Stability of Transonic Shock-Fronts in Three-Dimensional Conical Steady Potential Flow past a Perturbed Cone

For an upstream supersonic flow past a straight-sided cone in $\R^3$ whose vertex angle is less than the critical angle, a transonic (supersonic-subsonic) shock-front attached to the cone vertex can be formed in the flow. In this paper we analyze the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. We establish that the self-similar transonic shock-front solution is conditionally stable in structure with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. In particular, it is proved that the slope of the shock-front tends asymptotically to the slope of the unperturbed self-similar shock-front downstream at infinity.


Introduction
We study the stability of transonic shock-fronts in three-dimensional steady potential flow past a perturbed cone. The steady potential equations with cylindrical together with Bernoulli's law: where κ ∞ := 1 2 u 2 ∞ + 1 γ−1 ρ γ−1 ∞ is determined by the upstream flow state at infinity, i.e., the density ρ ∞ and velocity (u ∞ , 0), and y is the distance of the flow location in R 3 to the x-axis. In (1.2), we have used the pressure-density relation: so that the sound speed c = ρ (γ−1)/2 . For an upstream supersonic flow past a straight-sided cone, a shock-front is formed in the flow. When the vertex angle of the cone is less than the critical angle, the shock-front may be self-similar and attached to the cone vertex. There are two kinds of admissible shock-fronts depending on the downstream condition at infinity (cf. Courant-Friedrichs [18], Chapter VI): transonic (supersonic-subsonic) shockfronts and supersonic-supersonic shock-fronts. In this paper, we are interested in the stability of the transonic shock-front, behind which the flow is completely subsonic (see Fig. 1). More precisely, for fixed upstream density ρ ∞ > 0 at infinity, our problem is to understand the stability of self-similar transonic shock-front when the speed of the upstream flow velocity (u ∞ , 0) is large, equivalently, when the Mach number M ∞ := u∞ c∞ is large. ∞ . Therefore, without loss of generality, we can drop "˜" for notational convenience hereafter to assume that u ∞ = 1, the Bernoulli constant is Then we have Under this scaling, the problem reduces to the stability problem for self-similar transonic shock-fronts in transonic flow past a perturbed cone, governed by (1.1)-(1.2) with the Bernoulli constant (1.5), when the Mach number M ∞ of the upcoming flow is sufficiently large, or equivalently, the density ρ ∞ is sufficiently small.
Conical flow (i.e. cylindrically symmetric flow with respect to an axis, say, the xaxis) occurs in many physical situations. For instance, it occurs at the conical nose of a projectile facing a supersonic stream of air (cf. [18]). The study of supersonicsupersonic shock-fronts was initiated in Gu [22], Schaeffer [30], and Li [24] first for the wedge case; also see Chen [11,12,13], Zhang [34,35], and Chen-Zhang-Zhu [10] for the recent results. The stability of conical supersonic-supersonic shock-fronts has been studied in the recent years in Liu-Lien [26] in the class of BV solutions when the cone vertex angle is small, and Chen [14] and Chen-Xin-Yin [17] in the class of smooth solutions away from the conical shock-front when the perturbed cone is sufficiently close to the straight-sided cone.
The stability of transonic shock-fronts in three-dimensional steady flow past a perturbed cone has been a longstanding open problem. Some progress has been made for the wedge case in two-dimensional steady flow in Chen-Fang [16] and Fang [19]. In particular, in [16,19], it was proved that the transonic shock is conditionally stable under perturbation of the upstream flow and/or perturbation of wedge boundary. Also see [5,6,7,15,31,32,33] for steady transonic flow in multidimensional nozzles.
For the two-dimensional wedge case, the equations do not involve singular terms and the flow past the straight-sided wedge is piecewise constant. However, for the three-dimensional conical case, the governing equations have a singularity at the cone vertex and the flow past the straight-sided cone is self-similar, but is no longer piecewise constant. These cause additional difficulties for the stability problem. In this paper, we develop techniques to handle the singular terms in the equations and the singularity of the solutions.
Our main results indicate that the self-similar transonic shock-front is conditionally stable with respect to the conical perturbation of the cone boundary and the upstream flow in appropriate function spaces. That is, it is proved that the transonic shock-front and downstream flow in our solutions are close to the unperturbed self-similar transonic shock-front and downstream flow under the conical perturbation, and the slope of the shock-front asymptotically tends to the slope of the unperturbed self-similar shock at infinity.
In order to achieve these results, we first formulate the stability problem as a free boundary problem and then introduce a coordinate transformation to reduce the free boundary problem into a fixed boundary value problem for a singular nonlinear elliptic system. We develop an iteration scheme that consists of two iteration mappings: one is for an iteration of approximate transonic shock-fronts; and the other is for an iteration of the corresponding boundary value problems for the singular nonlinear systems for given approximate shock-fronts. To ensure the well-definedness and contraction property of the iteration mappings, it is essential to establish the well-posedness for a corresponding singular linearized elliptic equation, especially the stability with respect to the coefficients of the equation, and obtain the estimates of its solutions reflecting their singularity at the cone vertex and decay at infinity. The approach is to employ key features of the equation, introduce appropriate solution spaces, and apply a Fredholm-type theorem in Maz'ya-Plamenevskiǐ [28] to establish the existence of solutions by showing the uniqueness in the solution spaces.
The organization of this paper is as follows. In Section 2, we exploit the behavior of self-similar transonic shocks and corresponding transonic flows past straight-sided cones, governed by (1.1)-(1.2) with Bernoulli constant (1.5). In Section 3, we first formulate the stability problem as a free boundary problem, then introduce a coordinate transformation to reduce the free boundary problem into a fixed boundary value problem, and finally state the main theorem (Theorem 3.1) of this paper and its equivalent theorem (Theorem 3.2).
In Section 4, we establish the well-posedness for a singular linear elliptic equation, which will play an important role for establishing the main theorem, Theorem 3.1. In Section 5, we develop our iteration scheme for the stability problem, which includes two steps: one is an iteration of approximate transonic shock-fronts; and the other is the iteration of the corresponding nonlinear boundary value problems for given approximate shock-fronts. In Sections 6-7, we prove that the two iteration mappings in the iteration scheme are both well-defined, contraction mappings, based on the well-posedness theory for a singular linear elliptic equation established in Section 4. This implies that there exists a unique fixed point of each iteration mapping leading to the completion of the proof of the main theorem, Theorem 3.1.
We remark that all the results for the case γ > 1 is valid for the isothermal case γ = 1 as the limiting case when γ → 1, which can be checked step by step in the proofs.

