2. Schwartz very weak solutions and main result

The aim of this Section is to state the main result of this manuscript. As we already mentioned in the introduction, due to the Schwartz impossibility result about products of distributions, we need to define a suitable concept of solution. The approach that we shall adopt is the so-called very weak solutions (see [Reference Garetto13, Reference Garetto and Ruzhansky14]).

We consider the operator (1.2) under the hypothesis (1.5) on the coefficients. We are interested in the Cauchy problem (1.4), where the initial data satisfy $g \in \mathscr{S}'(\mathbb R^{n})$ and $f \in C([0,T]; \mathscr{S}'(\mathbb R^{n}))$. Once the main problem of the paper is posed, the next step is to define the notion of solution to (1.4) that we are going to deal with. To define it, we shall need some preliminaries.

Let ϕ and ψ be Schwartz functions such that $\phi(0) = 1 = \widehat{\psi}(0)$. Given a positive scale $\omega(\varepsilon)$, $\varepsilon \in (0,1]$, i.e. ω is positive, bounded, $\omega(\varepsilon) \to 0$ as $\varepsilon \to 0^{+}$ and $\omega(\varepsilon) \geq c \varepsilon^{r}$, for some $c, r \gt 0$, we define

\begin{equation*} \phi^{\omega(\varepsilon)}(x) = \phi(\omega(\varepsilon)x) \quad \text{and} \quad \psi_{\omega(\varepsilon)}(x) = \frac{1}{(\omega(\varepsilon))^{n}} \psi \left( \frac{x}{\omega(\varepsilon)} \right). \end{equation*}

Now, given any $u \in \mathscr{S}'(\mathbb R^n)$, the regularization

\begin{equation*} u_{\omega(\varepsilon)}(x) = \phi^{\omega(\varepsilon)}(x) \{\psi_{\omega(\varepsilon)} \ast u \}(x) \end{equation*}

satisfies

(2.1)\begin{equation} u_{\omega(\varepsilon)} \in \mathscr{S}(\mathbb R^n)\,\, \text{and} \,\, u_{\omega(\varepsilon)} \to u \,\, \text{in} \,\, \mathscr{S}'(\mathbb R^n)\,\, \text{as} \,\, \varepsilon \to 0^{+}. \end{equation}

See Appendix A at the end of the paper for more details on this kind of regularizations. The next proposition is standard, and for this reason, we are going to omit its proof.

It is convenient to describe $\mathscr{S}(\mathbb R^n)$ and $\mathscr{S}'(\mathbb R^n)$ in terms of suitable weighted Sobolev spaces. For $m, M \in \mathbb R$ we define

\begin{equation*} H^{m,M}(\mathbb R^n) = \{u \in \mathscr{S}'(\mathbb R^n) : \|u\|_{H^{m,M}(\mathbb R^n)}= \|\langle x \rangle^{M} \langle D \rangle^{m} u \|_{L^2} \lt \infty \}, \end{equation*}

where $\langle D \rangle^m$ stands for the Fourier multiplier given by the symbol $\langle \xi \rangle^{m}$. It is well-known that $H^{m,M}(\mathbb R^n)$ are Hilbert spaces and

\begin{equation*} \mathscr{S}(\mathbb R^n) = \bigcap_{m,M \in \mathbb R} H^{m,M}(\mathbb R^n), \quad \mathscr{S}'(\mathbb R^n) = \bigcup_{m,M \in \mathbb R} H^{m,M}(\mathbb R^n) \end{equation*}

with equivalent limit type topologies. For more details on weighted Sobolev spaces, see Proposition 2.3 and its corollaries in [Reference Parenti22], cf. also [Reference Cordes8].

Proposition 2.1 and (2.1) (cf. Theorem A.7) motivate the following Definition 2.2.

Remark 2.5. Let $\mathcal{M}_{\mathscr{S}}$ be the set of all $\mathscr{S}-$moderate nets and $\mathcal{N}_{\mathscr{S}}$ be the set of all $\mathscr{S}-$negligible nets. Then define the quotient space

\begin{equation*} \mathcal{G}_{\mathscr{S}} := \frac{\mathcal{M}_{\mathscr{S}}}{\mathcal{N}_{\mathscr{S}}}. \end{equation*}

The set $\mathcal{G}_{\mathscr{S}}$ is known as the Colombeau algebra associated to the locally convex space $\mathscr{S}(\mathbb R^n)$ (see definition 3.1 of [Reference Garetto12]). Next, for fixed mollifiers ϕ and ψ such that $\phi(0) = 1 = \widehat{\psi}(0)$, $(\partial^{\beta}\phi)(0) = 0$ for all β ≠ 0 and $\int x^{\alpha}\psi(x) dx = 0$ for all α ≠ 0, the map

\begin{equation*} \varPsi: \mathscr{S}'(\mathbb R^n) \to \mathcal{G}_{\mathscr{S}}, \quad \varPsi (u) = (\phi^{\varepsilon} (\psi_{\varepsilon} \ast u))_{\varepsilon \in (0,1]}, \quad \forall\,\, u \in \mathscr{S}'(\mathbb R^n), \end{equation*}

is a well-defined embedding (due to Theorem A.7 and Proposition 2.1). Then, the regularization $(\phi^{\varepsilon} (\psi_{\varepsilon} \ast u))_{\varepsilon}$ gives a way to include tempered distributions in the Colombeau algebra $\mathcal{G}_{\mathscr{S}}$. Notice that in $\mathcal{G}_{\mathscr{S}}$ we have a well-defined multiplication operation which is consistent with the product of functions in $\mathscr{S}(\mathbb R^n)$.

We are finally ready to introduce the concept of a very weak solution that we are interested in and the main result of this paper, namely Theorem 2.7.

To prove Theorem 2.7, we need to obtain a $\mathscr{S}$-very weak solution for (1.4), that is, we need to solve the regularized problem (1.7) in $\mathscr{S}(\mathbb R^{n})$. In order to do it, we shall employ the classical techniques developed in [Reference Kajitani and Baba17]. In the sequel, we shall fix the considered regularizations for the coefficients of (1.2) and for the data in (1.4). In what follows, we shall use the standard positive scale $\varepsilon$ for the sake of simplicity, and we shall replace it with a suitable positive scale $\omega(\varepsilon)$ just when it is needed.

2.1. Regularization of $c_j(t,x)$, $j=0,1,\ldots,n$

Let $\psi \in \mathscr{S}(\mathbb R^n)$ with $\int \psi = 1$ and $\phi \in \mathscr{S}(\mathbb R^{n})$ with $\phi(0) = 1$. We then define

(2.2)\begin{equation} c_{j,\varepsilon}(t,x) := \phi(\varepsilon x)(\psi_\varepsilon \ast_x c_j)(t,x). \end{equation}

Then we have $c_{j,\varepsilon} \in C([0,T];\mathscr{S}(\mathbb R^{n}))$ and the following estimate holds

(2.3)\begin{equation} |\langle x \rangle^{M} \partial^{\beta}_{x}c_{j,\varepsilon}(t,x)| \leq C_{\beta,M} \varepsilon^{-|\beta|-M-N}, \quad t \in [0,T], x \in \mathbb R^n, \beta \in \mathbb N_0^n, \end{equation}

where N > 0 is a number depending on the coefficients c_j, $j = 0, 1, \ldots, n$, and on the dimension.

