Semiclassical resolvent estimates for short-range L ∞ potentials. II

Abstract

We prove semiclassical resolvent estimates for real-valued potentials $V \in L^{\infty} (R^{n})$ , $n ⩾ 3$ , of the form $V = V_{L} + V_{S}$ , where $V_{L}$ is a long-range potential which is $C^{1}$ with respect to the radial variable, while $V_{S}$ is a short-range potential satisfying $V_{S} (x) = O ({⟨ x ⟩}^{- δ})$ with $δ > 1$ .

Keywords

Schrödinger operator resolvent estimates short-range potentials

1. Introduction and statement of results

The goal of this paper is to extend the semi-classical resolvent estimates obtained recently in [8,10] and [12] to a larger class of potentials. We are going to study the resolvent of the Schrödinger operator $\begin{matrix} P (h) = - h^{2} Δ + V (x) \end{matrix}$ where $0 < h ≪ 1$ is a semiclassical parameter, Δ is the negative Laplacian in $R^{n}$ , $n ⩾ 3$ , and $V \in L^{\infty} (R^{n})$ is a real-valued potential of the form $V = V_{L} + V_{S}$ , where $V_{L} \in C^{1} ([r_{0}, + \infty))$ with respect to the radial variable $r = | x |$ , $r_{0} > 0$ being some constant, is a long-range potential, while $V_{S}$ is a short-range potential satisfying $\begin{matrix} (1.1) & | V_{S} (x) | ⩽ C_{1} {(| x | + 1)}^{- δ} \end{matrix}$ with some constants $C_{1} > 0$ and $δ > 1$ . We suppose that there exists a decreasing function $p (r) > 0$ , $p (r) \to 0$ as $r \to \infty$ , such that $\begin{matrix} (1.2) & V_{L} (x) ⩽ p (| x |) for | x | ⩾ r_{0} . \end{matrix}$ We also suppose that $\begin{matrix} (1.3) & \partial_{r} V_{L} (x) ⩽ C_{2} {(| x | + 1)}^{- β} for | x | ⩾ r_{0} \end{matrix}$ with some constants $C_{2} > 0$ and $β > 1$ . As in [12] we introduce the quantity $\begin{matrix} g_{s}^{\pm} (h, θ) : = log {‖ {(| x | + 1)}^{- s} {(P (h) - E \pm i θ)}^{- 1} {(| x | + 1)}^{- s} ‖}_{L^{2} \to L^{2}} \end{matrix}$ where $L^{2} : = L^{2} (R^{n})$ , $0 < θ < 1$ , $s > 1 / 2$ is independent of h and $E > 0$ is a fixed energy level independent of h. Our first result is the following

Theorem 1.1.
Suppose the conditions ( 1.1 ), ( 1.2 ) and ( 1.3 ) are fulfilled with δ and β satisfying the condition $\begin{matrix} (1.4) & δ > 3, β > 3 . \end{matrix}$ Then there exist constants $C > 0$ and $h_{0} > 0$ independent of h and θ but depending on s and E such that for all $0 < h ⩽ h_{0}$ we have the bound $\begin{matrix} (1.5) & g_{s}^{\pm} (h, θ) ⩽ C h^{- 4 / 3} log (h^{- 1}) . \end{matrix}$ If the conditions ( 1.1 ), ( 1.2 ) and ( 1.3 ) are fulfilled with δ and β satisfying the condition $\begin{matrix} (1.6) & δ > 3, β = 3, \end{matrix}$ then we have the bound $\begin{matrix} (1.7) & g_{s}^{\pm} (h, θ) ⩽ C h^{- 4 / 3} {(log (h^{- 1}))}^{3 / 2} {(log log (h^{- 1}))}^{- 1} . \end{matrix}$

When $V_{S} \equiv 0$ and $V_{L}$ satisfying conditions similar to (1.2) and (1.3), it is proved in [4] when $n ⩾ 3$ and in [9] when $n = 2$ that $\begin{matrix} (1.8) & g_{s}^{\pm} (h, θ) ⩽ C h^{- 1} \end{matrix}$ with some constant $C > 0$ independent of h and θ. Previously, the bound (1.8) was proved for smooth potentials in [2] and an analog of (1.8) for Hölder potentials was proved in [11]. A high-frequency analog of (1.8) on Riemannian manifolds was also proved in [1] and [3]. When $V_{L} \equiv 0$ and $V_{S}$ satisfying the condition (1.1) with $δ > 3$ , the bound (1.5) has been recently proved in [12]. Previously, (1.5) was proved in [8] and [10] for real-valued compactly supported $L^{\infty}$ potentials. When $n = 1$ it has been recently shown in [6] (see also [7]) that we have the better bound (1.8) instead of (1.5). The method we use to prove Theorem 1.1 also allows us to get resolvent bounds when the conditions (1.4) and (1.6) are not satisfied, which however are much weaker than the bound (1.5). More precisely, we have the following
Theorem 1.2.
Suppose the conditions ( 1.1 ), ( 1.2 ) and ( 1.3 ) are fulfilled with δ and β satisfying either the condition $\begin{matrix} (1.9) & 1 < δ ⩽ 3, β > 1, \end{matrix}$ or the condition $\begin{matrix} (1.10) & δ > 3, 1 < β < 3 . \end{matrix}$ Then, there exist constants $C > 0$ and $h_{0} > 0$ independent of h and θ but depending on s and E such that for all $0 < h ⩽ h_{0}$ we have the bounds $\begin{matrix} (1.11) & g_{s}^{\pm} (h, θ) ⩽ C h^{- \frac{2}{3} - m_{1}} {(log (h^{- 1}))}^{ν} \end{matrix}$ if ( 1.9 ) holds, and $\begin{matrix} (1.12) & g_{s}^{\pm} (h, θ) ⩽ C h^{- \frac{4}{3} - \frac{1}{2} (3 - β) m_{2}} {(log (h^{- 1}))}^{ν} \end{matrix}$ if ( 1.10 ) holds, where $\begin{array}{l} m_{1} = max {\frac{7}{3 (δ - 1)}, \frac{4}{3 (β - 1)}} ⩾ \frac{7}{6}, \\ m_{2} = max {\frac{1}{δ - β}, \frac{4}{3 (β - 1)}} > \frac{2}{3}, \\ ν = max {\frac{1}{δ - 1}, \frac{1}{β - 1}} . \end{array}$

