2 Preliminaries

In this section, we recall some necessary definitions and results related to the STFT and Orlicz spaces. Let us start with some basic definitions.

Notice that, $\mathcal {F}_{\mathbb {Z}^n}=\mathcal {F}_{{\mathbb {T}}^n}^{-1}=\mathcal {F}_{{\mathbb {T}}^n}^{*} $ , and for $F \in \ell ^{2}(\mathbb {Z}^n)$ , $\left \|\mathcal {F}_{\mathbb {Z}^n} F\right \|_{L^{2}\left ({\mathbb {T}}^n\right )}=\|F\|_{\ell ^{2}(\mathbb {Z}^n)}$ . Moreover, $\mathcal {F}_{\mathbb {Z}^n}: \ell ^{2}(\mathbb {Z}^n) \rightarrow L^{2} ({\mathbb {T}}^n)$ is a surjective isomorphism.

Next, we define the Schwartz space $\mathcal {S}\left (\mathbb {Z}^{n}\right )$ on $\mathbb {Z}^{n}$ to be the space of rapidly decreasing functions $g: \mathbb {Z}^{n} \rightarrow \mathbb {C}$ . That is, $g \in \mathcal {S}\left (\mathbb {Z}^{n}\right )$ if for any $M<\infty ,$ there exists a constant $C_{g, M}$ for which

$$ \begin{align*}|g(k)| \leq C_{g, M} (1+|k|)^{-M}, \quad \text { for all } k \in \mathbb{Z}^{n}.\end{align*} $$

For $j \in \mathbb {N}_{0}= {\mathbb {N}} \cup \{0\}$ , we define the seminorms $p_{j}(g):=\sup \limits _{k \in \mathbb {Z}^{n}}(1+|k|)^{j}|g(k)|$ . Then, the topology on $\mathcal {S}\left (\mathbb {Z}^{n}\right )$ is given by seminorms $p_{j}$ . Also, we define the space of tempered distributions $\mathcal {S}^{\prime }\left (\mathbb {Z}^{n}\right )$ to be the space of all continuous linear functionals on $\mathcal {S}\left (\mathbb {Z}^{n}\right )$ .

Now, we define the STFT on ${\mathbb {Z}}^n \times {\mathbb {T}}^n$ . Let $f \in \ell ^2({\mathbb {Z}}^n)$ , and fix $k \in {\mathbb {Z}}^n$ , $w \in {\mathbb {T}}^n$ . For $m \in {\mathbb {Z}}^n$ , we define the translation operator $T_k$ by $T_k f(m)=f(m-k)$ and the modulation operator $M_w$ by $M_w f(m)=e^{2 \pi i w \cdot m} f(m)$ . For a fixed window function $g \in \mathcal {S}\left (\mathbb {Z}^{n}\right )$ , we define the STFT of a function $f \in \mathcal {S}^{\prime }\left (\mathbb {Z}^{n}\right )$ with respect to g to be the function on ${\mathbb {Z}}^n \times {\mathbb {T}}^n$ given by

$$ \begin{align*} V_g f (m, w)= \left\langle f, M_w T_m g \right\rangle = \sum_{k \in {\mathbb{Z}}^n} f(k) \overline{M_w T_m g (k)} = \sum_{k \in {\mathbb{Z}}^n} f(k) \overline{g(k-m)} e^{-2 \pi i w \cdot k}. \end{align*} $$

Let $\tilde {g}(k)=g(-k)$ , for $k \in {\mathbb {Z}}^n$ . Using the convolution on ${\mathbb {Z}}^n$ , we write $V_g f$ as

$$\begin{align*}V_g f (m, w)= e^{-2 \pi i w \cdot m} \left( f * M_w \overline{\tilde{g}} \right)(m). \end{align*}$$

The STFT on ${\mathbb {Z}}^n \times {\mathbb {T}}^n$ satisfies the following properties (see [Reference Dasgupta and Poria8]).

