| Russian Mathematical Surveys |
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Brief communications Boundedness of toroidal multilinear pseudodifferential operators with symbols in Hörmander classes D. B. Bazarkhanov Institute of Mathematics and Math Modeling, the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan, Almaty, Kazakhstan
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    Let $n\in \mathbb{N}$, $n\geqslant2$, and $\mathbb{N}_0=\mathbb{N} \cup \{0\}$. For $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n) \in \mathbb{R}^n$ we put $xy=x_1y_1+\cdots+x_ny_n$. In what follows $\mathcal{S}:=\mathcal{S}(\mathbb{R}^n)$ and $\mathcal{S}':=\mathcal{S}'(\mathbb{R}^n)$ are the Schwartz spaces of test functions and tempered distributions, respectively, $\widetilde{\mathcal{S}}':=\mathcal{S}(\mathbb{T}^n)$ is the space of all distributions $f\in\mathcal{S}'$ that are $1$-periodic in each variable (that is, such that $\langle f,\varphi(\cdot+\xi)\rangle=\langle f,\varphi\rangle$ for all $\varphi\in\mathcal{S}$ and $\xi\in\mathbb{Z}^m$), $\widetilde{\mathcal{S}}:=\mathcal{S}(\mathbb{T}^m)$ is the space of all infinitely differentiable functions on $\mathbb{T}^m$, $\widehat{\varphi}$ is the Fourier transform of $\varphi\in \mathcal{S}$, and $\widehat{u}$ is the Fourier coefficient of $u\in \widetilde{\mathcal{S}}$:
$$
\begin{equation*}
\widehat{\varphi}(\xi)= \int_{\mathbb{R}^n}\varphi(x)e^{-2\pi i\,\xi x}\,dx, \quad \xi\in\mathbb{R}^n, \qquad \widehat{u}(\xi)=\int_{\mathbb{T}^n} u(x) e^{-2\pi i\,\xi x}\,dx,\quad \xi\in\mathbb{Z}^n.
\end{equation*}
\notag
$$
We consider a multilinear pseudodifferential operator of the form
$$
\begin{equation}
T_a(u_1,\dots,u_N; x):=\int_{\Xi\in\mathbb{R}^{nN}} a(x,\xi^1,\dots,\xi^N) \prod_{\nu=1}^N\widehat{u}_\nu(\xi^\nu)e^{2\pi i \xi^\nu x}\,d\xi^\nu
\end{equation}
\tag{1}
$$
($x\in \mathbb{R}^n$, $\Xi=(\xi^1,\dots,\xi^N)$, $\xi^\nu\in\mathbb{R}^n$, and $u_\nu \in \mathcal{S}$, $\nu=1,\dots,N$) and its toroidal analogue
$$
\begin{equation}
\widetilde{T}_a(u_1,\dots,u_N;x):=\sum_{\Xi\in\mathbb{Z}^{nN}} a(x,\xi^1,\dots,\xi^N)\prod_{\nu=1}^N\widehat{u}_\nu(\xi^\nu) e^{2\pi i \xi^\nu x}
\end{equation}
\tag{2}
$$
($x\in \mathbb{T}^n,\, \Xi=(\xi^1, \dots, \xi^N),\, \xi^\nu\in\mathbb{Z}^n$ and $u_\nu \in \widetilde{\mathcal{S}},\, \nu=1, \dots, N$).