Self-similar transonic shocks and corresponding transonic flows past straight-sided cones
In this section, we exploit the behavior of self-similar transonic shocks and corresponding transonic flows past straight-sided cones, governed by (1.1)-(1.2) with Bernoulli constant (1.5).
Let the turning angle of the velocity field right behind the self-similar shock-front S be φ 1 and set b = tan φ 1 . Then v = bu for the velocity field (u, v) of the flow right across S. Assume that the angle between S and the upcoming velocity field (1, 0) is ω 1 and set τ = cot ω 1 . Then the Rankine-Hugoniot conditions on S are Using (2.1) and the relation v = bu, we have . Then a direct computation yields For γ > 1 and b > 0, we have Then the implicit function theorem implies that, in a neighborhood of (0, 0), τ can be expressed as a function of ν, that is, there exists a positive constant ν 0 such that Furthermore, there exist positive constants α 1 and α 2 such that, for any ν ∈ [0, ν 0 ], we have where O(1) depends only on γ and b. Thus, where q = √ u 2 + v 2 is the flow speed and O(1) depends only on γ and b. We now analyze the flow field between the self-similar shock-front S and the straight-sided cone. Let ω 0 be the vertex angle of the cone and κ = cot ω 0 . Since the equations and the boundary conditions are invariant under the scaling (x, y) → (αx, αy), α = 0, we seek self-similar solutions (u, v) = (u, v)(σ), σ = x/y, as in [18]. Then the flow field (u, v) between the shock-front S and the cone y = κx is determined by the following free boundary value problem: where ω 0 or κ is unknown and determined together with the solution, τ and (u S , v S ; ρ S ) are determined by the shock polar and the flow direction b right behind the shock-front S which are given in (2.2), and the density ρ is determined by Bernoulli's law with Bernoulli constant (1.5). By [18], there exists a vertex angle ω 0 = ω 0 (b) of the cone and the corresponding self-similar solution (u 0 , v 0 )(σ), σ ∈ [τ, k], between the shock-front and the cone as the solution of the free boundary value problem (2.7)-(2.9). We assume that the flow between the shock-front and the cone is subsonic, which is the case when M ∞ is large (equivalently, ρ ∞ is small). In this case, we employ (2.7) to obtain where q = √ u 2 + v 2 is the flow speed. It is easy to verify that and u 0 (σ), q 0 (σ), and the Mach number M 0 (σ) are strictly decreasing, while v 0 (σ) is strictly increasing, with respect to σ. Therefore, we have In the next sections, we develop a nonlinear iteration scheme and establish the stability of self-similar transonic shocks under perturbation of the upstream supersonic flow and the boundary surface of the straight-sided cone.

