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Local Well Posedness of the Euler–Korteweg Equations on \({{\mathbb {T}}^d}\)

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Abstract

We consider the Euler–Korteweg system with space periodic boundary conditions \( x \in {\mathbb {T}}^d\). We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data.

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Notes

  1. For \(\delta \) sufficiently small, if \( |j-k| \le \delta \left\langle j + k \right\rangle \) and \( |k| \le \delta \left\langle 2j - k \right\rangle \) then \( (j,k) = (0,0)\).

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A Bony–Weyl Calculus in Periodic Hölder Spaces

A Bony–Weyl Calculus in Periodic Hölder Spaces

In this Appendix we develop in a self-contained manner Weyl paradifferential calculus for space periodic symbols \( a(x, \xi ) \) which belong to the Banach scale of Hölder spaces \( W^{\varrho ,\infty } ({\mathbb {T}}^d) \). We mention [2] for paradifferential calculus on \( {\mathbb {T}}^d \) using the standard quantization, see [22] for the case of \({\mathbb {R}}^d \). The convenience of Weyl calculus for energy estimates was noted in [16], and then implemented in [11, 18]. The main results are the continuity Theorem A.7 and the composition Theorem A.8, which require mild regularity assumptions of the symbols in the space variable, and imply Theorems 2.4 and 2.5. We first provide some preliminary technical results.

Technical lemmas. In the following we denote by \(\partial _m\), \(m= 1, \ldots , d\) the discrete derivative, defined for functions \(f :{\mathbb {Z}}^d \rightarrow {\mathbb {C}}\) as

$$\begin{aligned} (\partial _m f)(n) := f(n) - f(n - \vec {e}_m), \quad n \in {\mathbb {Z}}^d, \end{aligned}$$
(A.1)

where \( \vec {e}_m \) denotes the usual unit basis vector of \( {\mathbb {N}}_0^d \) with 0 components expect the m-th one. Given a multi-index \(\beta \in {\mathbb {N}}^d_0 \), we set \( \partial ^\beta f := \partial _1^{\beta _1} \cdots \partial _d^{\beta _d} f \).

We shall use the Leibniz rule for finite differences in the following form: given \( k \in {\mathbb {N}}\), \( m = 1, \ldots , d \), there exist constants \( C_{k_1,k_2} \) (binomial coefficients) such that

$$\begin{aligned} (\partial _m^k) (f g) (n) = \sum _{k_1+k_2= k} C_{k_1,k_2} (\partial _m^{k_1} f) (n- k_2) (\partial _m^{k_2} g) (n). \end{aligned}$$
(A.2)

Moreover, when using discrete derivatives, the analogous of the integration by parts formula is given by the Abel resummation formula:

$$\begin{aligned} \sum _{n \in {\mathbb {Z}}^d} e^{{{\text {i}}}n \cdot z} \beta (x,n) = - \frac{1}{e^{{{\text {i}}}\vec {e}_m \cdot z}-1} \sum _{n \in {\mathbb {Z}}^d} e^{{{\text {i}}}n \cdot z} (\partial _m \beta ) (x,n), \quad \forall \, m = 1, \ldots , d . \end{aligned}$$
(A.3)

Lemma A.1

Let \(K :{\mathbb {T}}^d \rightarrow {\mathbb {C}}\) be a function satisfying, for constants A and B, the estimate

$$\begin{aligned} \left| K(y) \right| \lesssim A^d B \min \left( 1, \min _{1\le m \le d} \frac{1}{ |A \, 2 \sin \frac{y_m}{2}|^{(d+1)}}\right) , \quad \forall y \in {\mathbb {T}}^d. \end{aligned}$$
(A.4)

Then

$$\begin{aligned} \int _{{\mathbb {T}}^d} \left| K(y) \right| {\text {d}}y \lesssim B. \end{aligned}$$
(A.5)

Proof

If \( A \le 1 \) the bound (A.5) follows trivially integrating the first inequality in (A.4). Then we suppose \( A > 1 \). We split the integral in (A.5) as

$$\begin{aligned} \int _{{\mathbb {T}}^d} \left| K(y) \right| {\text {d}}y= \int \limits _{{\mathbb {T}}^d\cap \{|y|\le \frac{1}{A}\}} \left| K(y) \right| {\text {d}}y+\int \limits _{{\mathbb {T}}^d\cap \{ |y| > \frac{1}{A}\}} \left| K(y) \right| {\text {d}}y. \end{aligned}$$
(A.6)

We bound the first integral using the first inequality in (A.4), getting

$$\begin{aligned} \int \limits _{{\mathbb {T}}^d\cap \{|y|\le \frac{1}{A}\}} \left| K(y) \right| {\text {d}}y\lesssim A^dB \, {\text {meas}}\left( y \in [-\pi ,\pi ]^d :\ {|y|\le \frac{1}{A}} \right) \lesssim B. \end{aligned}$$
(A.7)

To bound the second integral in (A.6) we use that, for some \(c >0\), \( \max _{1\le m \le d} \big | \sin \big (\frac{y_m}{2}\big ) \big | \ge c|y| \), \( \forall y\in [-\pi ,\pi ]^d \), and therefore the second inequality in (A.4) implies

$$\begin{aligned} \int \limits _{{\mathbb {T}}^d\cap \{ |y|> \frac{1}{A}\}} \left| K(y) \right| {\text {d}}y\lesssim A^dB\int \limits _{\{y \in {\mathbb {R}}^d \, : \, |y|>\frac{1}{A}\}} \frac{{\text {d}}y}{|Ay|^{d+1}}{\mathop {\lesssim }\limits ^{z=Ay}} B\int \limits _{ \{|z|>1\}} \frac{{\text {d}}z}{|z|^{d+1}} \lesssim B. \end{aligned}$$
(A.8)

The bounds (A.7)–(A.8) and (A.6) imply (A.5). \(\square \)

The next lemma represents a Fourier multiplier operator acting on periodic functions as a convolution integral on \( {\mathbb {R}}^d \). The key step is the use of Poisson summation formula.

Lemma A.2

Let \( \chi \in {{{\mathcal {S}}}} ({\mathbb {R}}^d) \). Then the Fourier multiplier \( \chi _\theta ( D) := \chi ( \theta ^{-1} D) \), \( \theta \ge 1 \), acting on a periodic function \( u \in L^1 ({\mathbb {T}}^d, {\mathbb {C}}) \) can represented by

$$\begin{aligned} \chi _\theta ( D) u&= \int _{{\mathbb {R}}^d} u(y) \psi _{\theta } ( x- y ) {\text {d}}y = \int _{{\mathbb {R}}^d} u(x-y) \psi _{\theta } ( y ) {\text {d}}y \end{aligned}$$
(A.9)

where \( \psi _{\theta } (z) := \theta ^{d} \, \psi ( \theta z) \) and \( \psi \) denotes the anti-Fourier transform of \( \chi \) on \({\mathbb {R}}^d \).

Proof

For \( \theta \ge 1 \) we write

$$\begin{aligned} \chi \left( \frac{D}{\theta }\right) u= & {} \int _{{\mathbb {T}}^d} u(y) \, h_{\theta } (x-y) {\text {d}}y \qquad {\text {where}} \nonumber \\ h_\theta (z):= & {} \frac{1}{(2\pi )^d} \sum _{j \in {\mathbb {Z}}^d} \chi \left( \frac{j}{\theta }\right) e^{{{\text {i}}}j \cdot z}. \end{aligned}$$
(A.10)

Then the Fourier transform \( {\widehat{\psi _\theta }} (\xi ) = \int _{{\mathbb {R}}^d} \, \theta ^d \, \psi (\theta z) \, e^{- {{\text {i}}}z \cdot \xi } {\text {d}}z = \int _{{\mathbb {R}}^d} \psi (y)\, e^{- {{\text {i}}}y \cdot \frac{\xi }{\theta }} {\text {d}}y = {\widehat{\psi }} \left( \frac{\xi }{\theta } \right) = \chi \left( \frac{\xi }{\theta } \right) \), and, using Poisson summation formula, we write the periodic function \( h_\theta (z) \) in (A.10) as

$$\begin{aligned} h_\theta (z) = \frac{1}{(2\pi )^d}\sum _{j \in {\mathbb {Z}}^d} {\widehat{\psi _\theta }} (j) e^{{{\text {i}}}j \cdot z} = \sum _{j \in {\mathbb {Z}}^d} \psi _\theta ( z + 2 \pi j ). \end{aligned}$$