2.2. Regularization of the Cauchy data

Let $\mu, \nu \in \mathscr{S}(\mathbb R^{n})$ with $\int \mu = 1$ and $\nu(0) = 1$. Let $f_\epsilon(t,x) = \nu(\varepsilon x) (\mu_{\varepsilon} \ast_x f)(t,x)$ and $g_\epsilon(x) = \nu(\varepsilon x) \mu_{\varepsilon}(x) \ast g(x)$. Then we have $f_{\varepsilon} \in C([0,T]; \mathscr{S}(\mathbb R^{n}))$ and the following estimates

(2.4)\begin{equation} |\langle x \rangle^{M} \partial^{\beta}_{x}f_{\varepsilon}(t,x)|\leq C_{\beta,M} \varepsilon^{-|\beta|-M-\tilde{N}_f},\quad |\langle x \rangle^{M} \partial^{\beta}_{x} g_\epsilon(x)| \leq C_{\beta,M} \varepsilon^{-|\beta|-M-\tilde{N}_g}, \end{equation}

where $\tilde{N}_f \gt 0$ is a number depending on f and on the dimension n, $\tilde{N}_g \gt 0$ is a number depending on g and on the dimension n. By these estimates we immediately get that for all $m,M \in \mathbb N_0$ there exist C > 0, $N_f\in\mathbb N_0$ and $N_g\in\mathbb N_0$ such that

(2.5)\begin{equation} \| f_\varepsilon(t,\cdot) \|_{H^{m,M}}\leq C_{m,M} \varepsilon^{-N_f}, \quad \forall\, t\in[0,T],\ \epsilon\in (0,1] \end{equation}

(2.6)\begin{equation} \| g_\varepsilon \|_{H^{m,M}}\leq C_{m,M} \varepsilon^{-N_g},\quad \forall\, \epsilon\in (0,1]. \end{equation}

We are finally ready to define the family of regularized Cauchy problems that we shall study in the subsequent sections. We consider the family of regularized operators (1.8) for $\varepsilon \in (0, 1]$ and then the family of regularized Cauchy problems

(2.7)\begin{equation} \begin{cases} S_{\varepsilon} v(t,x) = f(t,x), \quad t \in [0,T],\, x \in \mathbb R^{n}, \\ v(0,x) = g(x), \quad \quad \,\, \, x \in \mathbb R^{n}, \end{cases} \end{equation}

where $f \in C([0,T];\mathscr{S}(\mathbb R^{n}))$, $g \in \mathscr{S}(\mathbb R^{n})$ and the coefficients of S_ɛ defined in (1.8) are given in (2.2). In the next sections, we will obtain a net of solutions $(u_{\varepsilon})_{\varepsilon \in (0,1]}$ where for every $\varepsilon$ the function $u_{\varepsilon} \in C([0,T]; \mathscr{S}(\mathbb R^{n}))$ is the unique solution of the Cauchy problem (2.7). Moreover, we will also derive energy estimates for the solutions u_ɛ expliciting how the constants depend on the parameter ɛ.

3. Solving the regularized problem (2.7)

We want to prove that there exists a unique solution in $\mathscr{S}(\mathbb R^{n})$ for the problem (2.7). Since $c_{j,\varepsilon} \in C([0,T];\mathscr{S}(\mathbb R^{n}))$ we know that (2.7) is well-posed in $H^{\infty}(\mathbb R^{n})$ without loss of derivatives, i.e., there exists a unique solution $u \in C([0,T];H^{\infty}(\mathbb R^{n}))$ satisfying (cf. Proposition 5.1 in [Reference Arias Junior, Ascanelli, Cappiello and Garetto3])

\begin{equation*} \|u_{\varepsilon}\|^{2}_{H^m} \leq C_{n,m} \exp\left\{C_{T,m,n} e^{\varepsilon^{-N-\theta_{n,m}}} \right\} \left\{\|g\|^{2}_{H^{m}} + \int_{0}^{t} \|f(\tau)\|^{2}_{H^{m}} d\tau \right\}, \end{equation*}

where N is a natural number depending on the coefficients c_j and $\theta_{n,m}$ is a natural number depending on the dimension $n$ and on the Sobolev index m. We want to conclude that the solution u is also rapidly decreasing at infinity. This motivates the following conjugation: for a fixed $s \in \mathbb N_0$, we define

(3.1)\begin{align} S_{\varepsilon, s} &:= \langle x \rangle^{s} S_{\varepsilon} \langle x \rangle^{-s} \nonumber \\ &= D_t + \sum_{j=1}^{n} D^{2}_{x_j} + \sum_{j=1}^{n} c_{j,\varepsilon}(t,x) D_{x_j} + 2is \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} D_{x_j} + c_{0,\varepsilon, s}(t,x), \end{align}

where

\begin{align*} c_{0,\varepsilon, s}(t,x) &= c_{0,\varepsilon}(t,x) + is \langle x \rangle^{-1} \sum_{j=1}^{n} c_{j,\varepsilon}(t,x) \frac{x_j}{\langle x \rangle} - s(s+2) \langle x \rangle^{-2} \frac{|x|^{2}}{\langle x \rangle^{2}} + ns \langle x \rangle^{-2}. \end{align*}

Observe that in $S_{\varepsilon, s}$ the new purely imaginary first-order term

(3.2)\begin{equation} 2is \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} D_{x_j} \end{equation}

appears. This term decays exactly like $\langle x \rangle^{-1}$ for $|x| \to \infty$. From the classical theory presented in [Reference Kajitani and Baba17] we know that the Cauchy problem for the operator $S_{\varepsilon, s}$ is well-posed in $H^{\infty}(\mathbb R^{n})$ and the decay $\langle x \rangle^{-1}$ on the first-order coefficients produces a finite loss of Sobolev regularity with respect to the initial data. Since (3.2) does not depend on ɛ, the loss of regularity produced by this new term will not depend on ɛ, but will depend on s. This intuitive idea leads to the following result.

Proposition 3.1. Let $\tilde{f} \in C([0,T]; \mathscr{S}(\mathbb R^{n}))$ and $\tilde{g} \in \mathscr{S}(\mathbb R^{n})$. There exists a unique solution $u$ in $C([0,T];H^{\infty}(\mathbb R^{n}))$ to the Cauchy problem

(3.3)\begin{equation} \begin{cases} S_{\varepsilon,s} u(t,x) = \tilde{f}(t,x), \quad t \in [0,T], x \in \mathbb R^{n}, \\ u(0,x) = \tilde{g}(x), \qquad \quad \, x \in \mathbb R^{n}, \end{cases} \end{equation}

and $u$ satisfies: for each $m \in \mathbb N_0$

(3.4)\begin{align} &\| u \|^{2}_{H^{m-c(s)}} \nonumber\\ &\quad \leq C_{m,s,n,T} \exp\left\{C_{m,s,n,T} e^{\varepsilon^{-J(N,s,m,n)}} \right\} \left\{\|\tilde{g}\|^{2}_{H^{m}} + \int_{0}^{t} \|\tilde{f}(\tau)\|^{2}_{H^{m}} d\tau \right\}, \end{align}

where $c(s) = Cs$ for some constant $C \gt 0$ and $J(N, s, m,n)$ is a natural number depending on $N,s,m$ and $n$.

Proposition 3.1 is crucial to prove the main result of the paper. Its proof is the most challenging part of the paper. However, since it is long and technical, we postpone it to Section 5 in order to address the reader as fast as possible to the proof of the main result.

The next theorem is a direct consequence of Proposition 3.1.

Theorem 3.2 For every $\varepsilon \in (0,1]$ consider the regularized Cauchy problem (2.7) with initial data $f \in C([0,T];\mathscr{S}(\mathbb R^{n}))$ and $g \in \mathscr{S}(\mathbb R^{n})$. Then there exists a unique solution $u_{\varepsilon} \in C([0,T]; \mathscr{S}(\mathbb R^{n}))$ for the problem (2.7). Moreover, the solution $u_{\varepsilon}$ satisfies for every $m,s\in\mathbb N_0$:

(3.5)\begin{align} &\|u_{\varepsilon}\|^{2}_{H^{m-c(s),s}} \nonumber\\ &\quad \leq \tilde{C}_{m,s,n,T} \exp\left\{C_{m,s,n,T} e^{\varepsilon^{-J(N,s,m,n)}} \right\} \left\{\|g\|^{2}_{H^{m,s}} + \int_{0}^{t} \|f(\tau)\|^{2}_{H^{m,s}} d\tau \right\}, \end{align}

where

1. (i) The constants $C_{m,s,n,T}$ and $\tilde{C}_{m,s,n,T}$ depend on the coefficients $c_j$, $j=0,1,\ldots,n$, and on the mollifiers $\psi, \phi$;

2. (ii) $N$ is a positive integer depending on the coefficients $c_0, c_1, \ldots, c_n$ and on the dimension;

3. (iii) $J(N,s,m,n)$ is a natural number depending on $N,s,m$ and $n$.

4. From the regularized problem to the original problem

In this section, we shall apply Theorem 3.2 to prove Theorem 2.7.