Stability Problem and Main Theorem
In this section we first formulate the stability problem as a free boundary value problem, then introduce a coordinate transformation to reduce the free boundary problem into a fixed boundary value problem, and finally state the main theorem (Theorem 3.1) of this paper and its equivalent theorem (Theorem 3.2).
3.1. Formulation of the stability problem. The stability problem can be formulated as the following free boundary problem.
Problem I: Free boundary problem. Determine the free boundary S = {x = φ(y)} and the velocity field (u, v) in the unbounded domain {φ(y) < x < ϕ −1 (y)} satisfying the equations: the free boundary conditions on S: and the slip boundary condition on the boundary surface of the perturbed cone, where the density ρ can be expressed as a function of the velocity (u, v) by Bernoulli's law: The equations in (3.1) can be rewritten in the matrix form: where U = (u, v) ⊤ and To solve the free boundary problem (Problem I), we introduce the following coordinate transformation: to fix the free boundary: Then the free boundary S becomes a fixed boundary Γ 1 = {ξ = η cot ω 1 }, and the domain {φ(y) < x < ϕ −1 (y)} becomes a fixed domain In transformation (3.7), φ as a function of y is unknown and can be also considered as a function of η in the following way: Then the transformation is written as In the case that ψ(η) is known, we can obtain the expression of φ(y) from (3.8). In fact, substituting ξ = η cot ω 1 into (3.8), we have Thus, In our case, ϕ ′ andψ should be small perturbations to tan ω 0 and cot ω 1 , respectively. Hence, we have dy dη > 0, and η can be also expressed as a function of y, i.e. η = η(y). Then φ(y) = ψ(η(y)) is what we need. Therefore, we consider the transformation with formulation (3.8) from now on. Then we have A direct calculation indicates that the Jacobian matrix of the transformation is .

3.2.
Weighted spaces for solutions. Based on the analysis of the self-similar transonic shock solutions in Section 2 and the behavior of solutions to elliptic equations at infinity, it is anticipated that the solutions have singularity at the origin and decay at infinity. Thus, we need the following weighted spaces as posed spaces to accommodate the features of solutions to our problem.
be an unbounded sector, where (r, θ) are the polar coordinates. Then the boundary of the domain D consists of two rays: For any k ∈ R, m = 0, 1, · · · , we define the following weighted Sobolev spaces W m,q (k) (D) as subspaces of u ∈ W m,q loc (D): where P(r, θ) = (t, θ) := (ln r, θ) is a coordinate transformation from (r, θ) to (t, θ). Define the norms for the trace of u on each ray Γ j of the boundary of D by It is easy to see that there exists a constant K, independent of u, such that and denote by C m (k) (D) the space of functions with norm · C m (k) (D) . When q > 2 and m ≥ 1, the well-known Sobolev imbedding theorem implies that For functions of single variable defined in R + , we can also define the following similar weighted norms: The main theorem of this paper is the following.  22) and the perturbed upstream flow field U − satisfies where Ω e := {η cot(ω 1 +δ 0 ) < ξ < η cot(ω 0 −δ 0 )} for some smallδ 0 > 0, Since Π ψ,ϕ or Π φ,ϕ is invertible, we conclude the following equivalent result from Theorem 3.1.  24) and the perturbed upstream flow field U − satisfies where Ω e := {y cot(ω 1 + δ 0 ) < x < y cot(ω 0 − δ 0 )} for some small δ 0 > 0. Moreover, the solution (U (x, y); φ(y)) satisfies φ(0) = 0 and the following estimates:  Hence, the self-similar transonic shock-front is conditionally stable with respect to the conical perturbation of the boundary surface of the cone and the upstream flow in the function spaces with restrictions on the downstream flow field both at the corner and at infinity.

