H| ≤ 1 2 |τ 2 − ξ 5 2 |, we can know the following facts: We perform the integration in (3.36) in terms of τ variable by using (3.23), then (3.36) is bounded by . Then, by performing integration in terms of ξ, we have 1.
We now prove Theorem 1.2.
Before proving Proposition 3.3, we bring the Strichartz estimates for the fifth-order dispersive equations.
Lemma 3.4 (Strichartz estimates for e t∂ 5 x operator [12]). Assume that −1 < σ ≤ 3 2 and 0 ≤ θ ≤ 1. Then there exists C > 0 depending on σ and θ such that D σθ 2 e t∂ 5 x ϕ L q t L p x ≤ C ϕ L 2 for ϕ ∈ L 2 , where p = 2 1−θ and q = 10 θ(σ+1) . In particular, we have Proof of Proposition 3.3. Similar mechanism as in the proof of Proposition 3.1 will be used. Let By the duality argument, (3.40) is equivalent to (3.45) ξ). We divide the frequency regions of integration . (3.46) into several regions associated to the relation of frequencies to prove (3.45). Case I. high-high-high ⇒ high (k 4 ≥ 10 and |k 1 − k 4 |, |k 2 − k 4 |, |k 3 − k 4 | ≤ 5). Without loss of generality, we may assume j 4 = j max . The change of variables yields in this case that (3.46) is bounded by The Fourier inversion formula, Minkowski inequality, (3.41) and the Cauchy-Schwarz inequality yield Using this, we estimate (3.47) by The choice of of 3 On the other hand, we see that the frequency summation includes only one infinite sum as We therefore have for s ≥ − 1 4 that (3.48) Case II. high-high-low ⇒ high (k 4 ≥ 10, |k 2 − k 4 |, |k 3 − k 4 | ≤ 5 and k 1 ≤ k 4 − 10) 7 . In this case, we know from (3.8) and (3.5) that j max ≥ 5k 4 . We further divide the case into two cases: k 1 = 0 and k 1 ≥ 1.
Case II-a. k 1 = 0. It suffices to consider By (3.10), we can control , and hence it suffices to show Without loss of generality, we may assume j 2 ≤ j 3 ≤ j 4 . When j 4 = j max , we know j 3 ≤ j sub . For s > −1, by choosing max( 3−2s 10 , 1 4 ) < b < 1 2 , we have LHS of (3.49) whenever α > 1 2 . When j 4 = j max , we know j 3 ≤ j 1 . Since j max ≥ 5k 4 , we have LHS of (3.49) Without loss of generality, we may assume that j 1 ≤ j 2 ≤ j 3 ≤ j 4 . Similarly as before, it suffices to show (3.50) 7 We may assume that ξ1 is the lowest frequency without loss of generality due to the symmetry.
If j 1 = j max , we apply the argument used in Case I to J 3 (f ♯ k 1 ,j 1 , f ♯ k 2 ,j 2 , f ♯ k 3 ,j 3 , f ♯ k 4 ,j 4 ) by changing the role of f k 1 ,j 1 and f k 4 ,j 4 . It is possible thanks to (3.9). Similarly as before, we have LHS of (3.50) Since the frequency summation includes only two infinite sums (but one of them is for low frequency mode), for s ≥ − 1 4 , by choosing 3 8 < b < 1 2 , we can have LHS of (3.50) (3.51) If j 1 = j max (we assume j 4 = j max ), we can obtain Then, similarly as the case when j 1 = j max , we have (3.51). Now it remains to show (3.52). It suffices to show for L 2 -functions g i : R → R ≥0 supported in I k i , i = 1, 2, 3, and g 4 : R 2 → R ≥0 supported in I j 4 × I k 4 , where G is defined as in (3.5). Indeed, if (3.53) holds true, then

The change of variables (ξ
Note that |ξ i | ∼ 2 k i , i = 1, 2, 3, still holds. A direct calculation gives and then the Cauchy-Schwarz inequality with respect to ξ 1 and ξ 2 , and the change of variable which completes the proof of (3.53). Thanks to (3.9), our assumption j 4 = j max does not lose the generality. Case III. high-high-high ⇒ low (k 3 ≥ 10, |k 1 − k 3 |, |k 2 − k 3 | ≤ 5 and k 4 ≤ k 3 − 10). In this case, we also have j max ≥ 5k 4 similarly as Case II. It suffices to show (3.54) The exact same argument as in Case II-b (by replacing the role of j 1 and j 4 ) can be applied to the left-hand side of (3.54) and hence, for s ≥ −1/4, by choosing 3 8 < b < 1 2 , we prove (3.54).
Case V-b. k 1 ≥ 1. Similarly as before, the worst bound of thanks to Lemma 3.2 (b-1). Hence, for s ≥ − 1 4 , by choosing 3 8 < b < 1 2 , we can obtain The low ×low × low ⇒low interaction component can be directly controlled by the Cauchy-Schwarz inequality, since the low frequency localized space D α with α > 1/2 allows the L 2 integrability with respect to τ -variables.
Collecting all, we therefore complete the proof of Proposition 3.3.