Next, we define the Orlicz spaces on ${\mathbb {Z}}^n \times {\mathbb {T}}^n$ . We first need to define the convex function. A function $\Phi :[0, \infty ] \rightarrow [0, \infty ]$ is called convex if

$$ \begin{align*}\Phi\left(x_1 y_1+x_2 y_2\right) \leq x_1 \Phi\left(y_1\right)+x_2 \Phi\left(y_2\right),\end{align*} $$

where $x_j, y_j \in \mathbb {R}$ satisfy $x_j, y_j \geq 0$ for $j=1,2$ and $x_1+x_2=1$ .

We recall the definition of the Young function and quasi-Young function (see [Reference Gumber, Rana, Toft and Üster19]).

Now, we recall the definition of the continuous and discrete Orlicz spaces (see [Reference Gumber, Rana, Toft and Üster19, Reference Maligranda and Mastylo21]).

Note that, if $\Phi (t):=t^p$ for some $p \geq 1$ , then $L^{\Phi } (\mathbb {R}^n)=L^p (\mathbb {R}^n)$ , the Lebesgue spaces of p the integrable functions on $\mathbb {R}^n$ . Hence, the continuous Orlicz spaces are the generalization of the Lebesgue spaces. Also, note that, if $\Phi (t)=t^p$ for some $p \geq 1$ , then we get $\ell ^{\Phi }(\mathbb {Z}^n)=\ell ^p (\mathbb {Z}^n)$ . Therefore, the discrete Orlicz spaces are the generalization of $\ell ^p (\mathbb {Z}^n)$ spaces. Next, we will define Orlicz spaces on ${\mathbb {Z}}^n \times {\mathbb {T}}^n$ parameterized with two (quasi-)Young functions.

In this article, we mainly assume that $\Phi $ , $\Phi _1$ , and $\Phi _2$ above are Young functions.

3 Orlicz modulation spaces on ${\mathbb {Z}}^n$

In this section, we define and study Orlicz modulation spaces on ${\mathbb {Z}}^n$ . Modulation spaces were first introduced by Feichtinger in [Reference Feichtinger, Krishna, Radha and Thangavelu14, Reference Feichtinger and Gröchenig15]. To define discrete Orlicz modulation spaces, it is essential to revisit the concept of modulation spaces defined on ${\mathbb {Z}}^n$ (see [Reference Dasgupta and Poria8]).

Definition 3.1 Let $1 \leq p \leq \infty $ and $g \in \mathcal {S}({\mathbb {Z}}^n)$ . We define the modulation space $M^{p}({\mathbb {Z}}^n)$ to be the space of all tempered distributions $f \in \mathcal {S'}({\mathbb {Z}}^n)$ for which $V_g f \in L^{p}({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . The norm on $M^{p}({\mathbb {Z}}^n)$ is

$$ \begin{align*} \Vert f \Vert_{M^{p}({\mathbb{Z}}^n)} = \Vert V_g f \Vert_{L^{p}({\mathbb{Z}}^n \times {\mathbb{T}}^n)} = \bigg( \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |V_g f(m,w)|^p \; dw \bigg)^{1/p} < \infty, \end{align*} $$

with the usual adjustments if p is infinite.

We have the following inclusions:

$$\begin{align*}\mathcal{S}({\mathbb{Z}}^n) \subset M^1({\mathbb{Z}}^n) \subset M^2({\mathbb{Z}}^n)=\ell^2({\mathbb{Z}}^n) \subset M^\infty({\mathbb{Z}}^n) \subset \mathcal{S'}({\mathbb{Z}}^n).\end{align*}$$

In particular, $M^p({\mathbb {Z}}^n) \hookrightarrow \ell ^p({\mathbb {Z}}^n)$ for $1 \leq p \leq 2$ , and $\ell ^p({\mathbb {Z}}^n) \hookrightarrow M^p({\mathbb {Z}}^n)$ for $2 \leq p \leq \infty $ . Moreover, for $p < \infty $ , $(M^{p}({\mathbb {Z}}^n))^{'} =M^{p'}({\mathbb {Z}}^n)$ , where $p'$ is the conjugate exponent of p. We have similar inclusion relations for modulation spaces on ${\mathbb {Z}}^n \times {\mathbb {T}}^n$ , which can be derived by employing techniques similar to those used in the study of modulation spaces on locally compact abelian groups (see [Reference Bastianoni and Cordero1, Reference Feichtinger, Krishna, Radha and Thangavelu14]). For more information on the properties and applications of modulation spaces, we refer to the book by Gröchenig [Reference Gröchenig18].