Well-posedness for a Singular Linear Elliptic Problem
In this section, we establish the well-posedness for a singular linear elliptic equation, which will play an essential role for establishing the main theorem, Theorem 3.1.

4.1.
Neumann problem for a singular second-order elliptic equation. Consider the following Neumann boundary value problem in Ω: We have the following proposition.
Moreover, we have the following estimate for the solution to problem (4.1): where the constant K is independent of ϕ, but depends only on q and ω 0 (actually cot ω 0 ).
To prove this proposition, we employ a criterion identified by Hartman-Wintner [23] for the uniqueness of solutions to the Dirichlet boundary value problem for systems of second-order differential equations. For self-containedness, we give a brief description here; for more details, see [23].
Lemma 4.1. Consider the following boundary value problem for the system of second-order differential equations for x ∈ R n : where A 1 (t) and A 2 (t) are n × n real matrices. Assume that there exists a matrix K(t) such that . Then problem (4.3) has only the trivial solution x ≡ 0.
Proof. Taking the inner product on the equations with x and integrating from t 0 to t 1 yields Combining the above two identities, we obtain t1 t0 Since N is positive definite, we conclude x ≡ 0.

(4.8)
Applying the Fourier transformation F t→λ with respect to t, we obtain a family of boundary value problems with complex parameter λ: (4.9) We now employ a Fredholm-type theorem, Theorem A.1 in Appendix, to find that, if the homogeneous problems of (4.9) (i.e. f = g 0 = g 1 = 0) have only the trivial solutionφ ≡ 0 for all λ with Imλ = −1, then, for any (f, g 0 , g 1 ) such that e t f ∈ W 0,q (D) and g j ∈ W 1−1/q,q (Σ j ), j = 0, 1, the boundary value problem (4.8) in the infinite strip D has a unique solution ϕ such that e −t ϕ ∈ W 2,q (D). Moreover, the solution ϕ satisfies the estimate: where K is independent of ϕ, but does depend on ω 0 because of the coefficient cot θ.
Then the results of Proposition 4.1 follow. Therefore, it suffices to verify that, in the case that f = g 1 = g 2 ≡ 0, the boundary value problems (4.9) with complex parameter λ, Imλ = −1, have only the trivial solutionφ ≡ 0. (4.11) Write y = y 1 + y 2 i and λ = µ − i. Then −λ 2 + λi = −µ 2 + 2 + 3µi, and (4.11) can be rewritten as the following boundary value problems of second-order differential equations with real coefficients: which implies that the symmetric matrix N is positive definite and hence satisfies the criterion (4.4). By Lemma 4.1, we obtain that y ≡ 0. That is,φ ≡ const. Then the equations in (4.9) yields thatφ ≡ 0 for any Imλ = −1. This completes the proof.
Proposition 4.1 can be directly applied to a special boundary value problem of first-order partial differential equations.

4.2.
A boundary value problem of a singular first-order elliptic system. Consider the boundary value problem for the first-order system: in Ω, (4.13) U ·α 0 = g 0 on Γ 0 , (4.14) where To solve this problem, we first construct a function Φ ∈ W 2,q (4.17) By virtue of [28], there exists a unique Φ with the following estimate: Then the boundary value problem (4.13)-(4.15) is reduced tô By the second equation of (4.19), there exists a potential function ϕ such that ∇ϕ = (∂ x ϕ, ∂ y ϕ) =Ũ . Then the boundary value problem (4.19)-(4.21) can be reformulated as a boundary value problem of a second-order elliptic equation: Now Proposition 4.1 yields that there exists a unique solution ϕ ∈ W 2,q (−1) (Ω) with the following estimate: where K depends only on ω 0 , but is independent of ϕ, F , and g j , j = 0, 1. Thus, there exists a unique solution U ∈ (W 1,q (0) (Ω)) 2 to problem (4.13)-(4.15) with the following estimate: whereK is independent of U , F , and g j , j = 0, 1, but depends only on ω 0 . With the argument above, we obtain the following corollary of Proposition 4.1: Then there exists a unique solution U ∈ (W 1,q (0) (Ω)) 2 to the boundary value problem (4.13)-(4.15). Moreover, the solution satisfies estimate (4.23).
Applying the continuity method, we can extend this result to a small "perturbed" boundary value problem for the first-order elliptic system.