Therefore the integral (A.10) is

$$\begin{aligned} \chi (\theta ^{-1} D) u&= \sum _{j \in {\mathbb {Z}}^d} \int _{{\mathbb {T}}^d} u(y) \, \psi _\theta ( x- y + 2 \pi j ) {\text {d}}y \nonumber = \sum _{j \in {\mathbb {Z}}^d} \int _{[0, 2\pi ]^d + 2 \pi j} u(y) \psi _\theta ( x- y ) {\text {d}}y \nonumber \\&= \int _{{\mathbb {R}}^d} u(y) \psi _\theta ( x- y ) {\text {d}}y = \int _{{\mathbb {R}}^d} u(x-y) \psi _\theta ( y ) {\text {d}}y \end{aligned}$$

proving (A.9). \(\square \)

We now give the definition and basic properties of the Hölder spaces \( W^{\varrho , \infty } ({\mathbb {T}}^d )\).

Definition A.3

(Periodic Hölder spaces) Given \(\varrho \in {\mathbb {N}}_0\), we denote by \( W^{\varrho , \infty } ({\mathbb {T}}^d ) \) the space of continuous functions \( u : {\mathbb {T}}^d \rightarrow {\mathbb {C}}\), \( 2 \pi \)-periodic in each variable \( (x_1, \ldots , x_d) \), whose derivatives of order \(\varrho \) are in \(L^\infty \), equipped with the norm \( \Vert u \Vert _{W^{\varrho ,\infty }} := \sum _{|\alpha | \le \varrho } \Vert \partial _x^\alpha u \Vert _{L^\infty } \), \( \alpha \in {\mathbb {N}}_0^d \). In case \( \varrho >0\), \(\varrho \notin {\mathbb {N}}\), we denote \( \lfloor \varrho \rfloor \) the integer part of \( \varrho \), and we define \( W^{\varrho , \infty } ({\mathbb {T}}^d ) \) as the space of functions u in \( C^{\lfloor \varrho \rfloor }({\mathbb {T}}^d,{\mathbb {C}}) \) whose derivatives of order \( \lfloor \varrho \rfloor \) are \( (\varrho - \lfloor \varrho \rfloor ) \)-Hölder-continuous, that is

$$\begin{aligned}{}[\partial _x^\alpha u]_{\varrho } := \sup _{x\ne y}\frac{|\partial _x^\alpha u(x)- \partial _x^\alpha u(y)|}{|x-y|^{\varrho - \lfloor \varrho \rfloor }} < + \infty , \ \ \ \forall |\alpha | = \lfloor \varrho \rfloor , \end{aligned}$$

equipped with the norm

$$\begin{aligned} \Vert u \Vert _{W^{\varrho ,\infty }} := {\sum _{|\alpha | \le \lfloor \varrho \rfloor } \Vert \partial _x^\alpha u \Vert _{L^\infty } + \sum _{|\alpha | = \lfloor \varrho \rfloor } [\partial _x^\alpha u]_{\varrho }}. \end{aligned}$$

For \( \varrho = 0 \) the norm \( \Vert \ \Vert _{W^{\varrho ,\infty }} = \Vert \ \Vert _{L^{\infty }} \).

The Hölder spaces \( W^{\varrho , \infty } ({\mathbb {T}}^d) \) can be described by the Paley–Littlewood decomposition of a function. Consider the locally finite partition on unity

$$\begin{aligned} 1 = \chi (\xi ) + \sum _{k \ge 1} \varphi (2^{-k}\xi ), \quad \varphi (z) := \chi (z) - \chi (2z), \end{aligned}$$
(A.11)

where \( \chi : {\mathbb {R}}^d \rightarrow {\mathbb {R}}\) is the cut-off function defined in (2.13). It induces the decomposition of a distribution \( u \in {{{\mathcal {S}}}}' ({\mathbb {T}}^d) \) as

$$\begin{aligned} u = \sum _{k \ge 0} \Delta _k u \quad {\text {where}} \quad \Delta _0:= \chi (D), \ \Delta _k := \varphi (2^{-k} D) = \chi _{2^k}(D)- \chi _{2^{k-1}}(D), \ k \ge 1.\nonumber \\ \end{aligned}$$
(A.12)

We also set

$$\begin{aligned} S_k := \sum _{0 \le j \le k} \Delta _j = \chi _{2^k}(D). \end{aligned}$$
(A.13)

The Paley–Littlewood theory of the Hölder spaces \( W^{\varrho ,\infty } ({\mathbb {T}}^d)\) follows as in \( {\mathbb {R}}^d \), see e.g. [22], once we represent the Fourier multipliers \( \Delta _k \) as integral convolution operators on \( {\mathbb {R}}^d \), by Lemma A.2. In particular the following smoothing estimates hold: for any \( \alpha \in {\mathbb {N}}^d_0 \), \( \varrho \ge 0 \),

$$\begin{aligned} \left\| \partial _x^\alpha S_k u \right\| _{L^\infty } \lesssim 2^{k(| \alpha | -\varrho )} \left\| u \right\| _{W^{\varrho ,\infty }}, \end{aligned}$$
(A.14)

and, for any \( \varrho >0 \),

$$\begin{aligned} \Vert u - \chi _\theta (D) u \Vert _{L^\infty } \lesssim \theta ^{-\varrho } \Vert u \Vert _{W^{\varrho ,\infty }}. \end{aligned}$$
(A.15)

In this way it results as in \( {\mathbb {R}}^d \) that the Hölder norms \( \left\| \ \right\| _{W^{\varrho , \infty }} \) satisfy interpolation estimates. In particular we shall use that, given \(\varrho , \varrho _1, \varrho _2 \ge 0 \),

$$\begin{aligned} \begin{aligned} \Vert u v \Vert _{W^{\varrho ,\infty }}&\lesssim \Vert u \Vert _{W^{\varrho ,\infty }} \Vert v \Vert _{L^{\infty }} + \Vert u \Vert _{L^{\infty }} \Vert v \Vert _{W^{\varrho , \infty }} \\ \left\| u \right\| _{W^{\varrho , \infty }}&\lesssim \left\| u \right\| _{W^{\varrho _1, \infty }}^\theta \, \left\| u \right\| _{W^{\varrho _2,\infty }}^{1-\theta }, \ \ \ \ \varrho = \theta \varrho _1 + (1-\theta ) \varrho _2, \ \theta \in (0,1). \end{aligned} \end{aligned}$$
(A.16)

Hölder estimates of regularized symbols.

In order to prove estimates of the regularized symbol \( a_\chi \) defined in (2.14) in Hölder spaces (Lemma A.5) we represent it as a convolution integral on \( {\mathbb {R}}^d \), by Lemma A.2,

$$\begin{aligned} a_\chi (x, \xi ) = \int _{{\mathbb {R}}^d} a(x-y, \xi ) \, \psi _{\epsilon \left\langle \xi \right\rangle }(y) \, {\text {d}}y \end{aligned}$$
(A.17)

where \(\psi _\theta (z) = \theta ^d \psi ( \theta z ) \) and \( \psi \) is the anti-Fourier transform of \( \chi \).

In the proof of Lemma A.5 we shall use the following estimate.