4.1. Existence

The regularized Cauchy data $f_\epsilon (t), g_\epsilon$ fulfil estimates (2.5), (2.6); moreover, from Theorem 3.2, the regularized problem (2.7) with data $f_\varepsilon(t)$ and g_ɛ has a unique solution $u_\varepsilon\in C([0,T];\mathscr S(\mathbb R^{n}))$ satisfying for every $m, s\in\mathbb N_0$

\begin{align*} &\|u_{\varepsilon}\|^{2}_{H^{m-c(s),s}}\\ &\quad \leq \tilde{C}_{m,s,n,T} \exp\left\{C_{m,s,n,T} e^{\varepsilon^{-J(N,s,m,n)}} \right\} \left\{\|g_\epsilon\|^{2}_{H^{m,s}} + \int_{0}^{t} \|f_\epsilon(\tau)\|^{2}_{H^{m,s}} d\tau \right\}, \\ &\quad \leq A_{m,s} \,\text{exp}(B_{m,s}e^{\varepsilon^{-N_{m,s}}}) \varepsilon^{-(N_f + N_g)}, \quad \varepsilon \in (0,1], t \in [0, T], \end{align*}

where $A_{m,s}, B_{m,s} \gt 0$ are independent from ɛ and $N_{m,s}\in \mathbb N_0$ is independent from ɛ but depends on the coefficients c_j, $j = 0, 1, \ldots, n$, on the dimension n and on Sobolev indices $m,s$. We thus obtain a $\mathscr{S}$-moderate net of solutions for the problem (1.4) by regularizing the coefficients c_j via mollifiers indexed by the positive scale

(4.1)\begin{equation} \omega(\varepsilon) = \{\log(\log(\log(\log(\varepsilon^{-1})))) \}^{-1}, \quad \varepsilon \in (0, 1). \end{equation}

Indeed, by inequality $\log(y) \leq C_{r} y^{\frac{1}{r}}$, for all $y \geq 2$ and all $r \geq 1$, we get

\begin{align*}\omega(\varepsilon)^{-N_{m,s}} &= \{\log(\log(\log(\log(\varepsilon^{-1}))))\}^{N_{m,s}} \leq C_{N_{m,s}}^{N_{m,s}} \log(\log(\log(\varepsilon^{-1})))\\ & = \log\left(\log(\log(\varepsilon^{-1}))^{C^{N_{m,s}}_{N_{m,s}}}\right)\end{align*}

and so we have

\begin{equation*} B_{m,s}e^{\omega(\varepsilon)^{-N_{m,s}}} \leq B_{m,s} \log(\log(\varepsilon^{-1}))^{C^{N_{m,s}}_{N_{m,s}}}\leq \tilde B_{m,s, N_{m,s}}\log(\varepsilon^{-1}) = \log(\varepsilon^{-D_{N_{m,s}}}), \end{equation*}

where $D_{N_{m,s}} \in \mathbb N_0$ is a large number depending on $N_{m,s}$ and on $B_{m,s}$. In this way, we obtain that for any $m,s \in \mathbb N_0$ we find constants $A_{m,s}, D_{N_{m,s}}$ such that

\begin{equation*} \|u_{\varepsilon}(t)\|^{2}_{H^{m-c(s),s}} \leq A_{m,s} \varepsilon^{-D_{N_{m,s}}-(N_f+N_g)}, \quad t \in [0,T], \, \varepsilon \in (0,1). \end{equation*}

Therefore, the Cauchy problem (1.4) admits a $\mathscr S'$-very weak solution.

4.2. Uniqueness

Suppose that we perturb the coefficients of the regularized operator S_ɛ in the following way

\begin{align*} S'_{\varepsilon} := D_t + \sum_{j=1}^{n} D^{2}_{x_j} + \sum_{j=1}^{n}(c_{j,\varepsilon}(t,x)+c'_{j,\varepsilon}(t,x)) D_{x_j} + (c_{0,\varepsilon}(t,x)+c'_{0,\varepsilon}(t,x)) \end{align*}

where $(c'_{j,\varepsilon})_{\varepsilon}$ are $\mathscr{S}$-negligible. Suppose moreover that the net $(u'_{\varepsilon})_\varepsilon$ solves the Cauchy problem

\begin{align*} \begin{cases} S'_{\varepsilon} u(t,x) = f_\varepsilon(t,x) +p_{\varepsilon}(t,x), \quad t \in [0,T],\, x \in \mathbb{R}^n, \\ u(0,x) = g_\varepsilon(x) + q_{\varepsilon}(x) \quad\qquad \,\,\,\,\, x \in \mathbb{R}^n, \end{cases}\end{align*}

where $(p_\varepsilon)_{\varepsilon}$ and $(q_\varepsilon)_{\varepsilon}$ are $\mathscr{S}$-negligible nets. We now want to compare the two very weak solutions u_ɛ and $u'_\varepsilon$. Since

\begin{align*}\begin{cases} S_{\varepsilon} u_\varepsilon(t,x) = f_{\varepsilon}(t,x), \quad t \in [0,T],\, x \in \mathbb{R}^n, \\ u_\varepsilon(0,x) = g_{\varepsilon}(x), \quad \quad \,\,\, x \in \mathbb{R}^n, \end{cases}\end{align*}

and

\begin{align*}\begin{cases} S'_{\varepsilon} u'_\varepsilon(t,x) = f_{\varepsilon}(t,x)+p_{\varepsilon}(t,x), \quad t \in [0,T],\, x \in \mathbb{R}^n, \\ u'_\varepsilon(0,x) = g_{\varepsilon}(x) + q_{\varepsilon}(x), \qquad\quad\,\, x \in \mathbb{R}^n, \end{cases}\end{align*}

it follows that

\begin{align*}\begin{cases} S_\varepsilon(u_\varepsilon-u'_\varepsilon)(t,x) = -p_{\varepsilon}(t,x)-(S_{\varepsilon}-S'_\varepsilon) u'_\varepsilon(t,x), \quad t \in [0,T],\, x \in \mathbb{R}^n, \\ (u_\varepsilon-u'_\varepsilon)(0,x) =-q_{\varepsilon}(x), \qquad \qquad \qquad \qquad \qquad \quad x \in \mathbb{R}^n. \end{cases}\end{align*}

By applying estimate (3.5), using the scale (4.1) to regularize the coefficients c_j, we have that for all $m, s \in \mathbb N_0$ there exist $\tilde{C} \gt 0$ and $\tilde{N}\in\mathbb N_0$ such that

(4.2)\begin{equation} \|u_{\varepsilon}-u'_\varepsilon\|^{2}_{H^{m-c(s),s}} \leq \tilde{C}\varepsilon^{-\tilde{N}} \left\{\|q_{\varepsilon}\|_{H^{m,s}} + \int_{0}^{t} \|p_{\varepsilon}(\tau)+(S_{\varepsilon}-S'_\varepsilon) u'_\varepsilon(\tau)\|^{2}_{H^{m,s}} d\tau\right\}. \end{equation}

Since $(p_\varepsilon)$ and $(q_\varepsilon)$ are both $\mathscr{S}$-negligible, the coefficients of the operator $(S_\varepsilon-S'_\varepsilon)_\varepsilon$ are $\mathscr{S}$-negligible and $(u'_{\varepsilon})_{\varepsilon}$ is $\mathscr{S}$-moderate, we conclude that the right-hand side of (4.2) can be estimated by any positive power of ɛ. Hence, $(u_{\varepsilon}-u'_{\varepsilon})_{\varepsilon}$ is $\mathscr{S}$-negligible.

5. Proof of Proposition 3.1

To solve (3.3), we take inspiration from the approach used in [Reference Kajitani and Baba17], which is based on a suitable change of variable that turns (3.3) into an auxiliary Cauchy problem well-posed in $L^2(\mathbb R^n)$. The change of variable we use is the composition of two pseudodifferential operators. In the next lines, we introduce this change of variable. A survival toolkit of what we need about the theory of pseudodifferential operators is reported in the Appendix B at the end of the paper.