Note that the definitions of the discrete Orlicz modulation spaces are independent of the choice of the window function $g \in \mathcal {S}({\mathbb {Z}}^n)$ . In addition, the difference between the spaces $M^{\Phi , \Psi }({\mathbb {Z}}^n)$ and $W^{\Phi , \Psi }({\mathbb {Z}}^n)$ is in terms of their topological differences. The topology of $M^{\Phi , \Psi }({\mathbb {Z}}^n)$ is induced by the mixed Orlicz space $L^{\Phi , \Psi }({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ norm, whereas the topology of $W^{\Phi , \Psi }({\mathbb {Z}}^n)$ is induced by the mixed Orlicz space $L_{*}^{\Phi , \Psi }({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ norm.

Next, we present some basic properties of discrete Orlicz modulation spaces. For some recent investigations of Orlicz modulation spaces, we refer to [Reference Gumber, Rana, Toft and Üster19, Reference Toft, Üster, Nabizadeh and Öztop27]. We use q as the conjugate exponent of p to define the dual space for a Lebesgue space. Similar to the theory of Lebesgue spaces, we can define the complementary function as a counterpart to the conjugate exponent. Also, in the theory of Orlicz spaces, the Young functions are classified using their growth properties. In particular, the $\Delta _2$ -condition plays an important role in defining the dual space of an Orlicz space (see [Reference Schnackers and Führ25]).

Next, we give a characterization of the dual space to the Orlicz space on ${\mathbb {Z}}^n \times {\mathbb {T}}^n$ . If $(\Phi , \Psi )$ is a complementary Young pair and $\Phi $ satisfies a local $\Delta _2$ -condition, then $\left ( L^{\Phi } (\mathbb {R}^n)\right )^*$ is isometrically isomorphic to $L^{\Psi } (\mathbb {R}^n)$ . Similarly, we can show that $\left (\ell ^{\Phi } ({\mathbb {Z}}^n)\right )^*$ is isometrically isomorphic to $\ell ^{\Psi } ({\mathbb {Z}}^n)$ . Let $\left (\Phi _i, \Psi _i \right )$ be complementary Young pairs which satisfy local $\Delta _2$ -condition and are strictly convex for $i = 1, 2$ . Then, $\left (L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)\right )^*$ is isometrically isomorphic to $L^{\Psi _1, \Psi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . The proofs of these properties can be obtained using a similar method as in [Reference Rao and Ren24].

Let $\left (\Phi _i, \Psi _i \right )$ be complementary Young pairs for $i = 1, 2$ . If $f \in \ell ^{\Phi _1} ({\mathbb {Z}}^n)$ and $g \in \ell ^{\Psi _1} ({\mathbb {Z}}^n)$ , then we have the following Hölder’s inequality for the Orlicz spaces:

(6) $$ \begin{align} \| fg \|_{\ell^{1} ({\mathbb{Z}}^n)} \leq \|f\|_{\ell^{\Phi_1} ({\mathbb{Z}}^n)} \|g\|_{\ell^{\Psi_1} ({\mathbb{Z}}^n)}. \end{align} $$

In addition, if we assume that $\Phi _2$ satisfies a local $\Delta _2$ -condition, then for $F \in L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n) $ and $G \in L^{\Psi _1, \Psi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , we have the following Hölder’s inequality for the Orlicz spaces:

(7) $$ \begin{align} \| FG \|_{L^{1} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \leq \|F\|_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \|G\|_{ L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align} $$