Lemma A.4

For any \( \beta \in {\mathbb {N}}_0^d \), \( u \in L^\infty ({\mathbb {T}}^d) \), we have

$$\begin{aligned} \Vert \partial _\xi ^\beta \chi _{\epsilon \langle \xi \rangle } (D) u \Vert _{L^\infty } \lesssim \left\langle \xi \right\rangle ^{-|\beta |} \Vert u \Vert _{L^\infty }. \end{aligned}$$
(A.18)

Proof

By (A.17) we have, for all \( \beta \in {\mathbb {N}}^d_0 \),

$$\begin{aligned} \partial _\xi ^\beta \chi _{\epsilon \left\langle \xi \right\rangle }(D) u = \int _{{\mathbb {R}}^d} u(x-y) \, \partial _\xi ^\beta \psi _{\epsilon \left\langle \xi \right\rangle }(y) \, {\text {d}}y. \end{aligned}$$
(A.19)

By the definition \(\psi _{\epsilon \left\langle \xi \right\rangle }(y) = (\epsilon \left\langle \xi \right\rangle )^d \, \psi (\epsilon \left\langle \xi \right\rangle y)\) and Faà di Bruno formula, we have that

$$\begin{aligned} \int _{{\mathbb {R}}^d} \big | \partial _\xi ^\beta \psi _{\epsilon \left\langle \xi \right\rangle }(y) \big | \, {\text {d}}y \lesssim \left\langle \xi \right\rangle ^{-|\beta |}, \quad \forall \xi \in {\mathbb {R}}^d. \end{aligned}$$
(A.20)

Then (A.18) follows by (A.19) and (A.20). \(\square \)

The next lemma provides estimates of the regularized symbol \( a_\chi \) in terms of the symbol a.

Lemma A.5

(Estimates on regularized symbols) Let \( m \in {\mathbb {R}}\), \( N \in {\mathbb {N}}_0 \).

  1. 1.

    If \(a \in \Gamma ^m_{L^\infty }\), \( m \in {\mathbb {R}}\), then \( a_\chi \) defined in (2.14) belongs to \(\Sigma ^m_{L^\infty }\) and

    $$\begin{aligned} \left| a_\chi \right| _{m, L^{\infty }, N} \lesssim \left| a \right| _{m, L^{\infty }, N}. \end{aligned}$$
    (A.21)
  2. 2.

    If \(a \in \Gamma _{H^{s_0-\varrho }}^m\), \(\varrho \ge 0\), \(s_0 > \frac{d}{2}\), then \(a_\chi \) belongs to \(\Gamma ^{m+\varrho }_{L^\infty }\) and

    $$\begin{aligned} | a_\chi |_{m+\varrho , L^\infty , N} \lesssim | a|_{m, {H^{s_0-\varrho }},N}. \end{aligned}$$
    (A.22)
  3. 3.

    If \(a \in \Gamma ^m_{W^{\varrho , \infty }}\), \(\varrho > 0 \), then, for any \(\beta \in {\mathbb {N}}_0^d \), \(\partial _\xi ^\beta a_\chi - (\partial _{\xi }^\beta a)_\chi \in \Sigma ^{m-|\beta | -\varrho }_{L^\infty }\) and

    $$\begin{aligned} \big | \partial _\xi ^\beta a_\chi - (\partial _{\xi }^\beta a)_\chi \big |_{m-|\beta |-\varrho , L^{\infty }, N} \lesssim \left| a \right| _{m, {W^{\varrho , \infty }}, N+ |\beta |}. \end{aligned}$$
    (A.23)
  4. 4.

    If \(a \in \Gamma ^m_{W^{\varrho , \infty }}\), \(\varrho \ge 0 \), then, for any \(\alpha \in {\mathbb {N}}_0^d\) with \(|\alpha | \ge \varrho \), \( \partial _x^\alpha a_\chi = (\partial _x^\alpha a)_\chi \in \Sigma ^{m+|\alpha | -\varrho }_{L^\infty }\) and

    $$\begin{aligned} \left| \partial _x^\alpha a_\chi \right| _{m+|\alpha | -\varrho , L^{\infty }, N} \lesssim \left| a \right| _{m, {W^{\varrho , \infty }}, N}. \end{aligned}$$
    (A.24)
  5. 5.

    If \(a \in \Gamma ^m_{W^{\varrho , \infty }}\), \(\varrho > 0 \), then, \(a- a_\chi \in \Gamma ^{m-\varrho }_{L^\infty }\) and

    $$\begin{aligned} \left| a- a_\chi \right| _{m-\varrho , L^{\infty }, N} \lesssim \left| a \right| _{m, {W^{\varrho , \infty }}, N}. \end{aligned}$$
    (A.25)

Proof

Proof of (A.21). Differentiating (2.14) for any \( \beta \in {\mathbb {N}}_0^d \), we have

$$\begin{aligned} \partial _\xi ^\beta a_\chi (x,\xi ) = \sum _{\beta _1+\beta _2 = \beta } C_{\beta _1, \beta _2} \partial _{\xi }^{\beta _1} \chi _{\epsilon \langle \xi \rangle }(D) \, \partial _{\xi }^{\beta _2} a(\cdot , \xi ). \end{aligned}$$

Then (2.7) and (A.18) directly imply (A.21).

Proof of (A.22) By the Cauchy–Schwartz inequality

$$\begin{aligned} \left| a_\chi (x,\xi ) \right|&= \Big | \sum _{n\in {\mathbb {Z}}^d} \chi _\epsilon \left( \frac{n}{\langle \xi \rangle }\right) \ {\widehat{a}}(n,\xi ) \, e^{{\text {i}}n\cdot x} \Big | \le \sum _{n\in {\mathbb {Z}}^d} \chi _\epsilon \Big (\frac{n}{\langle \xi \rangle }\Big ) \, \frac{\left\langle n \right\rangle ^{\varrho }}{\left\langle n \right\rangle ^{s_0 }} \, \left\langle n\right\rangle ^{s_0-\varrho } \, \left| {\widehat{a}}(n,\xi ) \right| \\&\lesssim \Big ( \sum _{n \in {\mathbb {Z}}^d} \chi _\epsilon ^2 \Big (\frac{n}{\langle \xi \rangle } \Big )\frac{\left\langle n\right\rangle ^{2\varrho }}{\left\langle n \right\rangle ^{2s_0}} \Big )^{1/2} \left\| a(\cdot , \xi ) \right\| _{H^{s_0-\varrho }} \lesssim \left\langle \xi \right\rangle ^{m+\varrho } \left| a \right| _{m, H^{s_0-\varrho }, 0}. \end{aligned}$$

The case \(N\ge 1\) follows in the same way.

Proof of (A.23). First, for any \( \xi \in {\mathbb {R}}^d \), we define \( k \in {\mathbb {N}}\) such that \( 2^{k-1} \le 2 \epsilon \langle \xi \rangle \le 2^k \). Then, by the properties of the cut-off function \( \chi \) in (2.13) and the projector \( S_k \) in A.13 we have

$$\begin{aligned} \partial _\xi ^\beta \chi _\epsilon \left( \frac{\eta }{\langle \xi \rangle } \right) = \Big (\partial _\xi ^\beta \chi _\epsilon \Big ( \frac{\eta }{\langle \xi \rangle } \Big )\Big ) \, S_k, \quad \ \forall \eta \in {\mathbb {R}}^d, \quad \forall \beta \in {\mathbb {N}}^d_0. \end{aligned}$$
(A.26)

Differentiating (2.14) and using (A.26) we get

$$\begin{aligned} \partial _\xi ^\beta a_\chi - (\partial _\xi ^\beta a)_\chi = \sum _{\beta _1 + \beta _2 = \beta , \beta _1 \ne 0} C_{\beta _1 \beta _2} \, \partial _\xi ^{\beta _1} \chi _{\epsilon \left\langle \xi \right\rangle }(D) \, S_k \, \partial _\xi ^{\beta _2} a(\cdot , \xi ), \end{aligned}$$

and, using (A.18) and (A.14)

$$\begin{aligned} \big \Vert \big (\partial _\xi ^\beta a_\chi - (\partial _\xi ^\beta a)_\chi \big )(\cdot , \xi ) \big \Vert _{L^\infty }&{\mathop {\lesssim }\limits ^{}}\sum _{\beta _1 + \beta _2 = \beta , \beta _1 \ne 0} \left\langle \xi \right\rangle ^{-|\beta _1|}\, 2^{-k \varrho } \big \Vert \partial _\xi ^{\beta _2} a(\cdot , \xi ) \big \Vert _{W^{\varrho , \infty }} \\&\lesssim \left\langle \xi \right\rangle ^{m-|\beta |}\, 2^{-k \varrho } \left| a \right| _{m, W^{\varrho , \infty }, |\beta |} \lesssim \left\langle \xi \right\rangle ^{m-|\beta |-\varrho }\, \left| a \right| _{m, W^{\varrho , \infty }, |\beta |} \end{aligned}$$

because \(\left\langle \xi \right\rangle \lesssim 2^k\). This proves (A.23) for \(N = 0\). For \(N \ge 1\) the estimate is similar.