Clearly, this theorem implies the following
Corollary 1.3.
Suppose that $V_{L} \equiv 0$ and let $V = V_{S}$ satisfy the condition ( 1.1 ) with $1 < δ ⩽ 3$ . Then, there exist constants $C > 0$ and $h_{0} > 0$ independent of h and θ but depending on s and E such that for all $0 < h ⩽ h_{0}$ we have the bound $\begin{matrix} (1.13) & g_{s}^{\pm} (h, θ) ⩽ C h^{- \frac{2 δ + 5}{3 (δ - 1)}} {(log (h^{- 1}))}^{\frac{1}{δ - 1}} . \end{matrix}$

To prove the above theorems we follow the same strategy as in [12] which in turn is inspired by the paper [10]. It consists of using Carleman estimates with phase and weight functions, denoted by φ and μ below, depending only on the radial variable r and the parameter h, which have very weak regularity. It turns out that it suffices to choose φ belonging only to $C^{1}$ and μ only continuous. Thus we get derivatives $φ^{″}$ and $μ^{'}$ belonging to $L^{\infty}$ , which proves sufficient for the Carleman estimates to hold. Note that higher derivatives of φ and μ are not involved in the proof of the Carleman estimates (see the proof of Theorem 3.1 below). In order to be able to prove the Carleman estimates the functions φ and μ must satisfy some conditions (see the inequalities (2.3) and (2.9) below). On the other hand, to get as good resolvent bounds as possible we are looking for a phase function φ such that $max φ$ is as small as possible. The construction of such phase and weight functions is carried out in Section 2 following that one in [12]. However, here the construction is more complicated due to the more general class the potential belongs to.

It is not clear if the bounds (1.5), (1.7), (1.11) and (1.12) are optimal for $L^{\infty}$ potentials. In any case, they seem hard to improve unless one manages to construct a better phase function. By contrast, the optimality of the bound (1.8) for smooth potentials is well known (e.g. see [5]).
2. The construction of the phase and weight functions revisited

We will follow closely the construction in Section 2 of [12] making some suitable modifications in order to adapte it to the more general class of potentials we consider in the present paper. We will first construct the weight function μ as follows: $\begin{matrix} μ (r) = \{\begin{matrix} {(r + 1)}^{2 k} - 1 & for 0 ⩽ r ⩽ a, \\ {(a + 1)}^{2 k} - 1 + {(a + 1)}^{- 2 s + 1} - {(r + 1)}^{- 2 s + 1} & for r ⩾ a, \end{matrix} \end{matrix}$ where $a = h^{- m}$ with $\begin{matrix} m = \{\begin{matrix} m_{0} (1 + 2 ϵ λ) & if (1.4) holds, \\ m_{0} + ϵ λ / 2 + ϵ T_{1} & if (1.6) holds, \\ m_{1} + ν ϵ λ + ϵ T_{2} & if (1.9) holds, \\ m_{2} + ν ϵ λ + ϵ T_{3} & if (1.10) holds, \end{matrix} \end{matrix}$ where $ϵ = {(log \frac{1}{h})}^{- 1}$ , $λ = log log \frac{1}{h}$ , $m_{0} = max {\frac{2}{3}, \frac{1}{δ - 3}}$ , $m_{1}$ , $m_{2}$ and ν are as in Theorem 1.2, while $T_{j} > 0$ , $j = 1, 2, 3$ , are parameters independent of h to be fixed in the proof of Lemma 2.3. Furthermore, $\begin{matrix} k = \{\begin{matrix} 1 - ϵ & if (1.4) holds, \\ 1 - \frac{ϵ λ}{2 m_{0}} - ϵ t_{1} & if (1.6) holds, \\ \frac{2}{3 m_{1}} - \frac{2 ν ϵ λ}{3 m_{1}^{2}} - ϵ t_{2} & if (1.9) holds, \\ \frac{β - 1}{2} - \frac{(β - 1) ν ϵ λ}{2 m_{2}} - ϵ t_{3} & if (1.10) holds, \end{matrix} \end{matrix}$ and $\begin{matrix} (2.1) & s = \frac{1 + ϵ}{2} \end{matrix}$ where $t_{j} > 1$ , $j = 1, 2, 3$ , are parameters independent of h to be fixed in the proof of Lemma 2.3. Clearly, the first derivative (in sense of distributions) of μ satisfies $\begin{matrix} (2.2) & μ^{'} (r) = \{\begin{matrix} 2 k {(r + 1)}^{2 k - 1} & for 0 ⩽ r < a, \\ (2 s - 1) {(r + 1)}^{- 2 s} & for r > a . \end{matrix} \end{matrix}$ The following properties of the functions μ and $μ^{'}$ are essential to prove the Carleman estimates in the next section.

Lemma 2.1.
For all $r > 0$ , $r \neq a$ , we have the inequalities $\begin{array}{l} (2.3) & 2 r^{- 1} μ (r) - μ^{'} (r) ⩾ 0, \\ (2.4) & μ^{'} (r) ⩾ ϵ {(r + 1)}^{- 2 s}, \\ (2.5) & \frac{μ (r)}{μ^{'} (r)} ≲ ϵ^{- 1} a^{2 k} {(r + 1)}^{2 s}, \\ (2.6) & \frac{μ {(r)}^{2}}{μ^{'} (r)} ≲ ϵ^{- 1} a^{4 k} {(r + 1)}^{2 s} . \end{array}$
Proof.
It is easy to see that for $r < a$ (2.3) follows from the inequality $\begin{matrix} f (r) : = 1 + (1 - k) r - {(r + 1)}^{1 - 2 k} ⩾ 0 \end{matrix}$ for all $r ⩾ 0$ and $0 ⩽ k ⩽ 1$ . It is obvious for $1 / 2 ⩽ k ⩽ 1$ , while for $0 ⩽ k < 1 / 2$ we have $\begin{matrix} f^{'} (r) = 1 - k - (1 - 2 k) {(r + 1)}^{- 2 k} ⩾ k ⩾ 0 . \end{matrix}$ Hence in this case the function f is increasing, which implies $f (r) ⩾ f (0) = 0$ as desired. For $r > a$ the left-hand side of (2.3) is bounded from below by $\begin{matrix} 2 r^{- 1} ({(a + 1)}^{2 k} - 1 - s) > 0 \end{matrix}$ provided a is taken large enough. The lower bound (2.4) is an immediate consequence of (2.1) and (2.2), while the bounds (2.5) and (2.6) follow from (2.4) and the fact that $μ = O (a^{2 k})$ . □