The proofs of inequalities (6) and (7) can be obtained using a similar method as in [Reference Rao and Ren24]. If $\Phi $ is continuous, then the Schwartz class $\mathcal {S}({\mathbb {Z}}^n)$ is embedded into the Orlicz space $\ell ^{\Phi } ({\mathbb {Z}}^n)$ . Also, if the complementary function $\Psi $ is continuous then the functions in the Orlicz space define tempered distributions on ${\mathbb {Z}}^n$ . More precisely, let $\left (\Phi _i, \Psi _i \right )$ be complementary Young pairs and $\Phi _i$ be continuous for $i = 1, 2$ , then we have the following inclusions:

$$ \begin{align*}\mathcal{S}({\mathbb{Z}}^n) \subset \ell^{\Phi_1} ({\mathbb{Z}}^n) \subset \mathcal{S'}({\mathbb{Z}}^n),\end{align*} $$

if $\Psi _1 $ is continuous. Also,

$$ \begin{align*}\mathcal{S}({\mathbb{Z}}^n \times {\mathbb{T}}^n) \subset L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n) \subset \mathcal{S'}({\mathbb{Z}}^n \times {\mathbb{T}}^n),\end{align*} $$

if $\Psi _1$ and $ \Psi _2 $ are continuous. If $\Phi _1$ and $\Phi _2$ are (quasi-)Young functions, then $L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ is translation invariant, which leads to the fact that $M^{\Phi _1, \Phi _2}({\mathbb {Z}}^n)$ is translation and modulation invariant (see [Reference Toft, Üster, Nabizadeh and Öztop27]). Using the fact that $\left (\ell ^{\Phi } ({\mathbb {Z}}^n)\right )^*$ is isometrically isomorphic to $\ell ^{\Psi } ({\mathbb {Z}}^n)$ , we can extend the convolution relation $\ell ^{1} ({\mathbb {Z}}^n) * \ell ^{p} ({\mathbb {Z}}^n) \subset \ell ^{p} ({\mathbb {Z}}^n)$ to the Orlicz spaces. Here, we present the following convolution relations.

Proof (1) Let $\left (\Phi _i, \Psi _i \right )$ be complementary Young pairs which satisfy local $\Delta _2$ -condition and are strictly convex for $i = 1, 2$ . Then, $\left (L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)\right )^*$ is isometrically isomorphic to $L^{\Psi _1, \Psi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . If $G \in L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , then $T_{(l,x)} G \in L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $\| T_{(l,x)} G \|_{L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)}= \| G \|_{L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)}$ . Let $H \in L^{\Psi _1, \Psi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . Using Hölder’s inequality (7), we obtain

$$ \begin{align*} |\langle F * G, H \rangle| & = \left| \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} F*G(m,w) \; \overline{H(m,w)} \; dw \right| \\ & \leq \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \left( \sum_{l \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \left| G(m-l, w-x) \right|\; |F(l,x)| \; dx \right) |H(m,w)| \; dw \\ & = \sum_{l \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \left( \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \left| T_{(l,x)} G(m, w) \right|\; |H(m,w)| \; dw \right) |F(l,x)| \; dx \\ & \leq \sum_{l \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |F(l,x)| \; \| T_{(l,x)} G \|_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \|H\|_{L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; dx \\ & = \| G \|_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \|H\|_{L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \sum_{l \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |F(l,x)| \; dx \\ & = \|F\|_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \| G \|_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \|H\|_{L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align*} $$

By duality, we get

$$ \begin{align*} \|F * G\|_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} & = \sup\left\{|\langle F * G, H \rangle|:\|H\|_{L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \leq 1 \right\} \\ & \leq \|F\|_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \|G\|_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align*} $$

(2) The proof follows similarly as in the first part of the proof by choosing $\Phi _1 = \Phi _2 = \Phi $ .