In [Reference Kajitani and Baba17] the authors proved that for every $M_1, M_2 \gt 0, $ there exist real-valued functions $\tilde\lambda_1, \tilde\lambda_2$ satisfying for $x \in \mathbb R^n, |\xi|\geq 1, \alpha, \beta \in \mathbb N_0^n$, β ≠ 0:

\begin{gather*} |\partial^{\alpha}_{\xi}\tilde{\lambda}_{1}(x,\xi)| \leq M_{1}C_{\alpha} |\xi|^{-|\alpha|} \log(2+|\xi|),\\ |\partial^{\alpha}_{\xi}\tilde{\lambda}_{2}(x,\xi)| \leq M_{2} C_{\alpha} |\xi|^{-|\alpha|},\\ |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} \tilde{\lambda}_{\ell}(x,\xi)| \leq M_{\ell} C_{\alpha,\beta} |\xi|^{-|\alpha|}, \quad \ell=1,2, \end{gather*}

and

\begin{equation*} \sum_{j=1}^{n} \xi_j \partial_{x_j} \tilde{\lambda}_{\ell}(x,\xi) \leq -M_{\ell} \langle x \rangle^{-\ell} \chi\left( \frac{\langle x \rangle}{|\xi|} \right) |\xi|, \qquad \ell=1,2, \end{equation*}

where $\chi \in \mathcal{C}^{\infty}_{c}(\mathbb R)$ such that $\chi(t) = 0$ for $|t| \gt 1$, $\chi(t) = 1$ for $|t| \lt \frac{1}{2}$, $t \chi'(t) \leq 0$ and $0 \leq \chi \leq 1$.

For a large parameter $h \geq 1$ to be chosen later on, we consider

(5.1)\begin{equation} \lambda_{\ell}(x,\xi) = \tilde{\lambda}_\ell(x,\xi) (1-\chi)\left( \frac{|\xi| }{h } \right), \qquad \ell=1,2. \end{equation}

Then, since $\langle \xi \rangle_{h} = \sqrt{h^2 + |\xi|^2 } \leq \sqrt{5}h$ on the support of $(1-\chi)(h^{-1}|\xi|)$, we have

\begin{equation*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} \lambda_{\ell}(x,\xi)| \leq \begin{cases} M_{1} C_{\alpha} \log(\langle \xi \rangle_h) \langle \xi \rangle^{-|\alpha|}_{h}, \quad \ell = 1, \beta = 0,\\ M_{2} C_{\alpha} \langle \xi \rangle^{-|\alpha|}_{h}, \quad \ell=2, \beta = 0, \\ M_{\ell} C_{\alpha,\beta} \langle \xi \rangle^{-|\alpha|}_{h}, \quad \ell= 1,2, \beta \neq 0, \end{cases} \end{equation*}

where the constants C_α and $C_{\alpha,\beta}$ do not depend on $M_{\ell}$ and on h. To define the change of variable, we shall use the symbols $e^{\lambda_\ell(x,\xi)}, \ell =1,2.$ The following estimates are followed by a standard application of Faà di Bruno formula:

\begin{gather*} |\partial^{\alpha}_{\xi}e^{\pm \lambda_1(x,\xi)}| \leq C_{\alpha}e^{M_1} (\log \langle \xi \rangle_h)^{|\alpha|} \langle \xi \rangle^{- |\alpha|}_{h} e^{\pm \lambda_1(x,\xi)},\\ |\partial^{\alpha}_{\xi}\partial^{\beta}_{x}e^{\pm \lambda_1(x,\xi)}| \leq C_{\alpha,\beta} e^{M_1} \langle \xi \rangle^{- |\alpha|}_{h} e^{\pm \lambda_1(x,\xi)}. \end{gather*}

On the other hand,

\begin{equation*} e^{\pm \lambda_1}(x,\xi) \leq e^{M_1C_{0} \log \langle \xi \rangle_h} \leq \langle \xi \rangle^{C_0M_1}. \end{equation*}

Hence, it follows that there exists a number $c \gt 0$ such that

\begin{equation*} e^{\pm \lambda_1} \in S^{cM_1}(\mathbb R^{2n}) \end{equation*}

and

\begin{equation*} |\partial^{\alpha}_{\xi}\partial^{\beta}_{x}e^{\pm \lambda_1(x,\xi)}| \leq C_{\alpha,\beta} e^{M_1} \langle \xi \rangle^{cM_1 - |\alpha|}_{h}. \end{equation*}

Concerning $e^{\lambda_2(x,\xi)}$ we have that it belongs to $S^0(\mathbb R^{2n}).$

We shall use the operators $e^{\lambda_\ell}(x,D)$ to handle the first-order terms of the operator $S_{\varepsilon,s}$ defined in (3.1). These terms are given by

(5.2)\begin{equation} \sum_{j=1}^{n} c_{j,\varepsilon}(t,x) D_{x_j} \end{equation}

and

(5.3)\begin{equation} 2is \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} D_{x_j}. \end{equation}

Since $c_{j,\varepsilon}(t,x)$ decays faster than $\langle x \rangle^{-1}$, we have that (5.2) does not produce any loss of Sobolev regularity. On the other hand, the term (5.3) decays exactly as $\langle x \rangle^{-1}$, so it produces a loss of regularity, but this loss will be independent on the parameter ɛ, since (5.3) does not depend on ɛ. Our idea is then to use $e^{\lambda_{2}}(x,D)$ to handle the term (5.2) and $e^{\lambda_{1}}(x,D)$ to control the new first-order term (5.3) produced by the conjugation by $\langle x \rangle^{s}$. Namely, we shall consider a change of variable of the form

(5.4)\begin{equation} \begin{cases} e^{\lambda_2} \circ e^{\lambda_1} \circ S_{\varepsilon,s} \circ \{e^{\lambda_1}\}^{-1} \circ \{e^{\lambda_2}\}^{-1} \circ e^{\lambda_2} \circ e^{\lambda_1} u(t,x) = e^{\lambda_2} \circ e^{\lambda_1}\tilde{f}(t,x), \\ e^{\lambda_{2}} \circ e^{\lambda_1} u(0,x) = e^{\lambda_2} \circ e^{\lambda_1} \tilde{g}(x). \end{cases} \end{equation}

In the remaining part of this section, we shall derive a priori energy estimates to the auxiliary problem (5.4), see Proposition 5.4. Then the proof of Proposition 3.1 will be a consequence of the derived energy estimates and the mapping properties of $e^{\lambda_1}(x,D)$ and $e^{\lambda_2}(x,D)$.

5.1. Invertibility of $e^{\lambda_1}(x,D)$ and conjugation theorem

The reverse operator $^{R}\{e^{\lambda_1}(x,D)\}$ was introduced in [Reference Kajitani and Wakabayashi18] (see Proposition 2.13) as the transpose operator of $e^{\lambda_1}(x,-D)$. Since λ ₁ is real-valued, in this particular case, the reverse operator coincides with the L ² adjoint of $e^{\lambda_1}(x,D)$. We have that

\begin{equation*} e^{\lambda_1}(x,D) \circ\, ^{R}\{e^{-\lambda_1}(x,D)\} = \mu(x,D), \end{equation*}

where

\begin{equation*} \mu(x,\xi) = Os- \iint e^{-iy\eta} e^{\lambda_1(x, \xi+\eta) - \lambda_1(x+y,\xi+\eta)} dyd{\kern-1pt\bar{}\kern1.5pt } \eta. \end{equation*}

So, Taylor’s formula and usual computations give

(5.5)\begin{equation} \mu(x,\xi) = \sum_{|\alpha| \lt N} \frac{1}{\alpha!} \partial^{\alpha}_{\xi}\{e^{\lambda_1(x,\xi)} D^{\alpha}_{x} e^{-\lambda_1(x,\xi)}\} + r_{N}(x,\xi), \end{equation}

where

\begin{align*} r_N(x,\xi) &= N \sum_{|\gamma| = N} \int_0^1 (1-\theta)^{N-1} Os\\ &\quad - \iint e^{-iy\eta} \partial^{\alpha}_{\xi} \{e^{\lambda_1(x, \xi+\theta\eta)} D^{\gamma}_{x}e^{-\lambda_1(x+y,\xi+\theta\eta)} \} dyd{\kern-1pt\bar{}\kern1.5pt } \eta \, d\theta. \end{align*}

The advantage of using the reverse operator is that in the above asymptotic formula, the terms

\begin{equation*} \partial^{\alpha}_{\xi}\{e^{\lambda_1(x,\xi)} D^\alpha_x e^{-\lambda_1(x,\xi)}\} \end{equation*}

appear, and they are of order exactly $-|\alpha|$ (i.e. the x-derivatives in the above asymptotic formula allow us to avoid the log growth appearing in the symbol estimates of λ ₁). The symbols $e^{\pm \lambda_1}$ are of finite order, so we can estimate r_N in the following way

\begin{equation*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} r_{N}(x,\xi)| \leq C_{\alpha, \beta, N, M_1,n} \langle \xi \rangle^{2cM_1 - N}_h. \end{equation*}