Proof of (A.24). For any \( \xi \in {\mathbb {R}}^d \), we define \( k \in {\mathbb {N}}\) such that \( 2^{k-1} \le 2 \epsilon \langle \xi \rangle \le 2^k \). By (2.14) and (A.26) with \(\beta = 0\), we write \( a_\chi (\cdot , \xi ) = \chi _{\epsilon \langle \xi \rangle } (D) a(\cdot , \xi ) = \chi _{\epsilon \langle \xi \rangle } (D) S_k a(\cdot , \xi ) \), and then

$$\begin{aligned} \Vert \partial _x^\alpha a_\chi (\cdot , \xi ) \Vert _{L^\infty }&= \Vert \chi _{\epsilon \langle \xi \rangle } (D) \partial _x^\alpha S_k a(\cdot , \xi ) \Vert _{L^\infty } {\mathop {\lesssim }\limits ^{(A.18)}}\Vert \partial _x^\alpha S_k a(\cdot , \xi ) \Vert _{L^\infty } \\&\quad {\mathop {\lesssim }\limits ^{(A.14)}}2^{k (|\alpha |-\varrho )} \Vert a(\cdot , \xi ) \Vert _{W^{\varrho ,\infty }} {\mathop {\lesssim }\limits ^{ \langle \xi \rangle \sim 2^k }}\langle \xi \rangle ^{|\alpha |-\varrho }\\&\quad \Vert a(\cdot , \xi ) \Vert _{W^{\varrho ,\infty }} \lesssim \langle \xi \rangle ^{m+ |\alpha |-\varrho } |a|_{m,W^{\varrho ,\infty },0} \end{aligned}$$

by (2.7). This proves (A.24) with \(N = 0\). For \(N \ge 1\) the estimate is similar.

Proof of (A.25). For any \( \beta \in {\mathbb {N}}_0^d \) we write \( \partial _\xi ^\beta (a - a_\chi ) = \big [ \partial _\xi ^\beta a - (\partial _\xi ^\beta a)_\chi \big ] + \big [ (\partial _\xi ^\beta a)_\chi - \partial _\xi ^\beta a_\chi \big ] \). The first term is bounded, using (A.15) with \( \theta = \epsilon \langle \xi \rangle \), as

$$\begin{aligned} \big \Vert \big ( \partial _\xi ^\beta a - (\partial _\xi ^\beta a)_\chi \big )(\cdot , \xi ) \big \Vert _{L^\infty } \lesssim \langle \xi \rangle ^{-\varrho } \Vert \partial _\xi ^\beta a (\cdot , \xi ) \Vert _{W^{\varrho ,\infty }} \lesssim |a|_{m,W^{\varrho ,\infty },|\beta |} \langle \xi \rangle ^{m-\varrho -|\beta |} \end{aligned}$$

The second term satisfies the same bound by (A.23). This proves (A.25). \(\square \)

Change of quantization. In order to prove the boundedness Theorem A.7 and the composition Theorem A.8, it is convenient to pass from the Weyl quantization of a symbol \( a(x, \xi )\), defined in (2.15), to the standard quantization which is defined, given a symbol \( b(x, \xi ) \), as

$$\begin{aligned} {\text {Op}} {(b)}[u]&:= \sum _{j \in {\mathbb {Z}}^d} \Big ( \sum _{k \in {\mathbb {Z}}^d} {\widehat{b}} \left( j-k, k\right) \, u_k \Big ) e^{{\text {i}}j \cdot x } = \sum _{k \in {\mathbb {Z}}^d} b (x, k) \, u_k e^{{\text {i}}k \cdot x }. \end{aligned}$$
(A.27)

We have the change of quantization formula

$$\begin{aligned} {{\text {Op}}}^W\!\left( a\right) = {{\text {Op}}}\left( b\right) \qquad \Leftrightarrow \qquad {\widehat{b}}(n, \xi ) := {\widehat{a}} \big ( n, \xi + \frac{n}{2} \big ). \end{aligned}$$
(A.28)

In the next lemma we estimate the norms of b in terms of those of a. We remind that \( \Sigma ^m_{\mathscr {W}}\) denotes the set of spectrally localized symbols, i.e. satisfying (2.6).

Lemma A.6

(Change of quantization) Let \(a \in \Sigma ^m_{L^\infty }\), \(m \in {\mathbb {R}}\). If \( \delta > 0 \) in (2.6) is small enough, then (cfr. A.28)

$$\begin{aligned} b(x,\xi ) := \sum _{n\in {\mathbb {Z}}^d} {\widehat{a}} \big ( n,\xi + \frac{n}{2} \big ) \, e^{{\text {i}}n\cdot x} \end{aligned}$$
(A.29)

is a symbol in \( \Sigma ^m_{L^\infty }\) satisfying

$$\begin{aligned} | b|_{m,L^\infty , N}\lesssim | a |_{m,L^\infty , N+d+1}, \quad \forall N\in {\mathbb {N}}_0. \end{aligned}$$
(A.30)

Proof

Since a satisfies (2.6) with \(\delta \) small enough, it follows that b satisfies (2.6). In order to prove (A.30) we differentiate (A.29) obtaining that, for any \( \beta \in {\mathbb {N}}_0^d \),

for some \( \epsilon = \epsilon (\delta ') > 0 \), where in the last equality we used that the sum is actually restricted over the indexes for which \(|n|\le \delta ' \langle \xi \rangle \), \(\delta ' \in (0,1)\). Then we represent \(\partial _\xi ^\beta b\) as the integral

$$\begin{aligned} \partial _\xi ^\beta b(x,\xi )= & {} \int _{{\mathbb {T}}^d} K(x, y)\, {\text {d}}y,\nonumber \\ K(x,y):= & {} \frac{1}{(2\pi )^d} \sum _{n \in {\mathbb {Z}}^d} (\partial _\xi ^\beta a) \big ( x-y,\xi + \frac{n}{2} \big ) \,\chi _\epsilon \left( \frac{n}{\left\langle \xi \right\rangle }\right) \, e^{{\text {i}}n\cdot y}. \end{aligned}$$
(A.31)

We are going to estimate the \( L^1 \)-norm of \( K(x, \cdot ) \) using Lemma A.1. First note that, since \( a \in \Sigma ^m_{L^\infty }\), we have \(\left\langle \xi + \frac{n}{2}\right\rangle \sim \left\langle \xi \right\rangle \) on the support of , and then we bound (A.31) as

$$\begin{aligned} |K(x,y)| \lesssim \sum _{|n| \le \delta ' \left\langle \xi \right\rangle } | a|_{m, L^\infty , |\beta |} \left\langle \xi \right\rangle ^{m-|\beta | } \lesssim | a |_{m, L^\infty , |\beta | }\, \langle \xi \rangle ^{d + m - |\beta |}, \end{aligned}$$
(A.32)

uniformly in x. Moreover, using Abel resummation formula (A.3) and the Leibniz rule (A.2) for finite differences, we get, for any \( h = 1, \ldots , d \),

$$\begin{aligned} K(x,y) = \frac{1}{\left( e^{{\text {i}}y_h}-1 \right) ^{d+1}} \sum _{ k_1 +k_2 = d+1} C_{k_1, k_2} \sum _{|n| \le \delta ' \left\langle \xi \right\rangle } \partial _h^{k_1} (\partial _\xi ^\beta a) \big ( x-y,\xi + \frac{n}{2} \big ) \, \partial _h^{k_2} \chi _\epsilon \left( \frac{n}{\left\langle \xi \right\rangle }\right) \, e^{{\text {i}}n\cdot y}. \end{aligned}$$