28)
where K is independent of U , F , and g j , j = 0, 1, but depends only on ω 0 .
Proof. Denote the boundary value problems (4.13)-(4.15) and (4.24)-(4.26) by linear bounded operatorsT and T respectively from (Γ j ). By Proposition 4.3,T is invertible andT −1 is also a linear bounded operator. Let T s = (1 − s)T + sT , s ∈ [0, 1]. By virtue of (4.27), we have where K is a constant. SinceT U = T s U + s(T − T )U , by Proposition 4.3, Choosingǫ sufficiently small such that Kǫ < 1, we have where K is independent of U and s ∈ [0, 1], but depends only on ω 0 . Then, applying the continuity method, Proposition 4.3, and the uniform estimates (4.29), we completes the proof.

Iteration Scheme
Our iteration scheme for the stability problem consists of two iteration mappings: One is for an iteration of approximate transonic shock-fronts; and the other is for an iteration of the corresponding nonlinear boundary value problems for given approximate shock-fronts.
Let q > 2 and ψ 0 (η) = η cot ω 1 . Define Let M S and M are positive constants to be determined later. In order to find the perturbed shock solution to the fixed boundary value problem (3.12), (3.2), (3.13), and (3.4) of the self-similar shock solution (U 0 ; U − 0 ; ψ 0 ), our strategy is as follows: Let ε 0 > 0 be a small constant to be determined later and 0 < ε ≤ ε 0 . Given an approximate boundary ψ ∈ Σ MS ε , solve the nonlinear boundary value problem (3.12), (3.2), and (3.4) to obtain a perturbed solution U ψ of U 0 . Then we use one of the Rankine-Hugoniot conditions, (3.13), to update the approximate boundary and obtain new ψ * : This defines an iteration mapping: J S : ψ → ψ * . To prove Theorem 3.1, it suffices to verify that there exist positive constants M S and ε 0 such that J S is a well-defined, contraction mapping in Σ MS ε for any 0 < ε ≤ ε 0 .
Since the initial value problem (5.2) is easier, we will focus mainly on the nonlinear boundary value problem (3.12), (3.2), and (3.4) for given ψ ∈ Σ MS ε , which requires another nonlinear iteration: For given δU ∈ O Mε , a linearized boundary value problem will be solved in the weighted Sobolev space W 1,q (0) (Ω) to obtain a unique solution δU * that is defined as an iteration mapping J : δU → δU * . By showing that there exist positive constants M and ε 0 such that J is a well-defined contraction mapping in O Mε for any 0 < ε ≤ ε 0 , we conclude that the nonlinear problem (3.12), (3.2), and (3.4) is uniquely solvable in the weighted Sobolev space W 1,q (0) (Ω) as a perturbation to the background self-similar transonic shock solution.
In particular, the linearized problem to (3.12), (3.2), and (3.4) in the iteration J is in Ω, between Γ 0 and Γ 1 described in Section 2, and We denote this linearized problem as a linear operator T : δU → (F ; g 0 , g 1 ) for Then where O(1) depends only on γ and b. Then Therefore, there exist constants ν 0 and ε 0 such that, for any 0 < ν ≤ ν 0 and 0 < ε ≤ ε 0 , whereÂ andB are the matrices in (4.16) andǫ is the constant in Proposition 4.3.

Proof of Main Theorem I: Fixed Point of the Iteration Map J
In this section, we first prove that there exists a unique fixed point of the iteration mapping J introduced in Section 5. To achieve this, we prove that J is a welldefined, contraction mapping.
We will need the following lemma. These can be seen by the following direct calculations: and, for any constant s = 0, 6.1. Well-definedness of the iteration mapping J . We first show that there exist positive constants M and ε 0 such that, for any 0 < ε ≤ ε 0 , J is well-defined in O Mε with the help of estimate (5.9). By Lemma 6.1, we have The other terms in the expression of F can be estimated analogously. Hence, we have It is easy to see that Furthermore, we have and . With a direct calculation, we have Then Since ε ≪ ν 1 γ−1 , we obtain that, for any 1 < γ ≤ 2, where U s = sU + (1 − s)U 0 and K is independent of ν and ε.
Hereafter, we fix M =K. Then the mapping J is well-defined in O Mε .