We now turn to the construction of the phase function $φ \in C^{1} ([0, + \infty))$ such that $φ (0) = 0$ and $φ (r) > 0$ for $r > 0$ . We define the first derivative of φ by $\begin{matrix} φ^{'} (r) = \{\begin{matrix} τ {(r + 1)}^{- k} - τ {(a + 1)}^{- k} & for 0 ⩽ r ⩽ a, \\ 0 & for r ⩾ a, \end{matrix} \end{matrix}$ where $\begin{matrix} (2.7) & τ = τ_{0} h^{- 1 / 3} \end{matrix}$ with some parameter $τ_{0} ≫ 1$ independent of h to be fixed in Lemma 2.3 below. Clearly, the first derivative of $φ^{'}$ satisfies $\begin{matrix} φ^{″} (r) = \{\begin{matrix} - k τ {(r + 1)}^{- k - 1} & for 0 ⩽ r < a, \\ 0 & for r > a . \end{matrix} \end{matrix}$
Lemma 2.2.
For all $r ⩾ 0$ we have the bounds $\begin{matrix} (2.8) & h^{- 1} φ (r) ≲ \{\begin{matrix} h^{- 4 / 3} log (h^{- 1}) & if (1.4) holds, \\ h^{- 4 / 3} {(log (h^{- 1}))}^{3 / 2} {(log log (h^{- 1}))}^{- 1} & if (1.6) holds, \\ h^{- \frac{2}{3} - m_{1}} {(log (h^{- 1}))}^{ν} & if (1.9) holds, \\ h^{- \frac{4}{3} - \frac{1}{2} (3 - β) m_{2}} {(log (h^{- 1}))}^{ν} & if (1.10) holds . \end{matrix} \end{matrix}$
Proof.
Since $k < 1$ we have $\begin{matrix} max φ = \int_{0}^{a} φ^{'} (r) d r ⩽ τ \int_{0}^{a} {(r + 1)}^{- k} d r ⩽ \frac{τ}{1 - k} {(a + 1)}^{1 - k} ≲ {(1 - k)}^{- 1} h^{- \frac{1}{3} - m (1 - k)} \end{matrix}$ and $\begin{matrix} {(1 - k)}^{- 1} ≲ \{\begin{matrix} ϵ^{- 1} & if (1.4) holds, \\ {(ϵ λ)}^{- 1} & if (1.6) holds, \\ 1 & if (1.9) holds, \\ 1 & if (1.10) holds . \end{matrix} \end{matrix}$ Since $\begin{matrix} m (1 - k) = \{\begin{matrix} O (ϵ) & if (1.4) holds, \\ ϵ λ / 2 + O (ϵ) & if (1.6) holds, \\ m_{1} - \frac{2}{3} + ν ϵ λ + O (ϵ) & if (1.9) holds, \\ \frac{1}{2} (3 - β) m_{2} + ν ϵ λ + O (ϵ) & if (1.10) holds, \end{matrix} \end{matrix}$ and taking into account that $h^{- ϵ λ} = ϵ^{- 1}$ and $h^{- ϵ} = e$ , we get (2.8) from the above bounds. □