Duhamel boundary forcing operator
In this section, we introduce the Duhamel boundary forcing operator, which was introduced by Colliander and Kenig [11] and further developed by several researchers [14,4,3,5], which helps to construct the solution operator involving the boundary forcing conditions. We, particularly, refer to [5] for the fifth-order KdV-type equation. For f ∈ C ∞ 0 (R + ), define the boundary forcing operator L 0 of order 0 By the change of variable and (2.5), we represent (4.2) by Moreover, a straightforward calculation gives We state the several lemmas associated to L 0 f defined as in (4.2). We refer to [5] and references therein for the proofs.
x), k = 0, 1, 2, 3, is continuous in x ∈ R and has the decay property in terms of the spatial variable as follows: x) is continuous in x for x = 0 and is discontinuous at x) also has the decay property in terms of the spatial variable |∂ 4 In the following, we give the generalization of the boundary forcing operator L 0 f and its properties introduced in [5].
Let Re λ > 0 and g ∈ C ∞ 0 (R + ) be given. Define where * denotes the convolution operator. Note that L λ ± is for the right / left half-line problem, respectively. With (in the sense of distribution), we represent each of them by g (t, y)dy.
g (t, y)dy (4.6) and respectively. It, thus, immediately satisfies (in the sense of distributions) and Lemma 4.2 (Spatial continuity and decay properties for L λ ± g(t, x) [5]). Let g ∈ C ∞ 0 (R + ) and M be as in (4.1). Then, we have Moreover, L −4 ± g(t, x) is continuous in x ∈ R \ {0} and has a step discontinuity of size M g(t) x) satisfies the following decay bounds: for all x ≥ 0 and m ≥ 0, and The unitary group associated to (4.7) as Recall L λ + in (4.6) for the right half-line problem. Let a j and b j be constants depending on λ j , j = 1, 2, given by Let us choose γ 1 and γ 2 satisfying We choose an appropriate λ j , j = 1, 2, such A is invertible. Then, u defined by See Section 3 in [5] for more details.

Nonlinear version. The Duhamel inhomogeneous solution operator
By choosing a suitable γ 1 and γ 2 depending on not only f and g, but also e t∂ 5 x φ(x) and Dw, u defined by We refer to [5] for more details.

Energy estimates
We are going to give the fundamental energy estimates. (c) (Bourgain spaces) For b ≤ 1 2 and α > 1 2 , we have . The implicit constants do not depend on 0 < T ≤ 1 but ψ.
Remark 5.1. In contrast to IVP, the time localization may restrict the regularity range (both upper-and lower-bounds) due to the (derivatives) time trace estimates. Similar phenomenon can be seen in X s,b estimates (see, in particular, Lemma 2.11 in [29]), while the modulation exponent b is affected to be taken by the cut-off function. See (b) and (c) in Lemma 5.1 for the comparison.
Proof. The proofs of (a) and (c) are standard, hence we omit the details and refer to [29].
If T |τ | > 1, we know | ψ T (τ − η)| T 1−k |τ | −k , for any positive k. For k > 1, we have from (5.5) that LHS of (5.3) We remark that the smoothness of ψ guarantees not the good bound of T , but the integrability, in other words, we only need a large k for 11 When s+2−j 5 = 1 2 , the constant depending on T is log 1 Then, the right-hand side of (5.3) is bounded by the Cauchy-Schwarz inequality with respect to η and τ yields (5.6) φ H s . If T |η| > 1, we know | ψ T (τ − η)| T 1−k |η| k , for any positive k. On the region |τ | < 1/T , similarly as before, we have which implies from the Cauchy-Schwarz inequality that On the region |τ | ≥ 1/T , it suffices to control the Cauchy-Schwarz inequality and the change of variable yield LHS of (5.3) III. 1 2 |τ | < |η| < 2|τ |. In this case, we know in the integrand of the right-hand side of (5.3). We further assume |τ |, |η| > 1/T , otherwise we can use the same way to control (5.6). If τ · η < 0, we know | ψ T (τ − η)| T 1−k |τ | k . Since we have LHS of (5.3) φ H s similarly as (5.7). If τ · η > 0, we further divide the case into |τ − η| < 1/T and |τ − η| > 1/T . For the former case, let where M f (x) is the Hardy-Littlewood maximal function of f . Since M f L p f L p for 1 < p ≤ ∞ (see, in particular, [26]), we have For the latter case, the integration region in η can be reduced to τ + 1/T < η < 2τ for positive τ and η, since the exact same argument can be applied to the other regions. 12 Since | ψ T (τ − η)| T 1−k |τ − η| −k in this case, the left-hand side of (5.3) is bounded by where Φ is defined as in (5.8). Let ǫ = (k − 1)/2 for k > 1. Then, the change of variable, the Cauchy-Schwarz inequality and the Fubini theorem yield Therefore, we complete the proof of (b).
for small σ depending on τ ′ and ξ 5 . Since T ψ ′ (T τ ) is L 1 integrable with respect to τ , the Cauchy-Schwarz inequality yields On the other hand, on the region |τ ′ − ξ 5 | > T −1 , we use the L 1 integrability of ψ T , so that We denote by w = w 1 + w 2 , where for a characteristic function χ.
For w 1 , we use the Taylor expansion of e x at x = 0. Then, we can rewrite (5.12) for w 1 as where ψ k (t) = t k ψ(t) and we have from Lemma 5.1 (b) that We first consider II. Let Note that and Lemma 5.1 (b) that when − 9 2 + j ≤ s ≤ 1 2 + j. Now it remains to deal with I. Taking the Fourier transform to I with respect to t variable, we have and hence it suffices to control The argument is very similar used in the proof of Lemma 5.1 (b), while the relation among |τ |, |τ ′ | and |ξ| 5 should be taken into account carefully. Hence, we only give, here, a short idea on each case. We first split the region in τ as follows: Case I. |τ | ≤ 1, Case II. 1 < |τ | ≤ 1 T , Case III. 1 T < |τ |.