Now, we study a few properties of the discrete Orlicz modulation spaces. Note that, the definitions of these spaces are independent of the choice of the window function g. Also, if the Young function satisfies a local $\Delta _2$ -condition, these spaces are Banach spaces. Moreover, if the Young functions are also strictly convex, then the mixed-norm discrete Orlicz modulation spaces are Banach spaces (see [Reference Schnackers and Führ25, Theorems 6 and 7]).

The proof of this theorem can be obtained using a similar method as discussed in [Reference Schnackers and Führ25]. So, we skip the proof here. Next, we discuss the duality properties of the discrete Orlicz modulation spaces. If $\left (\Phi , \Psi \right )$ is a complementary Young pair, and $\Phi $ satisfies a local $\Delta _2$ -condition and continuous, then $(M^{\Phi }({\mathbb {Z}}^n))^* \cong M^{\Psi }({\mathbb {Z}}^n)$ under the duality relation

$$\begin{align*}\langle f,h \rangle = \left\langle V_{g_0} f, V_{g_0} h \right\rangle = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} V_{g_0} f(m,w) \; \overline{V_{g_0} h(m,w)} \; dw \end{align*}$$

for $f \in M^{\Phi }({\mathbb {Z}}^n)$ and $h \in M^{\Psi }({\mathbb {Z}}^n)$ , $g_0 \in \mathcal {S}({\mathbb {Z}}^n)$ . Note that the duality relation is independent of the choice of the window function $g_0$ . If $\left (\Phi _i, \Psi _i \right )$ are complementary Young pairs which satisfy local $\Delta _2$ -condition, strictly convex and continuous for $i = 1, 2$ , then $(M^{\Phi _1, \Phi _2}({\mathbb {Z}}^n))^* \cong M^{\Psi _1, \Psi _2}({\mathbb {Z}}^n)$ under the duality relation

$$\begin{align*}\langle f,h \rangle = \left\langle V_{g_0} f, V_{g_0} h \right\rangle = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} V_{g_0} f(m,w) \; \overline{V_{g_0} h(m,w)} \; dw \end{align*}$$

for $f \in M^{\Phi _1, \Phi _2}({\mathbb {Z}}^n)$ and $h \in M^{\Psi _1, \Psi _2}({\mathbb {Z}}^n)$ , $g_0 \in \mathcal {S}({\mathbb {Z}}^n)$ .

Theorem 3.6 If $\Phi _i$ and $ \Psi _i$ are (quasi-)Young functions such that

$$\begin{align*}\lim_{x \to 0^+} \frac{\Psi_i(x)}{\Phi_i(x)} \end{align*}$$

exist and are finite for $i = 1, 2$ , then

(10) $$ \begin{align} L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n) \hookrightarrow L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n) \quad \text{and} \quad M^{\Phi_1, \Phi_2}({\mathbb{Z}}^n) \hookrightarrow M^{\Psi_1, \Psi_2}({\mathbb{Z}}^n). \end{align} $$

Proof Conditions (1) and (3), and (2) and (3) are equivalent follows from Theorem 3.6 and [Reference Toft, Üster, Nabizadeh and Öztop27, Proposition 5.11]. Now, we prove that conditions (1) and (2) are equivalent. Using the definition of discrete Orlicz modulation spaces, we have

$$\begin{align*}\|f\|_{M^{\Phi_1, \Phi_2}({\mathbb{Z}}^n)} = \left\|V_g f\right\|_{L^{\Phi_1, \Phi_2}({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \text{and} \; \|f\|_{M^{\Psi_1, \Psi_2}({\mathbb{Z}}^n)} = \left\|V_g f\right\|_{L^{\Psi_1, \Psi_2}({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align*}$$

If $M^{\Phi _1, \Phi _2}({\mathbb {Z}}^n) \subseteq M^{\Psi _1, \Psi _2}({\mathbb {Z}}^n)$ , then

$$ \begin{align*} \|f\|_{M^{\Psi_1, \Psi_2}({\mathbb{Z}}^n)} \leq \|f\|_{M^{\Phi_1, \Phi_2}({\mathbb{Z}}^n)} & \Rightarrow \left\|V_g f\right\|_{L^{\Psi_1, \Psi_2}({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \leq \left\|V_g f\right\|_{L^{\Phi_1, \Phi_2}({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \\ & \Rightarrow L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n) \subseteq L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n). \end{align*} $$