Let $ϕ \in C_{0}^{\infty} ([1, 2])$ , $ϕ ⩾ 0$ , be a real-valued function independent of h such that $\int_{- \infty}^{\infty} ϕ (σ) d σ = 1$ . Given a parameter $b ≫ r_{0}$ to be fixed in the proof of Theorem 3.1 below, independent of h, set $\begin{matrix} ψ_{b} (r) = b^{- 1} \int_{r}^{\infty} ϕ (σ / b) d σ . \end{matrix}$ Clearly, we have $0 ⩽ ψ_{b} ⩽ 1$ and $ψ_{b} (r) = 1$ for $r ⩽ b$ , $ψ_{b} (r) = 0$ for $r ⩾ 2 b$ . For $r > 0$ , $r \neq a$ , set $\begin{matrix} A (r) = {(μ φ^{' 2})}^{'} (r) \end{matrix}$ and $\begin{array}{l} B (r) = & \frac{3 {(μ (r) (h^{- 1} C_{1} {(r + 1)}^{- δ} + h^{- 1} Q_{b} ψ_{b} (r) + | φ^{″} (r) |))}^{2}}{h^{- 1} φ^{'} (r) μ (r) + μ^{'} (r)} \\ + μ (r) (1 - ψ_{b} (r)) C_{2} {(r + 1)}^{- β} \end{array}$ where $Q_{b} ⩾ 0$ is some constant depending only on b. The following lemma will play a crucial role in the proof of the Carleman estimates in the next section. Lemma 2.3.
There exist constants $b_{0} = b_{0} (E) > 0$ , $τ_{0} = τ_{0} (b, E) > 0$ and $h_{0} = h_{0} (b, E) > 0$ so that for τ satisfying ( 2.7 ) and for all $b ⩾ b_{0}$ , $0 < h ⩽ h_{0}$ we have the inequality $\begin{matrix} (2.9) & A (r) - B (r) ⩾ - \frac{E}{2} μ^{'} (r) \end{matrix}$ for all $r > 0$ , $r \neq a$ .
Proof.
For $r < a$ we have $\begin{array}{l} A (r) & = - {(φ^{' 2})}^{'} (r) + τ^{2} \partial_{r} {(1 - {(r + 1)}^{k} {(a + 1)}^{- k})}^{2} \\ = - 2 φ^{'} (r) φ^{″} (r) - 2 k τ^{2} {(r + 1)}^{k - 1} {(a + 1)}^{- k} (1 - {(r + 1)}^{k} {(a + 1)}^{- k}) \\ ⩾ 2 k τ {(r + 1)}^{- k - 1} φ^{'} (r) - 2 k τ^{2} {(r + 1)}^{k - 1} {(a + 1)}^{- k} \\ ⩾ 2 k τ {(r + 1)}^{- k - 1} φ^{'} (r) - τ^{2} a^{- k} μ^{'} (r) . \end{array}$ Taking into account the definition of the parameters a and τ we conclude $\begin{matrix} (2.10) & A (r) ⩾ 2 k τ {(r + 1)}^{- k - 1} φ^{'} (r) - O (h^{k m - 2 / 3}) μ^{'} (r) \end{matrix}$ for all $r < a$ . Observe now that if (1.4) holds, we have $\begin{matrix} k m - 2 / 3 = m_{0} - 2 / 3 + 2 ϵ λ m_{0} - O (ϵ) ⩾ 4 ϵ λ / 3 - O (ϵ) ⩾ ϵ λ \end{matrix}$ provided λ is big enough. If (1.6) holds, we have $\begin{array}{l} k m - 2 / 3 & = (1 - \frac{ϵ λ}{2 m_{0}} - ϵ t_{1}) (m_{0} + ϵ λ / 2 + ϵ T_{1}) - 2 / 3 \\ = m_{0} - 2 / 3 + ϵ (T_{1} - m_{0} t_{1}) - O (ϵ^{2} λ^{2}) ⩾ ϵ m_{0} t_{1} \end{array}$ provided we take $T_{1} = 3 m_{0} t_{1}$ . If (1.9) holds, we have $\begin{array}{l} k m - 2 / 3 & = (\frac{2}{3 m_{1}} - \frac{2 ν ϵ λ}{3 m_{1}^{2}} - ϵ t_{2}) (m_{1} + ν ϵ λ + ϵ T_{2}) - 2 / 3 \\ = (\frac{2 T_{2}}{3 m_{1}} - m_{1} t_{2}) ϵ - O (ϵ^{2} λ^{2}) ⩾ ϵ m_{1} t_{2} \end{array}$ provided we take $T_{2} = 6 m_{1}^{2} t_{2}$ . If (1.10) holds, we have $\begin{array}{l} k m - 2 / 3 & = (\frac{β - 1}{2} - \frac{(β - 1) ν ϵ λ}{2 m_{2}} - ϵ t_{3}) (m_{2} + ν ϵ λ + ϵ T_{3}) - 2 / 3 \\ = \frac{(β - 1) m_{2}}{2} - \frac{2}{3} + (\frac{(β - 1) T_{3}}{2} - m_{2} t_{3}) ϵ - O (ϵ^{2} λ^{2}) ⩾ ϵ m_{2} t_{3} \end{array}$ provided we take $T_{3} = \frac{6 m_{2} t_{3}}{β - 1}$ . Using that $h^{ϵ λ} = ϵ$ , $h^{ϵ} = e^{- 1}$ , we conclude that $\begin{matrix} (2.11) & h^{k m - 2 / 3} ⩽ \{\begin{matrix} ϵ & if (1.4) holds, \\ e^{- m_{0} t_{1}} & if (1.6) holds, \\ e^{- m_{1} t_{2}} & if (1.9) holds, \\ e^{- m_{2} t_{3}} & if (1.10) holds . \end{matrix} \end{matrix}$ Taking ϵ small enough and $t_{1}$ , $t_{2}$ , $t_{3}$ big enough, we obtain from (2.10) and (2.11) that in all cases we have the estimate $\begin{matrix} (2.12) & A (r) ⩾ 2 k τ {(r + 1)}^{- k - 1} φ^{'} (r) - \frac{E}{4} μ^{'} (r) \end{matrix}$ for all $r < a$ . We will now bound the function B from above. Note that taking h small enough we can arrange that $2 b < a / 2$ . Let first $0 < r ⩽ \frac{a}{2}$ . Since in this case we have $\begin{matrix} φ^{'} (r) ⩾ \tilde{C} τ {(r + 1)}^{- k} \end{matrix}$ with some constant $\tilde{C} > 0$ , we obtain $\begin{array}{l} B (r) ≲ & \frac{μ (r) (h^{- 2} {\tilde{Q}}_{b} {(r + 1)}^{- 2 δ} + φ^{″} {(r)}^{2})}{h^{- 1} φ^{'} (r)} + μ (r) (1 - ψ_{b} (r)) {(r + 1)}^{- β} \\ ≲ & {\tilde{Q}}_{b} {(τ h)}^{- 1} \frac{μ (r) {(r + 1)}^{1 + k - 2 δ}}{φ^{'} {(r)}^{2}} τ {(r + 1)}^{- k - 1} φ^{'} (r) + h \frac{μ (r) φ^{″} {(r)}^{2}}{μ^{'} (r) φ^{'} (r)} μ^{'} (r) \\ + (1 - ψ_{b} (r)) {(r + 1)}^{2 k - β} \\ ≲ & {\tilde{Q}}_{b} τ^{- 3} h^{- 1} {(r + 1)}^{1 + 5 k - 2 δ} τ {(r + 1)}^{- k - 1} φ^{'} (r) + τ h μ^{'} (r) \\ + (1 - ψ_{b} (r)) {(r + 1)}^{1 - β} μ^{'} (r) \\ ≲ & {\tilde{Q}}_{b} τ_{0}^{- 3} τ {(r + 1)}^{- k - 1} φ^{'} (r) + (τ_{0} h^{2 / 3} + b^{- β + 1}) μ^{'} (r) \end{array}$ where ${\tilde{Q}}_{b} > 0$ is some constant depending only on b and we have used that $k < (2 δ - 1) / 5$ in all cases. Taking h small enough, depending on $τ_{0}$ , and b big enough, independent of h and $τ_{0}$ , we get the bound $\begin{matrix} (2.13) & B (r) ⩽ C {\tilde{Q}}_{b} τ_{0}^{- 3} τ {(r + 1)}^{- k - 1} φ^{'} (r) + \frac{E}{4} μ^{'} (r) \end{matrix}$ with some constant $C > 0$ . In this case we get (2.9) from (2.12) and (2.13) by taking $τ_{0}$ big enough depending on b and C but independent of h.

Let now $\frac{a}{2} < r < a$ . Observe first that $m (δ - 1) - 1 > 0$ in all cases. Indeed, when (1.4) or (1.6) holds we have $\begin{matrix} m (δ - 1) - 1 > m_{0} (δ - 1) - 1 ⩾ {(δ - 3)}^{- 1} (δ - 1) - 1 = 2 {(δ - 3)}^{- 1} . \end{matrix}$ When (1.9) holds we have $\begin{matrix} m (δ - 1) - 1 > m_{1} (δ - 1) - 1 ⩾ 4 / 3 . \end{matrix}$ When (1.10) holds we have $\begin{matrix} m (δ - 1) - 1 > m_{2} (δ - 1) - 1 ⩾ {(δ - β)}^{- 1} (δ - 1) - 1 = (β - 1) {(δ - β)}^{- 1} . \end{matrix}$ Then, using this together with (2.11), we get the bound $\begin{array}{l} B (r) & ≲ {(\frac{μ (r)}{μ^{'} (r)})}^{2} {(h^{- 1} {(r + 1)}^{- δ} + | φ^{″} (r) |)}^{2} μ^{'} (r) + {(r + 1)}^{- β + 1} μ^{'} (r) \\ ≲ (h^{- 2} {(r + 1)}^{2 - 2 δ} + τ^{2} {(r + 1)}^{- 2 k}) μ^{'} (r) + a^{- β + 1} μ^{'} (r) \\ ≲ (h^{- 2} a^{2 - 2 δ} + τ^{2} a^{- 2 k}) μ^{'} (r) + a^{- β + 1} μ^{'} (r) \\ ≲ (h^{2 m (δ - 1) - 2} + h^{2 k m - 2 / 3} + h^{m (β - 1)}) μ^{'} (r) \\ ≲ (h^{2 m (δ - 1) - 2} + h^{k m} + h^{m (β - 1)}) μ^{'} (r) \\ ⩽ \frac{E}{4} μ^{'} (r) \end{array}$ provided h is taken small enough. Again, this bound together with (2.12) imply (2.9).

