In page 24 of the paper Landau Damping: Paraproducts and Gevrey Regularity, the authors claimed an algebra property of Gevrey spaces, the formula (3.14), without giving a proof. So I'm asking for a hint here.

Let $\mathbb{T}=[0,2\pi]$. For a function $f=f(z,v): \mathbb{T}^d\times\mathbb R^d\to\mathbb C$, we define its Fourier transform by $$\hat f_k(\eta):=\frac1{(2\pi)^d}\int_{\mathbb{T}^d\times\mathbb R^d}e^{-ik\cdot z-iv\cdot\eta}f(z,v)\,dz\,dv,\qquad k\in\mathbb Z^d,\eta\in\mathbb R^d.$$ And then we define the Gevery space $\mathcal{G}^{\lambda,\sigma;\nu}(\mathbb{T}^d\times\mathbb R^d)$ for $\lambda>0, \sigma\in\mathbb R, \nu\in(0,1]$: $$\|f\|_{\mathcal{G}^{\lambda,\sigma;\nu}}^2\ :=\sum_{k\in\mathbb Z^d}\int_{\mathbb R^d}|\hat f_k(\eta)|^2\langle k,\eta\rangle^{2\sigma}e^{2\lambda\langle k,\eta\rangle^\nu}\,d\eta,$$ where $\langle k,\eta\rangle=(1+|k,\eta|^2)^\frac12$ and $|k,\eta|=|k_1|+|k_2|+\cdots+|k_d|+|\eta_1|+|\eta_2|+\cdots+|\eta_d|$.

For a function $\rho=\rho(t,x): \mathbb R\times\mathbb T^d\to\mathbb R$, denote $\hat\rho_k(t)$ for the Fourier coefficients of $\rho(t)$, i.e. $$\hat\rho_k(t)=\int_{\mathbb T^d}\rho(t,x)e^{-ik\cdot x}\,dx.$$ We define $$\|\rho(t)\|_{\mathcal{F}^{\lambda,\sigma;\nu}}^2\ :=\sum_{k\in\mathbb Z^d}|\hat\rho_k(t)|^2\langle k,kt\rangle^{2\sigma}e^{2\lambda\langle k,kt\rangle^\nu}.$$

The mentioned formula (3.14) is as follows:

Let $0<s<1, \sigma\geq 0,\lambda>0$, and $g\in\mathcal{G}^{\lambda,\sigma;s}(\mathbb{T}^d\times\mathbb R^d)$, $r(t)\in\mathcal{F}^{\lambda,\sigma;s}(\mathbb T^d)$. Then $r(t)g\in\mathcal{G}^{\lambda,\sigma;s}(\mathbb{T}^d\times\mathbb R^d)$, and $$\sum_{k\in\mathbb Z^d}\int_{\mathbb R^d}\left|\sum_{\ell\in\mathbb Z^d}e^{\lambda\langle k,\eta\rangle^s}\langle k,\eta\rangle^\sigma\hat r_\ell(t)\hat g_{k-\ell}(\eta-\ell t)\right|^2\,d\eta\lesssim\|r(t)\|_{\mathcal{F}^{\lambda,\sigma;s}}^2\ \ \ \|g\|_{\mathcal{G}^{\lambda,\sigma;s}}^2$$

I can only prove the case in which $\sigma>\frac d2$, by an application of Lemma 3.1 in page 20. For the general case, I guess the paraproduct decompositions will be useful. But I got very messy.

Any help will be appreciated!



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