Since (5.5) holds for any $N \in \mathbb N$, it follows that

\begin{align*} e^{\lambda_1}(x,D) \circ\, ^{R}\{e^{-\lambda_1}(x,D)\} &= I - \textrm{op}\left(i\sum_{j=1}^{n} \partial_{\xi_j} \{\partial_{x_j} \lambda_1(x,\xi)\} \right) + r_{-2}(x,D)\\ & = I + r_{-1}(x,D), \end{align*}

where r ₋₂ has order −2. Hence, if we assume that $h \geq h_0(M_1)$ we obtain that

(5.6)\begin{align} \{e^{\lambda_1}(x,D) \}^{-1} &= ^{R}\{e^{-\lambda_1}(x,D)\} \circ \sum_{j \geq 0} (-r(x,D))^j \nonumber\\ &= ^{R}\{e^{-\lambda_1}(x,D)\} \circ \textrm{op}\left( 1 - i \sum_{j=1}^{n} \partial_{\xi_j}\partial_{x_j}\lambda_1(x,\xi) + r_{-2}(x,\xi) \right), \end{align}

where $\sum_{j \geq 0} (-r)^j$ has symbol of order zero and $r_{-2}$ has symbol of order −2 satisfying

\begin{equation*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} r_{-2}(x,\xi)| \leq C_{\alpha, \beta,M_1, n} \langle \xi \rangle^{-2-|\alpha|}_{h}, \end{equation*}

for some $C \gt 0$ independent on M ₁ and h (cf. theorem $I.1$ at page 372 of [Reference Kumano-Go20]).

For a $p \in S^{m}(\mathbb R^{2n})$ such that

\begin{equation*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} p(x,\xi)| \leq C_{\alpha, \beta} \langle \xi \rangle^{m-|\alpha|}_{h}, \end{equation*}

we consider

\begin{equation*} |p|^{(m)}_{\ell} = \max_{|\alpha|, |\beta| \leq \ell} \sup_{x,\xi \in \mathbb R^n} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} p(x,\xi)| \langle \xi \rangle^{-m+|\alpha|}_{h}. \end{equation*}

Thus

\begin{equation*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} p(x,\xi)| \leq |p|^{(m)}_{|\alpha+\beta|} \langle \xi \rangle^{m-|\alpha|}_{h}. \end{equation*}

Now we need to understand the composition $ e^{\lambda_1}(x,D) \circ p(x,D) \circ\, \{e^{\lambda_1}(x,D)\}^{-1} . $ This is done in two steps in view of (5.6). The next theorem is a direct consequence of Theorem B.3.

Theorem 5.2 For any $N \in \mathbb N$ the symbol of $e^{\lambda_1}(x,D)\circ p(x,D) \circ\, ^{R}\{e^{-\lambda_1}(x,D)\}$ is given by

(5.7)\begin{equation} p(x,\xi)+\sum_{1\leq |\alpha+\beta| \lt N} \frac{1}{\alpha!\beta!} \partial^{\beta}_{\xi} \{\partial^{\alpha}_{\xi}e^{\lambda_1(x,\xi)} D^{\alpha}_{x}p(x,\xi) D^{\beta}_{x}e^{-\lambda_1(x,\xi)} \} + r_{N}(x,\xi), \end{equation}

where

\begin{equation*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} r_{N}(x,\xi)| \leq C_{\alpha, \beta, N, M_1, n} |p|^{(m)}_{|\alpha+\beta|+N+J} \langle \xi \rangle_h^{2cM_1 + m - N}, \end{equation*}

$J$ being a natural number depending on $M_1, m$ and on the dimension $n$.

Theorem 5.2 implies the following: if we know that p_ɛ satisfies

\begin{equation*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} p_\varepsilon(x,\xi)| \leq C_{\alpha, \beta} \varepsilon^{-N_1 - |\beta|} \langle \xi \rangle^{1-|\alpha|}_{h}, \end{equation*}

then taking $N(M_1) = \lfloor 2cM_1 + 1 \rfloor + 1$ we get that r_N is a symbol of order zero and satisfies

(5.8)\begin{equation} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} r_{N}(x,\xi)| \leq C_{\alpha, \beta, M_1, n} \varepsilon^{-N_1 - |\beta| - J(M_1,n)} \langle \xi \rangle_h^{-|\alpha|}, \end{equation}

where $J$ is a natural number depending on $M_1$ and $n$. Thus, (5.6) becomes

\begin{equation*} p_\varepsilon(x,\xi) + \underbrace{\sum_{1\leq |\alpha+\beta| \lt N} \frac{1}{\alpha!\beta!} \partial^{\beta}_{\xi} \{\partial^{\alpha}_{\xi}e^{\lambda_1(x,\xi)} D^{\alpha}_{x}p_{\varepsilon}(x,\xi) D^{\beta}_{x}e^{-\lambda_1(x,\xi)} \}}_{q_{\varepsilon}(x,\xi)} + r_{0}(x,\xi), \end{equation*}

where r ₀ satisfies (5.7). Now note that we can write q_ɛ in the following way:

\begin{align*} q_{\varepsilon} &= \sum_{j=1}^{n} \partial_{\xi_j} \lambda_1 D_{x_j} p_{\varepsilon} -\sum_{j=1}^{n} \partial_{\xi_j}\{p_\varepsilon D_{x_j}\lambda_1\} \\ &\quad + \sum_{2 \leq |\alpha+\beta| \lt N} \frac{1}{\alpha!\beta!} \partial^{\beta}_{\xi} \{\partial^{\alpha}_{\xi}e^{\lambda_1(x,\xi)} D^{\alpha}_{x}p_{\varepsilon}(x,\xi) D^{\beta}_{x}e^{-\lambda_1(x,\xi)} \} \\ &= \sum_{j=1}^{n} \partial_{\xi_j} \lambda_1 D_{x_j} p_{\varepsilon} + r_0, \end{align*}

for a new zero-order term r ₀ satisfying an estimate like (5.7). Hence

(5.9)\begin{equation} e^{\lambda_1}(x,D) \circ p_\varepsilon(x,D) \circ\, ^{R} \{e^{-\lambda_1}(x,D)\} = p_\varepsilon(x,D) + \textrm{op}\left(\sum_{j=1}^{n} \partial_{\xi_j} \lambda_1 D_{x_j} p_{\varepsilon}\right) + r_{0}(x,D). \end{equation}

Since $ \sum\limits_{j=1}^{n} \partial_{\xi_j} \lambda_1 D_{x_j} p_{\varepsilon} $ behaves like $\varepsilon^{-N_1-1} \log\langle \xi \rangle_{h}$, it cannot be regarded as a zero-order term.

5.2. Invertibility of $e^{\lambda_2}(x,D)$

Here, we just report results derived in [Reference Arias Junior, Ascanelli, Cappiello and Garetto3] at Section 6.1.

Lemma 5.3. For all $h \geq h_0(M_2,n) := A e^{(2C+1)M_2}$, for some $A, C \gt 0$ independent on M ₂, the operator $e^{\lambda_2}(x,D)$ is invertible and we have

\begin{equation*} \left(e^{\lambda_2}(x,D)\right)^{-1} = e^{-\lambda_2}(x,D) \circ \sum_{j \geq 0} (-r(x,D))^{j}, \end{equation*}

where

\begin{equation*} r(x,\xi) = i\sum_{k=1}^{n} \partial_{\xi_k} \lambda_2(x,\xi)\partial_{x_k}\lambda_2(x,\xi) + r_{-2}(x,\xi). \end{equation*}

and

(5.10)\begin{align} r_{-2}(x,\xi) &= \sum_{|\gamma|=2} \frac{2}{\gamma!} Os \nonumber\\ &\quad - \iint e^{-y\eta} \int_{0}^{1} (1-\theta) \partial^{\gamma}_{\xi} e^{\lambda_2(x,\xi+\theta\eta)} d\theta D^{\gamma}_{x} e^{-\lambda_2(x+y,\xi)} dyd{\kern-1pt\bar{}\kern1.5pt } \eta. \end{align}

Moreover,

\begin{equation*} \sum_{j \geq 0} (-r(x,D))^{j} = s(x,D), \end{equation*}

where $s(x,\xi)$ is a zero-order symbol and its seminorms do not depend on $h$ and on $M_2$.