Operators (1) were introduced by Meyer and Coifman in the mid-1970s and play an important role in harmonic analysis and its applications (see [1]–[3]). Let $X_1(\mathbb{I}^n),\dots,X_N(\mathbb{I}^n)$, and $Y(\mathbb{I}^n)$ (where $\mathbb{I} \in \{\mathbb{R},\mathbb{T}\}$) be function spaces on $\mathbb{I}^n$, with (quasi)norms $\|\cdot|\, X_1(\mathbb{I}^n)\|,\dots, \|\cdot|\, X_N(\mathbb{I}^n)\|$, and $\|\cdot\,|Y(\mathbb{I}^n)\|$, respectively. Assume that $\mathcal{S}(\mathbb{I}^n)\subset X_\nu(\mathbb{I}^n)$, $\nu=1,\dots,N$. Let $R_a:=T_a$ for $\mathbb{I}=\mathbb{R}$ and $R_a:=\widetilde{T}_a$ for $\mathbb{I}=\mathbb{T}$. If there is a positive constant $C$ such that the inequality $\|R_a(u_1,\dots,u_n) \mid Y(\mathbb{I}^n)\| \leqslant C\prod_{\nu=1}^N\|u_\nu \mid X_\nu(\mathbb{I}^n)|$ is valid for all $(u_1,\dots,u_N)\in \mathcal{S}(\mathbb{I}^n)^N$, then we say that the operator $R_a$ is bounded from $X_1(\mathbb{I}^n)\times \cdots \times X_N(\mathbb{I}^n)$ to $Y(\mathbb{I}^n)$ and write $R_a\colon X_1(\mathbb{I}^n)\times\cdots\times X_N(\mathbb{I}^n)\to Y(\mathbb{I}^n)$. The boundedness of operators (1) has intensively been studied for various symbols and functions spaces; see, for instance, [1]–[6] and the references there. The Hörmander class $S_{\rho\delta}^m(\mathbb{R}^n;N)$ ($m\in\mathbb{R}$, $0\leqslant\delta, \rho\leqslant 1$) consists of all symbols $a\in C^\infty(\mathbb{R}^{n(N+1)})$ for which the following condition is satisfied: for all $\alpha,\beta^1,\dots,\beta^N \in \mathbb{N}_0^n$ there exists $C=C(\alpha,\beta^1,\dots,\beta^N)>0$ such that the differential inequality
$$
\begin{equation*}
|\partial^\alpha_x \partial^{\beta^1}_{\xi^1}\cdots \partial^{\beta^N}_{\xi^N}a(x,\xi^1,\dots,\xi^N)|\leqslant C\langle \Xi\rangle^{m+\delta|\alpha|-\rho(|\beta^1|+\cdots+ |\beta^N|)}\quad (x\in\mathbb{R}^n, \ \ \Xi\in\mathbb{R}^{nN}),
\end{equation*}
\notag
$$
holds true. Here $\langle \Xi \rangle=(1+\sum_{\nu=1}^N\xi^\nu\xi^\nu)^{1/2}$.
The Hörmander class $S_{\rho\delta}^m(\mathbb{T}^n; N)$ ($m\in\mathbb{R}$, $0\leqslant\delta, \rho\leqslant 1$) of toroidal symbols $a \colon \mathbb{R}^{n}\times\mathbb{Z}^{nN}\to\mathbb{C}$ ($a(\,\cdot\,,\Xi) \in C^\infty(\mathbb{T}^{n})$ for each $\Xi\in\mathbb{Z}^{nN}$) is defined analogously — with $\Delta^{\beta^\nu}_{\xi^\nu}$ (a finite difference of order $\beta^\nu$ with step $1$) and $\mathbb{Z}^{nN}$ in place of $\partial^{\beta^\nu}_{\xi^\nu}$ and $\mathbb{R}^{nN}$. We fix $\phi\in \mathcal{S}$ such that $\displaystyle\int_{\mathbb{R}^n} \phi=1$. Put $\phi_t(x):=t^{-n}\phi(x/t)$. Below $\widetilde{X}_p:=\widetilde{H}_p$ for $0< p\leqslant 1$, $\widetilde{X}_p:=\widetilde{L}_p$ for $1< p\leqslant\infty$, $\widetilde{Y}_r:=\widetilde{L}_r$ for $0< r< \infty$, and $\widetilde{Y}_\infty:=\widetilde{\operatorname{BMO}}$. Here $\widetilde{L}_p:=L_p(\mathbb{T}^n)$ ($0< p\leqslant\infty$) is the Lebesgue space endowed with the standard (quasi)norm $\|\cdot |\, \widetilde{L}_p\|$, $\widetilde{\operatorname{BMO}}:=\operatorname{BMO}(\mathbb{T}^n)$ is the space of functions of bounded mean oscillation on $\mathbb{R}^n$ which are 1-periodic in each variable, and
$$
\begin{equation*}
\widetilde{H}_p:=H_p(\mathbb{T}^n):=\Bigl\{f\in\mathcal{S}'(\mathbb{T}^n)\bigm|\|f\,|\,\widetilde{H}_p\| :=\Bigl\|\,\sup_{t>0} |\widetilde{\phi}_t\ast f|\Bigm|\widetilde{L}_p\Bigr\|<\infty\Bigr\}
\end{equation*}
\notag
$$
is the (real) Hardy space, $0< p\leqslant1$ (the periodization $\widetilde{g}\colon \mathbb{T}^m \to \mathbb{C}$ of the function $g\colon \mathbb{R}^m \to \mathbb{C}$ is defined as the (formal) sum of the series $\sum_{\xi\in \mathbb{Z}^m}g(x+\xi)$).