Similarly, if $ L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n) \subseteq L^{\Psi _1, \Psi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n) $ , then

$$ \begin{align*} \left\|V_g f\right\|_{L^{\Psi_1, \Psi_2}({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \leq \left\|V_g f\right\|_{L^{\Phi_1, \Phi_2}({\mathbb{Z}}^n \times {\mathbb{T}}^n)} & \Rightarrow \|f\|_{M^{\Psi_1, \Psi_2}({\mathbb{Z}}^n)} \leq \|f\|_{M^{\Phi_1, \Phi_2}({\mathbb{Z}}^n)} \\ & \Rightarrow M^{\Phi_1, \Phi_2}({\mathbb{Z}}^n) \subseteq M^{\Psi_1, \Psi_2}({\mathbb{Z}}^n). \end{align*} $$

This completes the proof.

Note that the constant of the estimate in condition (3) is uniform in i, and the converse implication holds without additional assumptions. Next, we show that $M^{\Phi }({\mathbb {Z}}^n)$ is close to $M^{2}({\mathbb {Z}}^n)$ in some sense. In the following proposition, we consider the same Young function $\Phi $ as in [Reference Gumber, Rana, Toft and Üster19], since this function plays an important role in finding the inclusion relations between the Orlicz modulation space and modulation spaces (see [Reference Gumber, Rana, Toft and Üster19]).

Proposition 3.8 Let $\Phi $ be a Young function which satisfies

(11) $$ \begin{align} \Phi(x)= - x^2 \log x, \quad 0 \leq x \leq e^{-\frac{2}{3}}. \end{align} $$

Then,

(12) $$ \begin{align} M^{p}({\mathbb{Z}}^n) \subseteq M^{\Phi}({\mathbb{Z}}^n) \subseteq M^{2}({\mathbb{Z}}^n), \quad p<2, \end{align} $$

with continuous and dense inclusions.

Throughout the following section, we assume that the Young function $\Phi $ satisfies (11) so that we can use the inclusion relations in (12) in the proof of the main results.

4 Localization operators on Orlicz modulation spaces on ${\mathbb {Z}}^n$

Here, we study the localization operators on ${\mathbb {Z}}^n$ and prove their boundedness. Furthermore, we demonstrate the compactness of these operators and their inclusion in the Schatten–von Neumann class.

Definition 4.1 Let $\sigma \in L^1({\mathbb {Z}}^n \times {\mathbb {T}}^n) \cup L^\infty ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . For the symbol $\sigma $ and two window functions $g_1, g_2 \in \mathcal {S} (\mathbb Z^n)$ , the localization operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ is defined on $\ell ^2({\mathbb {Z}}^n)$ by

(13) $$ \begin{align} \mathfrak{L}^{g_1, g_2}_{\sigma}f(k)=\sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; V_{g_1}f(m,w) \; M_w T_m g_2(k) \; dw, \quad k \in {\mathbb{Z}}^n. \end{align} $$

For any $f, h \in \ell ^2({\mathbb {Z}}^n)$ , we rewrite the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ in a weak sense as

(14) $$ \begin{align} \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f , h \right\rangle_{\ell^2({\mathbb{Z}}^n)} = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; V_{g_1}f(m,w) \; \overline{V_{g_2}h(m,w)} \; dw. \end{align} $$

For $1 \leq p \leq \infty $ , we define $\mathcal {B}(\ell ^p({\mathbb {Z}}^n))$ to be the space of all bounded linear operators from $\ell ^p({\mathbb {Z}}^n)$ into itself. For $p=2$ , the space $\mathcal {B}(\ell ^2({\mathbb {Z}}^n))$ is the C $^*$ -algebra of bounded linear operator $\mathcal {A}$ from $\ell ^2({\mathbb {Z}}^n)$ into itself, equipped with the norm

$$\begin{align*}\Vert \mathcal{A} \Vert_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))}= \sup_{\Vert f \Vert_{\ell^2({\mathbb{Z}}^n)} \leq 1} \Vert \mathcal{A}(f) \Vert_{\ell^2({\mathbb{Z}}^n)}. \end{align*}$$