It remains to consider the case $r > a$ . Using (2.5) we get $\begin{array}{l} B (r) & ≲ \frac{{(μ (r) (h^{- 1} {(r + 1)}^{- δ}))}^{2}}{μ^{'} (r)} + {(r + 1)}^{- β} μ (r) \\ ≲ ϵ^{- 2} h^{- 2} a^{4 k} {(r + 1)}^{4 s - 2 δ} μ^{'} (r) + ϵ^{- 1} a^{2 k} {(r + 1)}^{2 s - β} μ^{'} (r) \\ ≲ (ϵ^{- 2} h^{- 2} a^{4 k + 4 s - 2 δ} + ϵ^{- 1} a^{2 k + 2 s - β}) μ^{'} (r) \\ ≲ (h^{2 m (δ - 2 k - 2 s) - 2 - 2 ϵ λ} + h^{m (β - 2 k - 2 s) - ϵ λ}) μ^{'} (r) \end{array}$ where we have used that $ϵ^{- 1} = h^{- ϵ λ}$ . When (1.4) holds we have $\begin{array}{l} m (δ - 2 k - 2 s) - 1 - ϵ λ & = (m_{0} + 2 ϵ λ m_{0}) (δ - 3 + ϵ) - 1 - ϵ λ \\ ⩾ m_{0} (δ - 3) - 1 + ϵ λ (2 (δ - 3) m_{0} - 1) ⩾ ϵ λ \end{array}$ and $\begin{matrix} m (β - 2 k - 2 s) - ϵ λ ⩾ m_{0} (β - 3) - O (ϵ λ) ⩾ m_{0} (β - 3) / 2 . \end{matrix}$ When (1.6) holds we have $\begin{array}{l} m (δ - 2 k - 2 s) - 1 - ϵ λ \\ = (m_{0} + ϵ λ / 2 + ϵ T_{1}) (δ - 3 + \frac{ϵ λ}{m_{0}} + ϵ (2 t_{1} - 1)) - 1 - ϵ λ \\ > (m_{0} + ϵ λ / 2) (δ - 3 + \frac{ϵ λ}{m_{0}}) - 1 - ϵ λ \\ > m_{0} (δ - 3) - 1 + (δ - 3) ϵ λ / 2 ⩾ (δ - 3) ϵ λ / 2 \end{array}$ and $\begin{array}{l} m (β - 2 k - 2 s) - ϵ λ \\ = (m_{0} + ϵ λ + ϵ T_{1}) (\frac{ϵ λ}{m_{0}} + ϵ (2 t_{1} - 1)) - ϵ λ ⩾ ϵ t_{1} m_{0} . \end{array}$ When (1.9) holds we have $\begin{array}{l} m (δ - 2 k - 2 s) - 1 - ϵ λ \\ = (m_{1} + ν ϵ λ + ϵ T_{2}) (δ - 1 - \frac{4}{3 m_{1}} + \frac{4 ν ϵ λ}{3 m_{1}^{2}} + ϵ (2 t_{2} - 1)) - 1 - ϵ λ \\ = m_{1} (δ - 1) - \frac{7}{3} + ((δ - 1) ν - 1) ϵ λ + ϵ ((δ - 1) T_{2} + m_{1} (2 t_{2} - 1)) - O (ϵ^{2} λ^{2}) ⩾ ϵ m_{1} t_{2} \end{array}$ and $\begin{array}{l} m (β - 2 k - 2 s) - ϵ λ \\ = (m_{1} + ν ϵ λ + ϵ T_{2}) (β - 1 - \frac{4}{3 m_{1}} + \frac{4 ν ϵ λ}{3 m_{1}^{2}} + ϵ (2 t_{2} - 1)) - ϵ λ \\ = m_{1} (β - 1) - \frac{4}{3} + ((β - 1) ν - 1) ϵ λ + ϵ ((β - 1) T_{2} + m_{1} (2 t_{2} - 1)) - O (ϵ^{2} λ^{2}) ⩾ ϵ m_{1} t_{2} . \end{array}$ When (1.10) holds we have $\begin{array}{l} m (δ - 2 k - 2 s) - 1 - ϵ λ \\ = (m_{2} + ν ϵ λ + ϵ T_{3}) (δ - β + \frac{(β - 1) ν ϵ λ}{m_{2}} + ϵ (2 t_{3} - 1)) - 1 - ϵ λ \\ = m_{2} (δ - β) - 1 + ((δ - 1) ν - 1) ϵ λ + ϵ ((δ - β) T_{3} + m_{2} (2 t_{3} - 1)) - O (ϵ^{2} λ^{2}) ⩾ ϵ m_{2} t_{3} \end{array}$ and $\begin{array}{l} m (β - 2 k - 2 s) - ϵ λ \\ = (m_{2} + ν ϵ λ + ϵ T_{3}) (\frac{(β - 1) ν ϵ λ}{m_{2}} + ϵ (2 t_{3} - 1)) - ϵ λ \\ = ((β - 1) ν - 1) ϵ λ + ϵ m_{2} (2 t_{3} - 1) - O (ϵ^{2} λ^{2}) ⩾ ϵ m_{2} t_{3} . \end{array}$ We conclude from the above inequalities that $\begin{matrix} (2.14) & h^{2 m (δ - 2 k - 2 s) - 2 - 2 ϵ λ} + h^{m (β - 2 k - 2 s) - ϵ λ} ⩽ \{\begin{matrix} ϵ^{2} + h^{γ} & if (1.4) holds, \\ ϵ^{γ} + e^{- m_{0} t_{1}} & if (1.6) holds, \\ 2 e^{- m_{1} t_{2}} & if (1.9) holds, \\ 2 e^{- m_{2} t_{3}} & if (1.10) holds, \end{matrix} \end{matrix}$ with some constant $γ > 0$ independent of h. It follows from (2.14) that taking h small enough and $t_{1}$ , $t_{2}$ and $t_{3}$ large enough, independent of h, we can arrange the bound $\begin{matrix} (2.15) & B (r) ⩽ \frac{E}{2} μ^{'} (r) . \end{matrix}$ Since in this case $A (r) = 0$ , the bound (2.15) clearly implies (2.9). □