For $0 \leqslant \rho < 1$ and $0 < p,q,r \leqslant \infty$ such that $1/p+1/q=1/r$, we defined the quantities $m_\rho(p,q)=(1-\rho)m_0(p, q)$ and $m_0(p,q)=-n\max\{1/2,1/p,1/q,1- 1/r,1/r-1/2\}$. Theorem 1. Let $0\leqslant\rho<1$, and let $0< p, q, r\leqslant\infty$ satisfy $1/p+1/q=1/r$. Then $\sup\{ m\in \mathbb{R} \mid \widetilde{T}_a \colon \widetilde{X}_p \times \widetilde{X}_q \to \widetilde{Y}_r$ for all $a\in S^m_{\rho\rho}(\mathbb{T}^n; 2)\}=m_\rho(p,q)$. Moreover, $\widetilde{T}_a\colon \widetilde{X}_p \times \widetilde{X}_q \to \widetilde{Y}_r$ for each $a\in S^{m_0(p,q)}_{0 0}(\mathbb{R}^n;2)$. If, in addition, $0\leqslant\rho<1$ and $\min\{p,q,r\}\geqslant1$, then $\widetilde{T}_a\colon \widetilde{X}_p \times \widetilde{X}_q \to \widetilde{Y}_r$ for any $a\in S^{m_\rho(p,q)}_{\rho\rho}(\mathbb{T}^n;2)$. Theorem 2. Let $N\geqslant2$, $0< r,p_1,\dots,p_N\leqslant\infty$, $1/r \leqslant 1/p_1+\cdots+1/p_N$, and $m\in\mathbb{R}$. Then $\widetilde{T}_a \colon \widetilde{X}_{p_1}\times \cdots \times \widetilde{X}_{p_N} \to \widetilde{Y}_r$ for all $a\in S^m_{0 0}(\mathbb{T}^n;N)$ if and only if $m\leqslant n\bigl(\min\{1/p,1/2\}-\sum_{\nu=1}^N\max\{1/p_\nu,1/2\}\bigr)$. Remark. Theorems 1 and 2 are natural analogues of an important result of Miyachi and Tomita [4], [5] and the main result in [6], respectively. Notice the (only known to us) result in [7] on toroidal multiplinear PDOs with symbols in Hörmander classes: if $1\leqslant r<\infty$ and $1\leqslant p_\nu \leqslant\infty$ satisfy $1/r=1/p_1+\cdots+1/p_N$ ($N\geqslant2$), then for each $a\in S_{10}^0(\mathbb{T}^n;N)$ The author would like to thank the Isaac Newton Institute in Cambridge, UK, for their support and hospitality during the programme “Discretization and recovery in high-dimensional spaces” (1–26 July 2024) where the work on this paper was undertaken. 
Citation in format AMSBIB
\Bibitem{Baz25} |
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