Proof of Main Theorem II: Fixed Point of the Iteration Map J S
In this section, we prove that there exists a unique fixed point of the iteration mapping J S introduced in Section 5 by showing that J S is a well-defined, contraction mapping, which completes the proof of the main theorem. 7.1. Well-definedness of the iteration mapping J S . Let J S (ψ) = ψ * . Write Then Thus, by (5.2), we obtain whereK is a constant independent of ν and ε.
We choose M S =KM hereafter. Then J S is well-defined in Σ MS ε in the case that the positive constants ν and ε are sufficiently small. To complete the proof, it suffices to verify that J S is a contraction mapping in Σ MS ε . 7.2. Contraction of the iteration mapping J S . Let J S (ψ j ) = ψ j * , j = 1, 2. Then we have T (δU j ) = (F ; g 0 , g 1 )(δU j ; ψ j ), j = 1, 2, Thus, we obtain (Γj ) , where K is independent of δU j and ψ j , j = 1, 2, but depends only on ω 0 (b) and γ. Since we have An analogous calculation yields Then, as the calculation for J , we have . Analogous calculation for the other terms of F (δU 2 ; ψ 2 )−F (δU 1 ; ψ 1 ) finally leads to where K depends only on ω 0 (b) and γ, but independent of ψ j , ν 0 , and ε 0 . Furthermore, we have Then an analogous calculation as for J yields where K 1 and K 2 depend on ω 0 (b) and γ. Then, by (7.2)-(7.5), we have Choose ν 0 and ε 0 sufficiently small. Then, for any 0 < ν ≤ ν 0 and 0 < ε ≤ ε 0 , we have Thus, where, for the last inequality, we have again chosen ν 0 and ε 0 to be sufficiently small. This implies that J S is a contraction mapping so that it has a unique fix point in Σ MS ε , which completes the proof of Theorem 3.1.

Appendix: A Fredholm-type Theorem
To be self-contained, in this appendix, we give a proof for a Fredholm-type theorem, Theorem A.1, a special case of Theorem 4.1 in Maz'ya-Plamenevskiǐ [28], following their ideas. Consider the boundary value problem of an elliptic equation of second-order in an infinite strip G := {(t, x) : x ∈ I := (x 0 , x 1 ), t ∈ R} with boundaries Σ 0 = {x = x 0 } and Σ 1 = {x = x 1 }: where a(x) ∈ C 1 (Ī), f ∈ W 0,q (−1) (G), and g j ∈ W 1−1/q,q (−1) (R), j = 0, 1. We assume q > 2 since only this case is really used in this paper. Obviously, the operator (L; B 0 , B 1 ) of the boundary value problem (A.1)-(A.3) acts continuously from the space Consider the boundary value problem with a complex parameter λ on the interval I: Moreover, the solution ϕ ∈ W 2,p (β) (G) of (A.1)-(A.3) satisfies the estimate: Remark A.1. In the case p = 2, this assertion is well-known (cf. [29]). In this case, a solution in the class W 2,2 (β) (G) can be represented in the form where R(λ) denotes the inverse operator of problem (A.4)-(A.6) and F t→λ is the Fourier transform with respect to the t-variable into the λ-variable. If it is additionally assumed that f ∈ W 0,2 (β1) (G), g j ∈ W 1−1/2,2 (β1) (R) and that, in the closed strip between the lines Im λ = β and Im λ = β 1 , there are no points of the spectrum of (A.4)-(A.6), then the function ϕ defined by (A.8) belongs to W 2,2 (β1) (G), and To prove Theorem A.1, we need two lemmas, which are all in [28]. Let A 0 , A 1 , and A 2 be Banach spaces of functions on R, in each of which multiplication by scalar functions in C ∞ c (R) is defined. Let {ζ k } ∞ −∞ be a partition of unity on R subordinate to the covering of R by the intervals (k−1)δ < t < (k+1)δ, where δ is a fixed positive number and ζ k ∈ C ∞ (R). Suppose that the norms · j in the spaces A j , j = 0, 1, 2, possess the following properties: For p ∈ [1, ∞], where the constant C does not depend on v.
(ii) Let A 2 ⊂ A 0 . Suppose further that, for all functions v in A 1 with compact support on R, where σ k = ζ k−1 + ζ k + ζ k+1 , k = 0, ±1, . . . . Then Proof. According to (A.10) and (A.13), we have Since the operator of discrete convolution with kernel e −lε ∞ l=−∞ acts continuously in l p , it follows that where ε is a positive number and {ζ l } ∞ −∞ is a partition of unity on R subordinate to the covering of R by the intervals l − 1 < t < l + 1.