3. Carleman estimates

In this section we will prove the following

Theorem 3.1.
Suppose ( 1.1 ), ( 1.2 ) and ( 1.3 ) are fulfilled and let s satisfy ( 2.1 ). Then, for all functions $f \in H^{2} (R^{n})$ such that ${(| x | + 1)}^{s} (P (h) - E \pm i θ) f \in L^{2}$ and for all $0 < h ⩽ h_{0}$ , we have the estimate $\begin{array}{l} {‖ {(| x | + 1)}^{- s} e^{φ / h} f ‖}_{L^{2}} \\ ⩽ C a^{2 k} {(ϵ h)}^{- 1} {‖ {(| x | + 1)}^{s} e^{φ / h} (P (h) - E \pm i θ) f ‖}_{L^{2}} \\ (3.1) & + C a^{k} τ {(\frac{θ}{ϵ h})}^{1 / 2} {‖ e^{φ / h} f ‖}_{L^{2}} \end{array}$ with a constant $C > 0$ independent of h, θ and f.
Proof.
We will adapt the proof of Theorem 3.1 of [12] to this more general case. We pass to the polar coordinates $(r, w) \in R^{+} \times S^{n - 1}$ , $r = | x |$ , $w = x / | x |$ , and recall that $L^{2} (R^{n}) = L^{2} (R^{+} \times S^{n - 1}, r^{n - 1} d r d w)$ . In what follows we denote by $‖ \cdot ‖$ and $⟨ \cdot, \cdot ⟩$ the norm and the scalar product in $L^{2} (S^{n - 1})$ . We will make use of the identity $\begin{matrix} (3.2) & r^{(n - 1) / 2} Δ r^{- (n - 1) / 2} = \partial_{r}^{2} + \frac{{\tilde{Δ}}_{w}}{r^{2}} \end{matrix}$ where ${\tilde{Δ}}_{w} = Δ_{w} - \frac{1}{4} (n - 1) (n - 3)$ and $Δ_{w}$ denotes the negative Laplace–Beltrami operator on $S^{n - 1}$ . Set $u = r^{(n - 1) / 2} e^{φ / h} f$ and $\begin{array}{l} P^{\pm} (h) = r^{(n - 1) / 2} (P (h) - E \pm i θ) r^{- (n - 1) / 2}, \\ P_{φ}^{\pm} (h) = e^{φ / h} P^{\pm} (h) e^{- φ / h} . \end{array}$ Using (3.2) we can write the operator $P^{\pm} (h)$ in the coordinates $(r, w)$ as follows $\begin{matrix} P^{\pm} (h) = D_{r}^{2} + \frac{Λ_{w}}{r^{2}} - E \pm i θ + V \end{matrix}$ where we have put $D_{r} = - i h \partial_{r}$ and $Λ_{w} = - h^{2} {\tilde{Δ}}_{w}$ . Since the function φ depends only on the variable r, this implies $\begin{matrix} P_{φ}^{\pm} (h) = D_{r}^{2} + \frac{Λ_{w}}{r^{2}} - E \pm i θ - φ^{' 2} + h φ^{″} + 2 i φ^{'} D_{r} + V . \end{matrix}$ We now write $V = {\tilde{V}}_{S} + {\tilde{V}}_{L}$ with $\begin{matrix} {\tilde{V}}_{S} (x) = V_{S} (x) + ψ_{b} (| x |) V_{L} (x) \end{matrix}$ and $\begin{matrix} {\tilde{V}}_{L} (x) = (1 - ψ_{b} (| x |)) V_{L} (x) . \end{matrix}$ For $r > 0$ , $r \neq a$ , introduce the function $\begin{matrix} F (r) = - ⟨ (r^{- 2} Λ_{w} - E - φ^{'} {(r)}^{2} + {\tilde{V}}_{L} (r, \cdot)) u (r, \cdot), u (r, \cdot) ⟩ + {‖ D_{r} u (r, \cdot) ‖}^{2} \end{matrix}$ where ${\tilde{V}}_{L} (r, w) : = {\tilde{V}}_{L} (r w)$ . It is easy to check that its first derivative is given by $\begin{array}{l} F^{'} (r) = & \frac{2}{r} ⟨ r^{- 2} Λ_{w} u (r, \cdot), u (r, \cdot) ⟩ + {({(φ^{'})}^{2} - {\tilde{V}}_{L})}^{'} {‖ u (r, \cdot) ‖}^{2} \\ - 2 h^{- 1} Im ⟨ P_{φ}^{\pm} (h) u (r, \cdot), D_{r} u (r, \cdot) ⟩ \\ \pm 2 θ h^{- 1} Re ⟨ u (r, \cdot), D_{r} u (r, \cdot) ⟩ + 4 h^{- 1} φ^{'} {‖ D_{r} u (r, \cdot) ‖}^{2} \\ + 2 h^{- 1} Im ⟨ ({\tilde{V}}_{S} + h φ^{″}) u (r, \cdot), D_{r} u (r, \cdot) ⟩ . \end{array}$ Thus, if μ is the function defined in the previous section, we obtain the identity $\begin{array}{l} μ^{'} F + μ F^{'} = & (2 r^{- 1} μ - μ^{'}) ⟨ r^{- 2} Λ_{w} u (r, \cdot), u (r, \cdot) ⟩ + (E μ^{'} + {(μ {(φ^{'})}^{2} - μ {\tilde{V}}_{L})}^{'}) {‖ u (r, \cdot) ‖}^{2} \\ - 2 h^{- 1} μ Im ⟨ P_{φ}^{\pm} (h) u (r, \cdot), D_{r} u (r, \cdot) ⟩ \\ \pm 2 θ h^{- 1} μ Re ⟨ u (r, \cdot), D_{r} u (r, \cdot) ⟩ + (μ^{'} + 4 h^{- 1} φ^{'} μ) {‖ D_{r} u (r, \cdot) ‖}^{2} \\ + 2 h^{- 1} μ Im ⟨ ({\tilde{V}}_{S} + h φ^{″}) u (r, \cdot), D_{r} u (r, \cdot) ⟩ . \end{array}$ Using that $Λ_{w} ⩾ 0$ together with (2.3) we get the inequality $\begin{array}{l} μ^{'} F + μ F^{'} ⩾ & (E μ^{'} + {(μ {(φ^{'})}^{2} - μ {\tilde{V}}_{L})}^{'}) {‖ u (r, \cdot) ‖}^{2} + (μ^{'} + 4 h^{- 1} φ^{'} μ) {‖ D_{r} u (r, \cdot) ‖}^{2} \\ - \frac{3 h^{- 2} μ^{2}}{μ^{'}} {‖ P_{φ}^{\pm} (h) u (r, \cdot) ‖}^{2} - \frac{μ^{'}}{3} {‖ D_{r} u (r, \cdot) ‖}^{2} \\ - θ h^{- 1} μ ({‖ u (r, \cdot) ‖}^{2} + {‖ D_{r} u (r, \cdot) ‖}^{2}) \\ - 3 h^{- 2} μ^{2} {(μ^{'} + 4 h^{- 1} φ^{'} μ)}^{- 1} {‖ ({\tilde{V}}_{S} + h φ^{″}) u (r, \cdot) ‖}^{2} \\ - \frac{1}{3} (μ^{'} + 4 h^{- 1} φ^{'} μ) {‖ D_{r} u (r, \cdot) ‖}^{2} . \end{array}$ In view of the assumptions (1.2) and (1.3) we have $\begin{array}{l} {(μ {\tilde{V}}_{L})}^{'} & = μ^{'} {\tilde{V}}_{L} + μ {\tilde{V}}_{L}^{'} = μ^{'} (1 - ψ_{b}) V_{L} - μ ψ_{b}^{'} V_{L} + μ (1 - ψ_{b}) V_{L}^{'} \\ ⩽ μ^{'} (1 - ψ_{b}) p (r) + μ b^{- 1} ϕ (r / b) p (r) + μ (1 - ψ_{b}) C_{2} {(r + 1)}^{- β} \\ ⩽ μ^{'} (1 - ψ_{b}) p (b) + O (r) b^{- 1} ϕ (r / b) p (b) μ^{'} + μ (1 - ψ_{b}) C_{2} {(r + 1)}^{- β} \\ ⩽ O (1) p (b) μ^{'} + μ (1 - ψ_{b}) C_{2} {(r + 1)}^{- β} \\ ⩽ \frac{E}{3} μ^{'} + μ (1 - ψ_{b}) C_{2} {(r + 1)}^{- β} \end{array}$ provided b is taken large enough. Observe also that the assumption (1.1) yields $\begin{matrix} | {\tilde{V}}_{S} | ⩽ | V_{S} | + ψ_{b} | V_{L} | ⩽ C_{1} {(r + 1)}^{- δ} + Q_{b} ψ_{b} \end{matrix}$ where $Q_{b} = {sup}_{| x | ⩽ 2 b} | V_{L} (x) |$ . Combining the above inequalities we get $\begin{array}{l} μ^{'} F + μ F^{'} \\ ⩾ (\frac{2 E}{3} μ^{'} + {(μ {(φ^{'})}^{2})}^{'}) {‖ u (r, \cdot) ‖}^{2} \\ - (3 μ^{2} {(μ^{'} + h^{- 1} φ^{'} μ)}^{- 1} {(h^{- 1} C_{1} {(r + 1)}^{- δ} + h^{- 1} Q_{b} ψ_{b} + | φ^{″} |)}^{2} \\ + μ (1 - ψ_{b}) C_{2} {(r + 1)}^{- β}) {‖ u (r, \cdot) ‖}^{2} \\ - \frac{3 h^{- 2} μ^{2}}{μ^{'}} {‖ P_{φ}^{\pm} (h) u (r, \cdot) ‖}^{2} - θ h^{- 1} μ ({‖ u (r, \cdot) ‖}^{2} + {‖ D_{r} u (r, \cdot) ‖}^{2}) \\ = (\frac{2 E}{3} μ^{'} + A (r) - B (r)) {‖ u (r, \cdot) ‖}^{2} \\ - \frac{3 h^{- 2} μ^{2}}{μ^{'}} {‖ P_{φ}^{\pm} (h) u (r, \cdot) ‖}^{2} - θ h^{- 1} μ ({‖ u (r, \cdot) ‖}^{2} + {‖ D_{r} u (r, \cdot) ‖}^{2}) . \end{array}$ Now we use Lemma 2.3 to conclude that $\begin{array}{l} μ^{'} F + μ F^{'} ⩾ & \frac{E}{6} μ^{'} {‖ u (r, \cdot) ‖}^{2} - \frac{3 h^{- 2} μ^{2}}{μ^{'}} {‖ P_{φ}^{\pm} (h) u (r, \cdot) ‖}^{2} \\ - θ h^{- 1} μ ({‖ u (r, \cdot) ‖}^{2} + {‖ D_{r} u (r, \cdot) ‖}^{2}) . \end{array}$ We integrate this inequality with respect to r and use that, since $μ (0) = 0$ , we have $\begin{matrix} \int_{0}^{\infty} (μ^{'} F + μ F^{'}) d r = 0 . \end{matrix}$ Thus we obtain the estimate $\begin{array}{l} \frac{E}{6} \int_{0}^{\infty} μ^{'} {‖ u (r, \cdot) ‖}^{2} d r ⩽ & 3 h^{- 2} \int_{0}^{\infty} \frac{μ^{2}}{μ^{'}} {‖ P_{φ}^{\pm} (h) u (r, \cdot) ‖}^{2} d r \\ (3.3) & + θ h^{- 1} \int_{0}^{\infty} μ ({‖ u (r, \cdot) ‖}^{2} + {‖ D_{r} u (r, \cdot) ‖}^{2}) d r . \end{array}$ Using that $μ = O (a^{2 k})$ together with (2.4) and (2.6) we get from (3.3) $\begin{array}{l} \int_{0}^{\infty} {(r + 1)}^{- 2 s} {‖ u (r, \cdot) ‖}^{2} d r ⩽ & C a^{4 k} {(ϵ h)}^{- 2} \int_{0}^{\infty} {(r + 1)}^{2 s} {‖ P_{φ}^{\pm} (h) u (r, \cdot) ‖}^{2} d r \\ (3.4) & + C θ {(ϵ h)}^{- 1} a^{2 k} \int_{0}^{\infty} ({‖ u (r, \cdot) ‖}^{2} + {‖ D_{r} u (r, \cdot) ‖}^{2}) d r \end{array}$ with some constant $C > 0$ independent of h and θ. On the other hand, we have the identity $\begin{matrix} Re \int_{0}^{\infty} ⟨ 2 i φ^{'} D_{r} u (r, \cdot), u (r, \cdot) ⟩ d r = \int_{0}^{\infty} h φ^{″} {‖ u (r, \cdot) ‖}^{2} d r \end{matrix}$ and hence $\begin{array}{l} Re \int_{0}^{\infty} ⟨ P_{φ}^{\pm} (h) u (r, \cdot), u (r, \cdot) ⟩ d r = & \int_{0}^{\infty} {‖ D_{r} u (r, \cdot) ‖}^{2} d r + \int_{0}^{\infty} ⟨ r^{- 2} Λ_{w} u (r, \cdot), u (r, \cdot) ⟩ d r \\ - \int_{0}^{\infty} (E + φ^{' 2}) {‖ u (r, \cdot) ‖}^{2} d r + \int_{0}^{\infty} ⟨ V u (r, \cdot), u (r, \cdot) ⟩ d r . \end{array}$ This implies $\begin{array}{l} θ {(ϵ h)}^{- 1} a^{2 k} \int_{0}^{\infty} {‖ D_{r} u (r, \cdot) ‖}^{2} d r \\ ⩽ C τ^{2} θ {(ϵ h)}^{- 1} a^{2 k} \int_{0}^{\infty} {‖ u (r, \cdot) ‖}^{2} d r \\ + γ \int_{0}^{\infty} {(r + 1)}^{- 2 s} {‖ u (r, \cdot) ‖}^{2} d r \\ (3.5) & + γ^{- 1} {(ϵ h)}^{- 2} a^{4 k} \int_{0}^{\infty} {(r + 1)}^{2 s} {‖ P_{φ}^{\pm} (h) u (r, \cdot) ‖}^{2} d r \end{array}$ for every $γ > 0$ . Taking γ small enough, independent of h, and combining the estimates (3.4) and (3.5), we get $\begin{array}{l} \int_{0}^{\infty} {(r + 1)}^{- 2 s} {‖ u (r, \cdot) ‖}^{2} d r \\ ⩽ C a^{4 k} {(ϵ h)}^{- 2} \int_{0}^{\infty} {(r + 1)}^{2 s} {‖ P_{φ}^{\pm} (h) u (r, \cdot) ‖}^{2} d r \\ (3.6) & + C θ {(ϵ h)}^{- 1} a^{2 k} τ^{2} \int_{0}^{\infty} {‖ u (r, \cdot) ‖}^{2} d r \end{array}$ with a new constant $C > 0$ independent of h and θ. Clearly, the estimate (3.6) implies (3.1). □