Then, using (2.7) and that \( \displaystyle { \big | \partial _h^k \chi _\epsilon \big ( \frac{n}{\left\langle \xi \right\rangle } \big ) \big | \lesssim \left\langle \xi \right\rangle ^{-k}} \), \(\forall h = 1, \ldots , d\), we estimate

$$\begin{aligned} | K(x,y)| \lesssim \frac{\left\langle \xi \right\rangle ^{m - (d+1) - |\beta |}}{|2\sin (y_h/2)|^{d+1}} \ |a|_{m, L^\infty , |\beta | + d+1} \sum _{|n| \le \delta ' \left\langle \xi \right\rangle } 1 \lesssim \, \frac{\left\langle \xi \right\rangle ^{d+m - |\beta |} \, |a|_{m, L^\infty , |\beta | + d+1}}{|\left\langle \xi \right\rangle \, 2\sin (y_h/2)|^{d+1}} \ \end{aligned}$$
(A.33)

uniformly in x. In view of (A.32)–(A.33) we apply Lemma A.1 with \(A = \langle \xi \rangle \) and \(B= \left\langle \xi \right\rangle ^{m - |\beta |} \, |a|_{m, L^\infty , |\beta | + d+1}\) obtaining

$$\begin{aligned} \big | \partial _\xi ^\beta b(x,\xi ) \big | \le \int _{{\mathbb {T}}^d} |K(x,y)| {\text {d}}y \lesssim \left\langle \xi \right\rangle ^{m - |\beta |} \, |a|_{m, L^\infty , |\beta | + d+1}, \quad \forall (x,\xi )\in {\mathbb {T}}^d \times {\mathbb {R}}^d, \end{aligned}$$

that proves (A.30). \(\square \)

Continuity.

We now prove boundedness estimates in Sobolev spaces of operators with spectrally localized symbols, requiring derivatives in \(\xi \) of the symbol and no derivatives in x.

Theorem A.7

(Continuity) Let \(a \in \Sigma ^m_{L^\infty }\) with \(m \in {\mathbb {R}}\). Then \({{\text {Op}}}\left( a\right) \) defined in (A.27) extends to a bounded operator from \( H^s \rightarrow H^{s-m}\), for any \( s \in {\mathbb {R}}\), satisfying

$$\begin{aligned} \left\| {{\text {Op}}}\left( a\right) u \right\| _{{s-m}} \lesssim \, \left| a \right| _{m, L^\infty , d+1} \, \left\| u \right\| _{{s}}. \end{aligned}$$
(A.34)

Moreover, if a fulfills (2.6) with \( \delta > 0 \) small enough, then the operator \({{\text {Op}}}^W\!\left( a\right) \) defined in (2.15) satisfies

$$\begin{aligned} \left\| {{\text {Op}}}^W\!\left( a\right) u \right\| _{s-m} \lesssim \, \left| a \right| _{m, L^{\infty }, 2(d+1)} \, \left\| u \right\| _{s}. \end{aligned}$$
(A.35)

Proof

We first recall the Littlewood–Paley characterization of the Sobolev norm

$$\begin{aligned} \Vert u \Vert _s^2 \sim \sum _{k \ge 0} 2^{2ks} \Vert \Delta _k u \Vert _0^2 \end{aligned}$$
(A.36)

where \( \Delta _k \) are defined in (A.12). The norm \( \Vert \ \Vert _0 = \Vert \ \Vert _{L^2} \). We first prove (A.34). Step 1: according to (A.11), we perform the Littlewood–Paley decomposition of \({{\text {Op}}}\left( a\right) \),

$$\begin{aligned} {{\text {Op}}}\left( a\right) v = \sum _{k \ge 0} {{\text {Op}}}\left( a_k\right) v, \end{aligned}$$
(A.37)

where

$$\begin{aligned} a_0 (x,\xi ) := a(x,\xi ) \chi (\xi ), \quad a_k(x,\xi ) := a(x,\xi ) \varphi (2^{-k}\xi ), \quad k \ge 1. \end{aligned}$$
(A.38)

In order to prove (A.34), it is sufficient to prove that

$$\begin{aligned} \left\| {{\text {Op}}}\left( a_k\right) v \right\| _{0} \lesssim |a|_{m, L^\infty , d+1} \, 2^{km} \, \left\| v \right\| _{0}, \quad \forall k \in {\mathbb {N}}_0, \quad \forall v \in L^2. \end{aligned}$$
(A.39)

Indeed, decomposing v in Paley–Littlewood packets as in (A.12),

$$\begin{aligned} v = \sum _{j \ge 0} \Delta _j v, \quad \Delta _0 = \chi (D), \ \Delta _j = \varphi (2^{-j} D ), \end{aligned}$$
(A.40)

which are almost orthogonal in \(L^2\) (namely \( \Delta _k \Delta _j = 0 \) for any \( |j-k| \ge 3 \)), using the fact that \( {{\text {Op}}}\left( a_k\right) v = {{\text {Op}}}\left( a\right) \Delta _k v \), and since the action of \({{\text {Op}}}\left( a_k\right) \) does not spread much the Fourier support of functions being a spectrally localized, according to (2.17), we have

$$\begin{aligned} \Vert {{\text {Op}}}\left( a\right) v \Vert _{s-m}^2&{\mathop {=}\limits ^{(A.37)}} \Big \Vert \sum _{k \ge 0} {{\text {Op}}}\left( a_k\right) v \Big \Vert _{s-m}^2 {\mathop {=}\limits ^{(A.40),(A.38)}} \Big \Vert \sum _{|j-k|< 3} {{\text {Op}}}\left( a_k\right) \Delta _j v \Big \Vert _{s-m}^2 \\&\sim \sum _{|j-k|< 3} 2^{2k(s-m)} \, \Vert {{\text {Op}}}\left( a_k\right) \Delta _j v \Vert _{0}^2 {\mathop {\lesssim }\limits ^{(A.39)}}|a|_{m, L^\infty , d+1}^2 \, \sum _{|j-k| < 3} \, 2^{2ks} \, \Vert \Delta _j v \Vert _{0}^2 \\&\lesssim |a|_{m, L^\infty , d+1}^2 \, \sum _{k \ge 0} 2^{2ks} \left\| \Delta _k v \right\| _{0}^2 {\mathop {\sim }\limits ^{(A.36)}}|a|_{m, L^\infty , d+1}^2 \Vert v \Vert _{s}^2. \end{aligned}$$

Step 2: By (A.38) and (A.27) we write \({{\text {Op}}}\left( a_k\right) \) as the integral operator

$$\begin{aligned} ({{\text {Op}}}\left( a_k\right) v)(x) = \int _{{\mathbb {T}}^d} K_k(x, x-y) \, v(y) \, {\text {d}}y \end{aligned}$$
(A.41)

with kernel

$$\begin{aligned} K_k (x,z) := \frac{1}{(2 \pi )^d} \sum _{\ell \in {\mathbb {Z}}^d} e^{{{\text {i}}}\ell \cdot z} \, a(x,\ell ) \, \varphi (2^{-k}\ell ). \end{aligned}$$
(A.42)

We shall deduce (A.39) by applying the Schur lemma: if

$$\begin{aligned} \sup _{x \in {\mathbb {T}}^d} \int _{{\mathbb {T}}^d} |K(x,x-y)| {\text {d}}y =: C_1< + \infty , \quad \sup _{y \in {\mathbb {T}}^d} \int _{{\mathbb {T}}^d} |K(x,x-y)| {\text {d}}x =: C_2 < + \infty \nonumber \\ \end{aligned}$$
(A.43)

then Schur lemma guarantees that the integral operator (A.41) is bounded on \( L^2 ({\mathbb {T}}^d) \) and

$$\begin{aligned} \Vert {{\text {Op}}}\left( a_k\right) v \Vert _0 \le (C_1 C_2)^{1/2} \Vert v \Vert _0. \end{aligned}$$
(A.44)

Let us prove (A.43) and estimate the constants \(C_1, C_2\). By (A.42) we have that

$$\begin{aligned} \left| K_k(x,z) \right|&\lesssim \sum _{\ell \in {\mathbb {Z}}^d} |a(x,\ell )| \varphi (2^{-k}\ell ) {\mathop {\lesssim }\limits ^{(2.7)}}| a |_{m, L^\infty , 0}\sum _{\ell \in {\mathbb {Z}}^d} \left\langle \ell \right\rangle ^m \varphi (2^{-k}\ell ) \nonumber \\&\lesssim 2^{k(d+m)} | a |_{m, L^\infty , 0}. \end{aligned}$$
(A.45)