We need to write $s(x,D)$ in the following convenient way:

\begin{align*} s(x, D) &= I - r(x,D) + \sum_{j \geq 2} (-r(x,D))^{j} = I - r(x,D) + (r(x,D))^{2} s(x,D) \\ &= I - \textrm{op}\left( i\sum_{k=1}^{n} \partial_{\xi_k} \lambda_2(x,\xi)\partial_{x_k}\lambda_2(x,\xi) \right) - r_{-2}(x,D) + (r(x,D))^{2} s(x,D). \end{align*}

So,

(5.11)\begin{equation} s(x,\xi) = 1 - i\sum_{k=1}^{n} \partial_{\xi_k} \lambda_2(x,\xi)\partial_{x_k}\lambda_2(x,\xi) + s_{-2}(x,\xi), \end{equation}

where $s_{-2}(x,\xi)$ has order −2 and satisfies

\begin{equation*} |\partial^{\alpha}_{\xi}\partial^{\beta}_{x} s_{-2}(x,\xi)| \leq C_{\alpha,\beta,n} e^{CM_2} \langle \xi \rangle^{-2-|\alpha|}_{h}, \end{equation*}

where the constants $C_{\alpha,\beta,n}$ and $C$ do not depend on $M_2$ and on $h$.

5.3. Conjugation of $S_{\varepsilon,s}$ by $e^{\lambda_1}(x,D)$

We recall that

\begin{align*} S_{\varepsilon, s} &:= \langle x \rangle^{s} S_{\varepsilon} \langle x \rangle^{-s} \\ &= D_t + \sum_{j=1}^{n} D^{2}_{x_j} + \sum_{j=1}^{n} c_{j,\varepsilon}(t,x) D_{x_j} + 2is \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} D_{x_j} + c_{0,\varepsilon, s}(t,x), \end{align*}

where

\begin{align*} c_{0,\varepsilon, s}(t,x) = c_{0,\varepsilon}(t,x) + is \langle x \rangle^{-1} \sum_{j=1}^{n} c_{1,j}(t,x) \frac{x_j}{\langle x \rangle} - s(s+2) \langle x \rangle^{-2} \frac{|x|^{2}}{\langle x \rangle^{2}} + ns \langle x \rangle^{-2}. \end{align*}

Let us set

\begin{equation*} S_{\varepsilon, s, \lambda_1} = e^{\lambda_1} \circ S_{\varepsilon, s} \circ \{e^{\lambda_1}\}^{-1}. \end{equation*}

Applying Theorem 5.2 to $\sum\limits_{j=1}^{n} D^{2}_{x_j}$ and formula (5.8) to the first-order terms of $S_{\varepsilon, s}$ we get

(5.12)\begin{align} S_{\varepsilon,s,\lambda_1} &= D_t + \sum_{j=1}^{n} D^{2}_{x_j} +2i\sum_{j=1}^{n} (\partial_{x_j}\lambda_1)(x,D) D_{x_j}\nonumber \\ &\quad +\sum_{j=1}^{n} c_{j,\varepsilon}(t,x)D_{x_j}+ 2is \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} D_{x_j} \nonumber \\ &\quad + \textrm{op}\left( \sum_{j=1}^{n} \partial_{\xi_j} \lambda_1 D_{x_j} p_{\varepsilon}\right) + d_{0,\varepsilon,s}(t,x,D), \end{align}

where

(5.13)\begin{equation} p_{\varepsilon}(t,x,\xi) = \sum_{j=1}^{n} c_{j,\varepsilon}(t,x)\xi_j + 2is \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} \xi_j, \end{equation}

and $d_{0,\varepsilon, s}$ is a zero-order term satisfying

(5.14)\begin{equation} |\partial^{\alpha}_{\xi}\partial^{\beta}_{x} d_{0,\varepsilon,s}(t,x,\xi)| \leq C_{\alpha,\beta,M_1,n,s} \varepsilon^{-J(N,M_1,n)-|\beta|} \langle \xi \rangle^{-|\alpha|}_{h}. \end{equation}

In order to simplify the expression for $S_{\varepsilon,s,\lambda_1}$ we set

\begin{equation*} p_{\varepsilon, \log} (t,x,\xi) = \sum_{j=1}^{n} \partial_{\xi_j} \lambda_1(x,\xi) D_{x_j} p_{\varepsilon}(t,x,\xi). \end{equation*}

We note that

\begin{equation*} |\partial^{\alpha}_{\xi}\partial^{\beta}_{x} p_{\varepsilon, \log} (t,x,\xi)| \leq M_1C_{\alpha,\beta,s,n} \varepsilon^{-N-|\beta|-1} \log \langle \xi \rangle_h \langle \xi \rangle^{-|\alpha|}_h, \end{equation*}

and we may write

(5.15)\begin{align} S_{\varepsilon,s,\lambda_1} &= D_t + \sum_{j=1}^{n} D^{2}_{x_j} + \sum_{j=1}^{n} c_{j,\varepsilon}(t,x)D_{x_j}\nonumber \\ &\quad +2is \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} D_{x_j} + 2i\sum_{j=1}^{n} (\partial_{x_j}\lambda_1)(x,D) D_{x_j} \nonumber \\ &\quad + p_{\varepsilon,\log}(t,x,D) + d_{0,\varepsilon,s}(t,x,D). \end{align}

5.4. Conjugation of $S_{\varepsilon, s, \lambda_1}$ by $e^{\lambda_2}(x,D)$

We shall denote

\begin{equation*} S_{\varepsilon, s, \lambda_1, \lambda_2} = e^{\lambda_2}(x,D) \circ S_{\varepsilon, s,\lambda_1} \circ \{e^{\lambda_2}(x,D)\}^{-1}. \end{equation*}

We recall that $e^{\lambda_2}$ has order zero. So, using Lemma 5.3, Theorem B.3, and repeating similar ideas of the previous subsection, we end up with

(5.16)\begin{align} S_{\varepsilon,s,\lambda_1, \lambda_2} &= D_t + \sum_{j=1}^{n} D^{2}_{x_j} + \sum_{j=1}^{n} c_{j,\varepsilon}(t,x)D_{x_j} + 2i\sum_{j=1}^{n} (\partial_{x_j}\lambda_2)(x,D) D_{x_j}\nonumber \\ &\quad + 2is \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} D_{x_j} + 2i\sum_{j=1}^{n} (\partial_{x_j}\lambda_1)(x,D) D_{x_j} \nonumber \\ &\quad + p_{\varepsilon,\log}(t,x,D) + d_{0,\varepsilon,s}(t,x,D), \end{align}

for a new zero-order term $d_{0,\varepsilon,s}(t,x,\xi)$ satisfying

(5.17)\begin{equation} |\partial^{\alpha}_{\xi}\partial^{\beta}_{x} d_{0,\varepsilon,s}(t,x,\xi)| \leq C_{\alpha,\beta,M_1,n,s} \varepsilon^{-J(N,M_1,n) - |\beta|} e^{CM_2}\langle \xi \rangle^{-|\alpha|}_{h}, \end{equation}

where $J(N, M_1, n)$ is a natural number depending on $N, M_1$ and on the dimension and the constants $C_{\alpha,\beta,M_1,n,s}$, $C$ do not depend on $\varepsilon, M_2$ and $h$.

5.5. Choices of M ₁, M ₂, h and L ² energy estimate

Inserting (5.12) in $p_{\varepsilon,\log}(t,x,\xi)$ we obtain

\begin{align*} p_{\varepsilon, \log}(t,x,\xi) &= \sum_{j=1}^{n} \sum_{\ell = 1}^{n} \partial_{\xi_j} \lambda_1(x,\xi) D_{x_j} c_{\ell,\varepsilon}(t,x) \xi_\ell + 2i \sum_{j=1}^{n} \sum_{\ell = 1}^{n} \partial_{\xi_j} \lambda_1(x,\xi) D_{x_{j}} \frac{x_\ell}{\langle x \rangle^{2}} \xi_\ell \nonumber \\ &= q_{1,\varepsilon}(t,x,\xi) + q_{2,\varepsilon}(t,x,\xi). \end{align*}

Since $c_{j,\varepsilon}(t,x)$ are Schwartz functions in x we obtain

\begin{align*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} q_{1,\varepsilon}(t,x,\xi)| &\leq M_1C_{\alpha, \beta, s, n} \varepsilon^{-N - |\beta| - 3} \log\langle \xi \rangle_h \langle \xi \rangle^{-|\alpha|}_{h} \langle x \rangle^{-2} \nonumber \\ &= M_1C_{\alpha, \beta, s, n } \varepsilon^{-N - |\beta| - 3} \langle \xi \rangle^{-1}_{h}\log\langle \xi \rangle_h \langle \xi \rangle^{1-|\alpha|}_{h}\langle x \rangle^{-2} \nonumber \\ &\leq M_1C_{\alpha, \beta, s, n} \varepsilon^{-N - |\beta| - 3} h^{-\frac{1}{2}} \langle \xi \rangle^{1-|\alpha|}_{h} \langle x \rangle^{-2}. \end{align*}

Analogously

\begin{align*} |\partial^{\alpha}_{\xi} \partial^{\beta}_{x} q_{2,\varepsilon}(t,x,\xi)| \leq M_1C_{\alpha, \beta, s, n} h^{-\frac{1}{2}} \langle \xi \rangle^{1-|\alpha|}_{h} \langle x \rangle^{-2}. \end{align*}

So, we can consider $q_{1,\varepsilon}$ and $q_{2,\varepsilon}$ as symbols of order 1 with decay $\langle x \rangle^{-2}$ and seminorms bounded by $M_1C_{\alpha, \beta, s, n}\varepsilon^{-N - |\beta| - 3}h^{-\frac{1}{2}}$.