4. Resolvent estimates

Theorems 1.1 and 1.2 can be obtained from Theorem 3.1 in the same way as in Section 4 of [12]. Here we will sketch the proof for the sake of completeness. Observe that it follows from the estimate (3.1) and Lemma 2.2 that for $0 < h ≪ 1$ and s satisfying (2.1) we have the estimate $\begin{matrix} (4.1) & {‖ {(| x | + 1)}^{- s} f ‖}_{L^{2}} ⩽ M {‖ {(| x | + 1)}^{s} (P (h) - E \pm i θ) f ‖}_{L^{2}} + M θ^{1 / 2} ‖ f ‖_{L^{2}} \end{matrix}$ where $M > 0$ is given by $\begin{matrix} log M = \{\begin{matrix} C h^{- 4 / 3} log (h^{- 1}) & if (1.4) holds, \\ C h^{- 4 / 3} {(log (h^{- 1}))}^{3 / 2} {(log log (h^{- 1}))}^{- 1} & if (1.6) holds, \\ C h^{- \frac{2}{3} - m_{1}} {(log (h^{- 1}))}^{ν} & if (1.9) holds, \\ C h^{- \frac{4}{3} - \frac{1}{2} (3 - β) m_{2}} {(log (h^{- 1}))}^{ν} & if (1.10) holds, \end{matrix} \end{matrix}$ with a constant $C > 0$ independent of h and θ. On the other hand, since the operator $P (h)$ is symmetric, we have $\begin{array}{l} θ ‖ f ‖_{L^{2}}^{2} & = \pm Im {⟨ (P (h) - E \pm i θ) f, f ⟩}_{L^{2}} \\ (4.2) & ⩽ {(2 M)}^{- 2} {‖ {(| x | + 1)}^{- s} f ‖}_{L^{2}}^{2} + {(2 M)}^{2} {‖ {(| x | + 1)}^{s} (P (h) - E \pm i θ) f ‖}_{L^{2}}^{2} . \end{array}$ We rewrite (4.2) in the form $\begin{matrix} (4.3) & M θ^{1 / 2} ‖ f ‖_{L^{2}} ⩽ \frac{1}{2} {‖ {(| x | + 1)}^{- s} f ‖}_{L^{2}} + 2 M^{2} {‖ {(| x | + 1)}^{s} (P (h) - E \pm i θ) f ‖}_{L^{2}} . \end{matrix}$ We now combine (4.1) and (4.3) to get $\begin{matrix} (4.4) & {‖ {(| x | + 1)}^{- s} f ‖}_{L^{2}} ⩽ 4 M^{2} {‖ {(| x | + 1)}^{s} (P (h) - E \pm i θ) f ‖}_{L^{2}} . \end{matrix}$ It follows from (4.4) that the resolvent estimate $\begin{matrix} (4.5) & {‖ {(| x | + 1)}^{- s} {(P (h) - E \pm i θ)}^{- 1} {(| x | + 1)}^{- s} ‖}_{L^{2} \to L^{2}} ⩽ 4 M^{2} \end{matrix}$ holds for all $0 < h ≪ 1$ and s satisfying (2.1). Finally, observe that if (4.5) holds for s satisfying (2.1), it holds for all $s > 1 / 2$ independent of h. Indeed, given an arbitrary $s^{'} > 1 / 2$ independent of h, we can arrange by taking h small enough that s defined by (2.1) is less than $s^{'}$ . Therefore the bound (4.5) holds with s replaced by $s^{'}$ as desired.

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