Then, applying \( (d+1)\)-times Abel resummation formula (A.3) to (A.42), we obtain, for any \( h = 1, \ldots , d \),

$$\begin{aligned} K_k (x,z) = \frac{1}{(2 \pi )^d} \frac{1}{(e^{{{\text {i}}}z_h}-1)^{d+1}} \sum _{\ell \in {\mathbb {Z}}^d} e^{{{\text {i}}}\ell \cdot z} \ \partial _h^{d+1} ( a (x,\ell )\varphi (2^{-k}\ell ) ) \end{aligned}$$

and we deduce, using (2.7), (A.2), \( \left| K_k(x,z) \right| \lesssim \left| 2 \sin (z_h/2) \right| ^{-d-1} |a|_{m, L^\infty ,d+1} 2^{k(m-1)} \) for any \( h = 1, \ldots , d \), thus

$$\begin{aligned} \left| K_k(x,z) \right| \lesssim 2^{k(d+m)} \, |a|_{m, L^\infty ,d+1} \, \ \min _{h =1, \ldots , d} \frac{1 }{( 2^{k} 2 \, | \sin (z_h/2)|)^{d+1} }. \end{aligned}$$
(A.46)

By (A.45), (A.46) we apply Lemma A.1 with \( A = 2^{k}\) and \(B= 2^{km} |a|_{m, L^\infty , d+1} \), deducing that

$$\begin{aligned} \int _{{\mathbb {T}}^d} |K_k\bigl (x,x-y\bigr )| {\text {d}}y&= \int _{{\mathbb {T}}^d} |K_k\bigl (x,z\bigr )| {\text {d}}z \lesssim 2^{km} \, |a|_{m, L^\infty ,d+1} \end{aligned}$$
(A.47)

uniformly for \( x \in {\mathbb {T}}^d \). Similarly

$$\begin{aligned} \int _{{\mathbb {T}}^d} | K_k\bigl (x,x-y\bigr )| {\text {d}}x&\lesssim 2^{km} \, |a|_{m, L^\infty ,d+1} \end{aligned}$$
(A.48)

uniformly for \( y \in {\mathbb {T}}^d \). Finally (A.47), (A.48), (A.44) prove (A.39) completing the proof of (A.34).

Proof of (A.35). By Lemma A.6 we have \({{\text {Op}}}^W\!\left( a\right) = {{\text {Op}}}\left( b\right) \) for a spectrally localized symbol \(b \in \Sigma ^m_{L^\infty }\) which fulfills estimate (A.30). Then (A.35) follows by (A.34). \(\square \)

Composition of paradifferential operators. We finally prove a composition result for paradifferential operators. The difference with respect to Theorem 6.1.1 and 6.1.4 in [22] is to have periodic symbols and the use of the Weyl quantization.

We shall use that, in view of the interpolation inequality (A.16), if \( a \in \Gamma ^m_{W^{\varrho ,\infty }} \) and \( b \in \Gamma ^{m'}_{W^{\varrho ,\infty }} \) then \( ab \in \Gamma ^{m+m'}_{W^{\varrho ,\infty }} \) and, for any \( N \in {\mathbb {N}}_0 \), any \(0 \le \varrho _1 \le \alpha \le \beta \le \varrho _2 \) such that \(\varrho _1 + \varrho _2 = \alpha + \beta \)

$$\begin{aligned}&\left| ab \right| _{m+m', {W^{\varrho ,\infty }}, N} \lesssim \left| a \right| _{m, {W^{\varrho ,\infty }}, N} \, \left| b \right| _{m', L^\infty , N} +\left| a \right| _{m, L^\infty , N} \, \left| b \right| _{m',{W^{\varrho ,\infty }}, N},\nonumber \\&| a|_{m, W^{\alpha , \infty }, N} \, |b|_{m', W^{\beta , \infty },N} \lesssim |a|_{m, W^{\varrho _1,\infty }, N} \, |b|_{m', W^{\varrho _2, \infty },N}+ |a|_{m, W^{\varrho _2,\infty }, N} \, |b|_{m', W^{\varrho _1, \infty },N}.\nonumber \\ \end{aligned}$$
(A.49)

Theorem A.8

(Composition) Let \(a \in \Gamma ^m_{W^{\varrho , \infty }}\), \(b \in \Gamma ^{m'}_{W^{\varrho , \infty }}\) with \(m, m' \in {\mathbb {R}}\) and \(\varrho \in (0,2]\). Then

$$\begin{aligned} {{\text {Op}}}^{\text {BW}}\!\left( a\right) {{\text {Op}}}^{\text {BW}}\!\left( b\right)&= {{\text {Op}}}^{\text {BW}}\!\left( a\#_\varrho b\right) + R^{-\varrho }(a,b) \end{aligned}$$
(A.50)

where the linear operator \(R^{-\varrho }(a,b):{\dot{H}}^s \rightarrow {\dot{H}}^{s-(m+m')+\varrho }\), \(\forall s \in {\mathbb {R}}\), satisfies

$$\begin{aligned} \left\| R^{-\varrho }(a,b)u \right\| _{{s -(m+m') +\varrho }} \lesssim \left( \left| a \right| _{m, W^{\varrho , \infty }, N} \, \left| b \right| _{m', L^\infty , N} + \left| a \right| _{m, L^\infty , N} \, \left| b \right| _{m', W^{\varrho , \infty }, N} \right) \left\| u \right\| _{s} \end{aligned}$$
(A.51)

with \( N \ge 3d+4\).

Proof

We give the proof in the case \( \varrho \in (1, 2] \). We first compute \({{\text {Op}}}^{\text {BW}}\!\left( a\right) {{\text {Op}}}^{\text {BW}}\!\left( b\right) \). Recalling the definition (2.16) we obtain

$$\begin{aligned} {{\text {Op}}}^{\text {BW}}\!\left( a\right) {{\text {Op}}}^{\text {BW}}\!\left( b\right) u&= {{\text {Op}}}^W\!\left( a_\chi \right) {{\text {Op}}}^W\!\left( b_\chi \right) \\&= \sum _{ j, k , \ell } {\widehat{a}}_\chi \Big (j-k, \frac{j+k}{2} \Big ) \, {\widehat{b}}_\chi \Big ( k-\ell , \frac{k + \ell }{2} \Big ) \, u_\ell \, e^{{\text {i}}j \cdot x}. \end{aligned}$$

We now perform a Taylor expansion of \( {\widehat{a}}_\chi \big (j-k, \frac{j+k}{2} \big ) \) in the second variable, around the point \(\frac{j+ \ell }{2}\). Writing \(j+ k = j+ \ell + (k - \ell )\), we obtain

$$\begin{aligned} {\widehat{a}}_\chi \Big (j-k, \frac{j+k}{2} \Big )&= {\widehat{a}}_\chi \Big (j-k, \frac{j+\ell }{2} \Big ) + \Big ( \frac{k-\ell }{2}\Big ) \cdot \partial _\xi {\widehat{a}}_\chi \Big (j-k, \frac{j+\ell }{2} \Big ) \\&\quad + \sum _{\alpha \in {\mathbb {N}}_0^d, |\alpha | = 2} \Big ( \frac{k-\ell }{2}\Big )^{\alpha } \int _0^1 (1-t) \, \partial _\xi ^\alpha {\widehat{a}}_\chi \Big (j-k, \frac{j+\ell + t(k- \ell )}{2} \Big ) {\text {d}}t. \end{aligned}$$