In order to derive an L ² energy estimate for $S_{\varepsilon,\lambda}$ we write

(5.18)\begin{align} iS_{\varepsilon,s,\lambda_1, \lambda_2} &= \partial_t + i\sum_{j=1}^{n} D^{2}_{x_j}\nonumber \\ &\quad + \sum_{j=1}^{n} ic_{j,\varepsilon}(t,x)D_{x_j} - 2\sum_{j=1}^{n} (\partial_{x_j}\lambda_2)(x,D) D_{x_j} \nonumber \\ &\quad -2s \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} D_{x_j} - 2\sum_{j=1}^{n} (\partial_{x_j}\lambda_1)(x,D) D_{x_j} \nonumber \\ &\quad +iq_{1,\varepsilon}(t,x,D) + iq_{2,\varepsilon}(t,x,D) + id_{0,\varepsilon,s}(t,x,D). \end{align}

We also recall that

\begin{align*} -2 \sum_{j=1}^{n} \xi_j \partial_{x_j} \lambda_{\ell}(x,\xi) \geq M_{\ell} \langle x \rangle^{-\ell} \chi\left( \frac{\langle x \rangle}{|\xi|} \right) (1-\chi)\left( \frac{|\xi| }{h } \right) |\xi|, \quad \ell = 1, 2. \end{align*}

Note that if a symbol $c(x,\xi)$ of order 1 decays like $\langle x \rangle^{-2}$, then we can split it as follows

\begin{align*} c(x,\xi) &= c(x,\xi)\chi\left( \frac{\langle x \rangle}{|\xi|} \right) (1-\chi)\left( \frac{|\xi| }{h } \right) \nonumber \\ &\quad + \underbrace{c(x,\xi) (1-\chi)\left( \frac{\langle x \rangle}{|\xi|} \right) (\chi + (1-\chi))\left( \frac{|\xi| }{h } \right) + c(x,\xi) \chi\left( \frac{\langle x \rangle}{|\xi|} \right) \chi\left( \frac{|\xi| }{h } \right)}_{\text{order zero}}. \end{align*}

So, outside of the support of $\chi\left( \frac{\langle x \rangle}{|\xi|} \right) (1-\chi)\left( \frac{|\xi| }{h } \right)$ we have that

\begin{equation*} iS_{\varepsilon,s,\lambda_1, \lambda_2} = \partial_t + i\sum_{j=1}^{n} D^{2}_{x_j} + \tilde{q}_{0}(t,x,D), \end{equation*}

where

\begin{equation*} |\partial^{\alpha}_{\xi}\partial^{\beta}_{x} \tilde{q}_{0}(t,x,\xi)| \leq C_{\alpha,\beta,M_1,n,s} \varepsilon^{-J(N,M_1,n) - |\beta|} e^{CM_2}\langle \xi \rangle^{-|\alpha|}_{h}. \end{equation*}

On the other hand, on the support of $\chi\left( \frac{\langle x \rangle}{|\xi|} \right) (1-\chi)\left( \frac{|\xi| }{h } \right)$ we can take $M_1(s)$ and $M_2(\varepsilon)$ large in order to get that the symbols of the first-order part are non-negative. Indeed, we have

\begin{align*} -\chi\left( \frac{\langle x \rangle}{|\xi|} \right) &(1-\chi)\left( \frac{|\xi| }{h } \right)\sum_{j=1}^{n} \mathop{\rm Im}\nolimits\,c_{j,\varepsilon}(t,x)\xi_j - 2\sum_{j=1}^{n} (\partial_{x_j}\lambda_2)(x,\xi) \xi_j \nonumber \\ &\geq \left( M_2 - C(c_j)\varepsilon^{-N-2} \right) |\xi| \langle x \rangle^{-2} \chi\left( \frac{\langle x \rangle}{|\xi|} \right) (1-\chi)\left( \frac{|\xi| }{h } \right) \end{align*}

and

\begin{align*} -\chi\left( \frac{\langle x \rangle}{|\xi|} \right) &(1-\chi)\left( \frac{|\xi| }{h } \right)2s \langle x \rangle^{-1} \sum_{j=1}^{n}\frac{x_j}{\langle x \rangle} \xi_j - 2\sum_{j=1}^{n} (\partial_{x_j}\lambda_1)(x,\xi) \xi_j \nonumber \\ &+ \chi\left( \frac{\langle x \rangle}{|\xi|} \right) (1-\chi)\left( \frac{|\xi| }{h } \right) (-\mathop{\rm Im}\nolimits\, q_{1,\varepsilon}(t,x,\xi) - \mathop{\rm Im}\nolimits\,q_{2,\varepsilon}(t,x,D)) \nonumber \\ &\geq \left( M_1 - C(s) - h^{-\frac{1}{2}}C_{s,n,M_1} \varepsilon^{-N-3} \right) |\xi| \langle x \rangle^{-1} \chi\left( \frac{\langle x \rangle}{|\xi|} \right) (1-\chi)\left( \frac{|\xi| }{h } \right). \end{align*}

Hence, choosing

\begin{equation*} M_2(\varepsilon) = \frac{C(c_j)\varepsilon^{-N-2}}{2}, \quad M_1(s) = \frac{C(s)}{2} \end{equation*}

and taking h large so that

\begin{equation*} h^{-\frac{1}{2}} C_{s,n,M_1}\varepsilon^{-N-3} \lt \frac{C(s)}{2}, \end{equation*}

we obtain that the real part of the first-order symbols in (5.18) are all non-negative.

Hence, we may apply the sharp Gårding inequality to get the following energy estimate:

(5.19)\begin{align} \partial_{t} \|u(t)\|^{2}_{L^2} &= 2 Re\, \langle \partial_t u, u \rangle_{L^2}\nonumber \\ &\leq \|S_{\varepsilon,s,\lambda_1,\lambda_2} u\|^{2}_{L^2} + \|u\|^{2}_{L^2} + C_{s,n,M_1} \varepsilon^{-J(N, M_1, s)} e^{CM_2} \|u\|^{2}_{L^2}. \end{align}

To obtain (5.19) with general $m-$norms we apply the same ideas to the operator

\begin{equation*} \langle D_x \rangle^{m} S_{\varepsilon,s,\lambda_1,\lambda_2} \langle D_x \rangle^{-m}. \end{equation*}

which is almost equal to $S_{\varepsilon,s,\lambda_1,\lambda_2}$ except for a zero-order term. So,

\begin{align*} \partial_{t} \|u(t)\|^{2}_{H^m} &\leq \|S_{\varepsilon,s,\lambda_1,\lambda_2} u\|^{2}_{H^m} + \|u\|^{2}_{H^m} + C_{m,s,n,M_1} \varepsilon^{-J(N,M_1,s,m)} e^{CM_2} \|u\|^{2}_{H^m}. \end{align*}

Gronwall’s inequality then gives (recalling that $M_1$ depends only on $s$ and $M_2 = C\varepsilon^{-N-2}$)

(5.20)\begin{align} \|u(t)\|^{2}_{H^m} &\leq C_{m,s,T,n}\exp \left\{C_{T,m,n} e^{\varepsilon^{-J(N,s,m)}} \right\} \nonumber\\ &\quad \times\left( \|u(0)\|^{2}_{H^{m}} + \int_{0}^{t}\|S_{\varepsilon,s,\lambda_1,\lambda_2}(\tau) u(\tau)\|^{2}_{H^m} d\tau \right). \end{align}

The next proposition is a consequence of (5.20).