We expand analogously \( {\widehat{b}}_\chi \big (k-\ell , \frac{k+ \ell }{2} \big ) \) around the point \(\frac{j+\ell }{2}\). Writing \( k + \ell = j+ \ell - (j-k)\), we obtain

$$\begin{aligned} {\widehat{b}}_\chi \Big (k-\ell , \frac{k+\ell }{2} \Big )&= {\widehat{b}}_\chi \Big (k-\ell , \frac{j+\ell }{2} \Big ) - \Big ( \frac{j-k}{2}\Big )\cdot \partial _\xi {\widehat{b}}_\chi \Big (k-\ell , \frac{j+\ell }{2} \Big )\\&\quad + \sum _{\beta \in {\mathbb {N}}_0^d, |\beta | = 2} \Big ( \frac{k-j}{2}\Big )^{\beta } \int _0^1(1-t) \, \partial _\xi ^\beta {\widehat{b}}_\chi \Big (k-\ell , \frac{j+\ell + t(k- j)}{2} \Big ) {\text {d}}t. \end{aligned}$$

Moreover, recalling (2.20) and (2.15), we write \({{\text {Op}}}^{\text {BW}}\!\left( a\#_\varrho b \right) u = {{\text {Op}}}^W\!\left( (ab + \frac{1}{2{\text {i}}}\{a, b\})_\chi \right) u \) and, by the previous expansions,

$$\begin{aligned} \Big ({{\text {Op}}}^{\text {BW}}\!\left( a\right) {{\text {Op}}}^{\text {BW}}\!\left( b\right) - {\text {Op}}^{BW} \Big ( ab + \frac{1}{2{\text {i}}}\{a,b\} \Big ) \Big ) u&= \sum _{i=1}^4 R_i(a,b) u \end{aligned}$$

where

$$\begin{aligned}&R_1(a,b)u := {\text {Op}}^{\mathrm{W}} \Big ( {a_\chi b_\chi - (ab)_\chi + \frac{1}{2{\text {i}}} \big ( \{a_\chi , b_\chi \} - (\{a,b\})_\chi \big )} \Big ) u \end{aligned}$$
(A.52)
$$\begin{aligned}&\quad R_2(a,b)u:= \sum _{j,k,\ell } {\widehat{b}}_\chi \Big (k-\ell , \frac{k+\ell }{2} \Big ) \!\! \sum _{|\alpha | = 2} \Big ( \frac{k-\ell }{2}\Big )^{\alpha } \!\!\nonumber \\&\qquad \qquad \qquad \qquad \int _0^1 \!\!(1-t) \, \partial _\xi ^\alpha {\widehat{a}}_\chi \Big (j-k, \frac{j+\ell + t(k- \ell )}{2} \Big ) {\text {d}}t u_\ell \, e^{{\text {i}}j \cdot x} \end{aligned}$$
(A.53)
$$\begin{aligned}&\qquad R_3(a,b) u := \sum _{j,k,\ell } \!\! - \Big ( \frac{k-\ell }{2}\Big )\! \cdot \partial _\xi {\widehat{a}}_\chi \Big (j-k, \frac{j+\ell }{2} \Big ) \Big ( \frac{j-k}{2}\Big )\! \cdot \!\!\nonumber \\&\qquad \qquad \qquad \qquad \int _0^1 \!\! \partial _\xi {\widehat{b}}_\chi \Big (k-\ell , \frac{j+\ell + t(k- j)}{2} \Big ) {\text {d}}t u_\ell \, e^{{\text {i}}j \cdot x} \end{aligned}$$
(A.54)
$$\begin{aligned}&\quad R_4(a,b) u := \sum _{j,k,\ell } {\widehat{a}}_\chi \Big (j-k, \frac{j+\ell }{2} \Big ) \!\! \sum _{|\beta | = 2} \Big ( \frac{k-j}{2}\Big )^{\beta } \!\!\nonumber \\&\qquad \qquad \qquad \qquad \int _0^1 \!\! (1-t) \, \partial _\xi ^\beta {\widehat{b}}_\chi \Big (k-\ell , \frac{j+\ell + t(k- j)}{2} \Big ) {\text {d}}t u_\ell \, e^{{\text {i}}j \cdot x}. \end{aligned}$$
(A.55)

We show now that the operators \(R_i(a,b)\), \( i = 1, \ldots , 4\) fulfill estimate (A.51).

Estimate of  \({R_1(a,b)}\). By exchanging the role of a and b it is enough to prove that the symbols \( \partial _\xi ^\alpha a_\chi \partial _x^\alpha b_\chi - (\partial _\xi ^\alpha a \,\partial _x^\alpha b)_\chi \), \(|\alpha |\le 1\), belong to \(\Sigma _{L^\infty }^{m+m'-\varrho }\) and then apply Theorem (A.7). The spectral localization property follows because of the cut-off \(\chi _\epsilon \) and \(\epsilon \) small. As \(\partial _x^\alpha \) commutes with the Fourier multiplier \( \chi _{\epsilon \left\langle \xi \right\rangle }(D) \) we have that \( \partial _x^\alpha b_\chi = (\partial _x^\alpha b)_\chi \) and we write \( \partial _{\xi }^\alpha a_\chi \, \partial _{x}^\alpha b_\chi - (\partial _{\xi }^\alpha a \, \partial _{x}^\alpha b)_\chi \) as

$$\begin{aligned}&(\partial _{\xi }^\alpha a)_\chi \left[ (\partial _{x}^\alpha b)_\chi - \partial _{x}^\alpha b\right] + \left[ (\partial _{\xi }^\alpha a)_\chi - \partial _{\xi }^\alpha a\right] \partial _x^\alpha b + \left[ \partial _\xi ^\alpha a\, \partial _x^\alpha b - (\partial _\xi ^\alpha a \, \partial _x^\alpha b)_\chi \right] \end{aligned}$$
(A.56)
$$\begin{aligned}&\quad +\left[ \partial _{\xi }^\alpha a_\chi - (\partial _{\xi }^\alpha a)_\chi \right] (\partial _{x}^\alpha b)_\chi . \end{aligned}$$
(A.57)

Consider first the term in (A.57). By Lemma A.5, \(\partial _{\xi }^\alpha a_\chi - (\partial _{\xi }^\alpha a)_\chi \in \Gamma ^{m-\varrho -|\alpha |}_{L^\infty }\) and \((\partial _{x}^\alpha b)_\chi \in \Gamma _{L^\infty }^{m'+|\alpha |}\) and by remark (v) after Definition 2.1, for any \( n \in {\mathbb {N}}_0 \),

$$\begin{aligned}&\left| \left[ \partial _{\xi }^\alpha a_\chi - (\partial _{\xi }^\alpha a)_\chi \right] (\partial _{x}^\alpha b)_\chi \right| _{m+m'-\varrho , L^\infty , n}\\&\quad \le \big | \partial _{\xi }^\alpha a_\chi - (\partial _{\xi }^\alpha a)_\chi \big |_{m-|\alpha |-\varrho , L^\infty , n} | (\partial _{x}^\alpha b)_\chi |_{m'+ |\alpha |, L^\infty , n} \\&\quad {\mathop {\lesssim }\limits ^{(A.23),(A.24)}}\left| a \right| _{m, W^{\varrho , \infty }, n+|\alpha |} \, \left| b \right| _{m', L^{\infty }, n}. \end{aligned}$$

Next consider the terms in (A.56). By remarks (iii), (iv) after Definition 2.1, we have \( \partial _{\xi }^\alpha a \in \Gamma ^{m-|\alpha |}_{W^{\varrho , \infty }}\subset \Gamma ^{m-|\alpha |}_{W^{\varrho - |\alpha |, \infty }}\), \(\partial _{x}^\alpha b \in \Gamma ^{m'}_{W^{\varrho -|\alpha |, \infty }}\), so we can apply Lemma A.5, property (A.49) and (2.10) to obtain

$$\begin{aligned} \left| (A.56) \right| _{m+m'-\varrho , L^\infty , n}&\lesssim \left| a \right| _{m, W^{\varrho -|\alpha |, \infty }, n+|\alpha |} \, \left| b \right| _{m', W^{|\alpha |, \infty }, N} + \left| a \right| _{m, L^\infty , n+|\alpha |} \, \left| b \right| _{m', W^{\varrho , \infty }, n} \nonumber \\&\lesssim \left| a \right| _{m, W^{\varrho , \infty }, n+1} \, \left| b \right| _{m', L^\infty , n+1} + \left| a \right| _{m, L^\infty , n+1} \, \left| b \right| _{m', W^{\varrho , \infty }, n+1} \end{aligned}$$
(A.58)

where to pass from the first to the second line we used the second interpolation inequality in (A.49). Altogether we have proved that the symbol in (A.52) belongs to \(\Sigma ^{m+m'-\varrho }_{L^\infty }\) and its seminorms are bounded by (A.58). Then Theorem (A.7) proves that \(R_1(a,b)\) fulfills estimate (A.51).

Estimate of \({R_{2}(a,b)}\). First we rewrite (A.53) as

$$\begin{aligned} R_{2}(a,b)u = \frac{1}{4} \sum _{j,\ell } \Big ( \int _0^1(1-t) \, \sum _{|\alpha |=2} {\widehat{f}}^\alpha _t (j-\ell , \ell ) \, {\text {d}}t \Big ) u_\ell \, e^{{\text {i}}j \cdot x} \end{aligned}$$

where

and \( D_{x_n} := \partial _{x_n} / {\text {i}}\) and \( D_x^\alpha := D_{x_1}^{\alpha _1} \cdots D_{x_d}^{\alpha _d} \). Then, recalling (A.27),

$$\begin{aligned} R_{2}(a,b) u = \frac{1}{4} \int _0^1(1-t) \sum _{|\alpha |=2} {{\text {Op}}}\left( f^\alpha _t \right) u \, {\text {d}}t \end{aligned}$$

where

(A.59)

We claim that \( {f_t^\alpha } (x, \xi ) \) is spectrally localized, namely

$$\begin{aligned} \exists \delta \in (0,1) :\ \ \ |n | \le \delta \left\langle \xi \right\rangle , \quad \forall (n, \xi ) \in {{\text {supp }}}\,{\widehat{f_t^{\alpha }}}. \end{aligned}$$
(A.60)

In fact on the support of \({\widehat{b}}_\chi \big (j, \xi +\frac{j}{2} \big )\) we have, for some \( \delta ' \in (0,1) \),

$$\begin{aligned} |j| \le \delta ' \langle \xi \rangle , \end{aligned}$$
(A.61)

whereas, on the support of \( \partial _\xi ^\alpha {\widehat{a}}_\chi \big (n -j, \xi + \frac{n + tj}{2} \big )\), \( t \in [0,1] \),

$$\begin{aligned}&| n - j | \le \delta \langle \xi \rangle + \delta \langle n \rangle + \delta \langle j \rangle {\mathop {\le }\limits ^{(A.61)}}(\delta + \delta \delta ' ) \langle \xi \rangle + \delta \langle n \rangle . \end{aligned}$$
(A.62)

The estimates (A.61)–(A.62) then give \( | n | \le | j | +| n - j | \le \delta ' \langle \xi \rangle + (\delta + \delta \delta ' ) \langle \xi \rangle + \delta \langle n \rangle \), which implies (A.60).

In order to apply Theorem (A.7) it remains to prove that, for any \( N \ge 3d+4 \),

$$\begin{aligned} | f_t^\alpha (x,\xi )|_{m+m'-\varrho , L^\infty , d+1}\lesssim | b |_{m', W^{\varrho ,\infty }, N} \ | a |_{m, L^\infty , N}, \end{aligned}$$
(A.63)

which implies, for any \(s\in {\mathbb {R}}\), \( u \in \dot{H}^s \), \( \Vert R_{2}(a,b)u\Vert _{{s-m-m'+\varrho }}\lesssim | b |_{m', W^{\varrho ,\infty }, N}\ | a |_{m, L^\infty , N}\, \Vert u\Vert _{s} \). Thus \(R_2(a,b)\) satisfies the estimate (A.51).

In order to prove (A.63) note that, differentiating (A.59), for any \( \beta \in {\mathbb {N}}_0^{d} \),

(A.64)

where \( C_{\beta _1, \beta _2} \) are binomial coefficients and

$$\begin{aligned} K_t^{\beta _1, \beta _2} (x,y,z):= & {} \frac{1}{(2\pi )^{2d}} \sum _{n,j} (\partial _\xi ^{\beta _1} D_x^\alpha b_\chi )\Big (x-z-y,\xi +\frac{j}{2} \Big )\nonumber \\&\partial _\xi ^{\alpha +\beta _2} a_\chi \Big (x-z, \, \xi + \frac{n + tj}{2} \Big ) e^{{\text {i}}(n \cdot z + j \cdot y)}. \end{aligned}$$
(A.65)

By (A.60) and (A.61) the sum over n in (A.59) is restricted to indexes satisfying

$$\begin{aligned} |n| \ll \left\langle \xi \right\rangle , \ |j|\ll \langle \xi \rangle , \qquad \text { and} \ \text {therefore} \qquad \langle \xi +\frac{j}{2}\rangle \sim \langle \xi + \frac{n +tj}{2}\rangle \sim \langle \xi \rangle . \end{aligned}$$

We deduce that the sum in (A.65) is bounded by

$$\begin{aligned} \big | K_t^{\beta _1, \beta _2}(x,y,z) \big |&\lesssim \langle \xi \rangle ^{2d + m + m' - |\beta | -\varrho } \, | \partial _\xi ^{\beta _1} D_x^\alpha b_\chi |_{m'-|\beta _1|+|\alpha |-\varrho , L^\infty , 0} \nonumber \\&\qquad \quad \times | \partial _\xi ^{\alpha +\beta _2} a_\chi |_{m -|\alpha |-|\beta _2|, L^\infty , 0} \nonumber \\&{\mathop {\lesssim }\limits ^{(2.10),(A.24),(A.21)}}\langle \xi \rangle ^{2d + m + m' - |\beta |- \varrho } \, |b |_{m', W^{\varrho , \infty }, |\beta |} \, | a |_{m, L^\infty , 2+|\beta |}, \end{aligned}$$
(A.66)

recalling that \(|\alpha | = 2 \). We also estimate \(K_t^{\beta _1, \beta _2} (x, y, z)\) applying Abel resummation formula (A.3) in the sum (A.65), in the index n and in the index j separately, obtaining, using (A.24), (A.21), (A.2) and (2.10),

$$\begin{aligned} \big | K_t^{\beta _1, \beta _2}(x,y,z) \big |&\lesssim \langle \xi \rangle ^{2d+m+m'-|\beta |-\varrho } \, | b|_{m', W^{\varrho ,\infty } , 2d+1+|\beta |} | a |_{m, L^\infty , 2d+3+|\beta |}\nonumber \\&\quad \times \min _{1\le h \le d} \Big ( \left| \langle \xi \rangle 2 \sin \frac{y_h}{2} \right| ^{-(2d+1)}, \ \left| \langle \xi \rangle 2\sin \frac{z_h}{2} \right| ^{-(2d+1)} \Big ). \end{aligned}$$
(A.67)

In view of (A.66)–(A.67) and \( |\beta | \le d+ 1 \), we apply Lemma A.1 with \(d \leadsto 2d\), choosing \(A=\langle \xi \rangle \), \( B=\langle \xi \rangle ^{m+m'-|\beta |-\varrho } \, | b|_{m', W^{\varrho ,\infty } , 2d+1+|\beta |} \, | a |_{m, L^\infty , 2d+3+|\beta |} \) and we obtain

$$\begin{aligned} \Vert \partial _\xi ^{\beta } f^\alpha _t(\cdot ,\xi ) \Vert _{L^\infty }\lesssim & {} \int _{{\mathbb {T}}^{2d}} |K_t^{\beta _1, \beta _2} (x, y, z)| \, {\text {d}}y\, {\text {d}}z \lesssim \langle \xi \rangle ^{m+m'-\varrho -|\beta |} \, | b|_{m', W^{\varrho ,\infty } , 3d+2} \\&| a |_{m, L^\infty , 3d+4} \end{aligned}$$

proving (A.63).

The proof that \(R_3(a,b)\) and \(R_4(a,b)\) satisfy the estimate (A.51) follows similarly. \(\square \)

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Berti, M., Maspero, A. & Murgante, F. Local Well Posedness of the Euler–Korteweg Equations on \({{\mathbb {T}}^d}\). J Dyn Diff Equat 33, 1475–1513 (2021). https://doi.org/10.1007/s10884-020-09927-3

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