Proposition 5.4. Let $f \in C([0,T]; H^m(\mathbb R^{n}))$ and $g \in H^{m}(\mathbb R^{n})$. There exists a unique solution $u$ in $C([0,T];H^{m}(\mathbb R^{n}))$ to the Cauchy problem

(5.21)\begin{equation} \begin{cases} S_{\varepsilon,s,\lambda_1, \lambda_2} u(t,x) = f(t,x), \quad t \in [0,T], x \in \mathbb R^{n},\\ u(0,x) = g(x), \qquad \quad \qquad x \in \mathbb R^{n}, \end{cases} \end{equation}

and $u$ satisfies (5.20).

Proof of Proposition 3.1

We observe that $e^{\lambda_2}(x,D): H^{m} \to H^{m}$ and $e^{\lambda_1}(x,D): H^m \to H^{m-\frac{c(s)}{2}}$, for some c(s) depending on s. Hence $e^{\lambda_2}(x,D)$ does not produce any loss of regularity and $e^{\lambda_1}(x,D)$ produces a loss depending on s. Then, Proposition 3.1 is a consequence of Proposition 5.4 and the mapping properties of $e^{\lambda_1}(x,D)$ and $e^{\lambda_2}(x,D)$.

6. Consistency with the classical theory in the case of regular coefficients

Suppose that the coefficients of the operator $S$ are Schwartz functions, more precisely:

\begin{equation*} c_j \in C([0,T];\mathscr{S}(\mathbb R^{n})), \quad j = 0, 1, \ldots, n. \end{equation*}

In this situation the Cauchy problem (1.4) with data $f \in C([0,T];\mathscr{S}(\mathbb R^{n}))$ and $g \in \mathscr{S}(\mathbb R^{n})$ admits a unique solution $u \in C([0,T]; \mathscr{S}(\mathbb R^{n}))$ satisfying an energy estimate like (3.5) (with a constant which does not depend on $\varepsilon$). The goal of this section is to verify that the net obtained in Theorem 3.2 converges to u in the $\mathscr{S}(\mathbb R^{n})$ topology.

Let u_ɛ be the solution of the Cauchy problem (2.7). Then $u - u_{\varepsilon}$ solves the following Cauchy problem

(6.1)\begin{equation} \begin{cases} S\{u-u_{\varepsilon}\}(t,x) = (f-f_{\varepsilon}) + Q_{\varepsilon}u_{\varepsilon}(t,x), \quad t \in [0,T],\, x \in \mathbb R^{n},\\ \{u-u_{\varepsilon}\}(0,x) = (g-g_{\varepsilon}), \qquad \qquad \qquad \quad x \in \mathbb R^{n}, \end{cases} \end{equation}

where

\begin{equation*} Q_{\varepsilon} = \sum_{j=1}^{n} (c_{j,\varepsilon}(t,x) - c_{j}(t,x))D_{x_j} + (c_{0,\varepsilon}(t,x)-c_0(t,x)). \end{equation*}

So, we have the following estimate: for any $m, s \in \mathbb N_0$

\begin{align*} &\|\{u-u_{\varepsilon}\}(t)\|^{2}_{H^{m-c(s), s}} \\ &\quad \leq C_{m,s} \left\{\|g-g_{\varepsilon}\|^{2}_{H^{m,s}} + \int_{0}^{t} \|(f-f_\varepsilon)(\tau) + Q_{\varepsilon} u_{\varepsilon}(\tau)\|^{2}_{H^{m,s}} d\tau \right\} \\ &\quad \leq C_{m,s}\|g-g_{\varepsilon}\|^{2}_{H^{m,s}} + C_m \int_{0}^{t} \|(f-f_\varepsilon)(\tau)\|^{2}_{H^{m,s}} d\tau \\ &\qquad + C_{m,s} \int_{0}^{t} \sum_{j=1}^{n} \max_{|\alpha| \leq m} \sup_{x \in \mathbb R^{n}}|D^{\alpha}_{x}c_{j,\varepsilon}(t,x) - D^{\alpha}_{x}c_{j}(t,x)|^{2} \|u_{\varepsilon}(\tau)\|^{2}_{H^{m+1,s}} d\tau \\ &\qquad + C_{m,s} \int^{t}_{0} \max_{|\alpha| \leq m}\sup_{x\in\mathbb R^n}|D^{\alpha}_{x}c_{0,\varepsilon}(t,x)-D^{\alpha}_{x}c_0(t,x)|^{2} \|u_{\varepsilon}(\tau)\|^{2}_{H^{m,s}} d\tau. \end{align*}

Now we observe the following:

1. (i) Since, in this case, we are regularizing regular functions, it is easy to conclude that the regularizations $c_{j,\varepsilon}$ satisfy uniform estimates with respect to $\varepsilon$. So, in this particular case, we obtain estimate (3.5) uniformly with respect to $\varepsilon$. Hence, for any $m,s \in \mathbb N_0$,

   \begin{equation*} \|u_{\varepsilon}(\tau)\|^{2}_{H^{m,s}} \leq C_{m,s}(f,g), \quad \forall \varepsilon \in (0,\varepsilon_0). \end{equation*}

2. (ii) We have the following convergence: for any $\beta \in \mathbb N^n_0$ and $j = 0, 1, \ldots, n$ it holds

   \begin{equation*} \sup_{t \in [0,T],\, x \in \mathbb R^{n}} |\partial^{\beta}_{x}c_{j,\varepsilon}(t,x) - \partial^{\beta}_{x}c_{j}(t,x)| \to 0, \quad \text{as} \, \varepsilon \to 0. \end{equation*}

3. (iii) The regularized data satisfy

   \begin{equation*} \|g-g_\varepsilon\|_{H^{m,s}} \to 0, \quad \sup_{t \in [0,T]} \|f(t)-f_{\varepsilon}(t)\|_{H^{m,s}} \to 0, \quad \text{when}\,\, \varepsilon \to 0, \end{equation*}

   for all $m, s \in \mathbb N_0$.

We therefore conclude $u_{\varepsilon} \to u$ in $C([0,T];\mathscr{S}(\mathbb R^{n}))$. Summing up, we have proven the following result.

7. Classical Schwartz well-posedness result

As we mentioned in the introduction, from the theory developed in this paper, we obtain a result concerning the well-posedness in Schwartz spaces of the Cauchy problem (1.4), which is new and interesting per se. In this section, we shall state it and explain how to prove it from the ideas used throughout the manuscript.

Let $S$ be the Schrödinger type operator given in (1.2) with the following hypotheses on the coefficients:

(7.1)\begin{equation} c_j \in C([0,T]; \mathcal{B}^{\infty}(\mathbb R^n)), \quad |\mathop{\rm Im}\nolimits\, c_j(t,x)| \leq C \langle x \rangle^{-1}, \quad j = 0, 1, \ldots, n. \end{equation}

Then the following result holds.

Theorem 7.1 Suppose the coefficients of $S$ satisfy (7.1). Let $f \in C([0,T]; \mathscr{S}(\mathbb R^n))$ and $g \in \mathscr{S}(\mathbb R^n)$ and consider the Cauchy problem (1.4). Then there exists a unique solution $u \in C([0,T]; \mathscr{S}(\mathbb R^n))$. Moreover, the solution $u$ satisfies the following energy inequality

\begin{equation*} \| u(t) \|^{2}_{H^{m-c(s), s}} \leq C_{m,s,n,T} \left\{\|g\|^{2}_{H^{m,s}} + \int_{0}^{t} \|f(\tau)\|^{2}_{H^{m,s}} d\tau \right\}, \quad s,m \in \mathbb R, \, t \in [0,T], \end{equation*}

for some constant $c(s)$ depending only on $s$. In particular, the Cauchy problem (1.4) is well-posed in $\mathscr{S}(\mathbb R^n)$.

To prove the above theorem, the idea is to consider the operator

\begin{equation*} S_s = \langle x \rangle^{s} \circ S \circ \langle x \rangle^{-s}, \end{equation*}

and to repeat the steps of Section 5. We just remark that since we do not have to worry with a parameter ɛ and the coefficients have rapid decay, the sole conjugation by $e^{\lambda_1}(x,D)$ is enough to prove the above result via an auxiliary Cauchy problem (cf. again Section 5).