To define the Schatten–von Neumann class $S_p$ on ${\mathbb {Z}}^n$ , we need to first recall the definition of singular values of an operator. For a compact operator $\mathcal {A} \in \mathcal {B}(\ell ^2({\mathbb {Z}}^n))$ , the singular values of $\mathcal {A}$ are the eigenvalues of the positive self-adjoint operator $|\mathcal {A}|=\sqrt {\mathcal {A}^* \mathcal {A}}$ and denoted by $\{ s_n(\mathcal {A}) \}_{n \in \mathbb {N}}$ . For $1 \leq p < \infty $ , we define the Schatten–von Neumann class $S_p$ to be the space of all compact operators whose singular values lie in $\ell ^p$ , and equipped with the norm

$$\begin{align*}\Vert \mathcal{A} \Vert_{S_p}= \left(\sum_{n=1}^\infty (s_n(\mathcal{A}))^p \right)^{1/p}. \end{align*}$$

For $p=\infty $ , $S_\infty $ is the class of all compact operators with the norm $\Vert \mathcal {A} \Vert _{S_\infty } :=\Vert \mathcal {A} \Vert _{\mathcal {B}(\ell ^2({\mathbb {Z}}^n))}$ . For $p = 1$ , the trace of an operator $\mathcal {A} \in S_1$ is defined by

$$\begin{align*}tr(\mathcal{A}) = \sum_{n=1}^\infty \langle \mathcal{A} v_n, v_n \rangle_{\ell^2({\mathbb{Z}}^n)}, \end{align*}$$

where $\{v_n \}_n$ is any orthonormal basis of $\ell ^2({\mathbb {Z}}^n)$ . In addition, if $\mathcal {A}$ is positive, then

$$\begin{align*}tr(\mathcal{A})=\Vert \mathcal{A} \Vert_{S_1} .\end{align*}$$

For a compact operator $\mathcal {A}$ on the Hilbert space $\ell ^2({\mathbb {Z}}^n)$ if the positive operator $\mathcal {A}^* \mathcal {A} \in S_1$ , then we call the operator $\mathcal {A}$ as a Hilbert–Schmidt operator. For any orthonormal basis $\{v_n \}_n$ of $\ell ^2({\mathbb {Z}}^n)$ , we have

$$\begin{align*}\Vert \mathcal{A}\Vert_{HS}^2 :=\Vert \mathcal{A}\Vert_{S_2}^2= \Vert \mathcal{A}^* \mathcal{A} \Vert_{S_1}= tr(\mathcal{A}^* \mathcal{A}) = \sum_{n=1}^\infty \Vert \mathcal{A} v_n\Vert^2_{\ell^2({\mathbb{Z}}^n)}. \end{align*}$$

4.1 Boundedness and compactness of $\mathfrak {L}^{g_1, g_2}_{\sigma }$

Here, we consider $g_1, g_2 \in M^{\Phi }({\mathbb {Z}}^n)$ , and prove the results related to the boundedness and compactness of $\mathfrak {L}^{g_1, g_2}_{\sigma }$ .

Proposition 4.2 Let $\Phi $ be a Young function, $\sigma \in L^\infty ({\mathbb {Z}}^n \times {\mathbb {T}}^n),$ and $g_1, g_2 \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, the operator $\mathfrak {L}^{g_1, g_2}_{\sigma } \in \mathcal {B}(\ell ^2({\mathbb {Z}}^n))$ , and

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{L^\infty({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Proof Let $f, h \in \ell ^2({\mathbb {Z}}^n)$ . Applying Hölder’s inequality, we get

$$ \begin{align*} \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f, h \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right| & \leq \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\sigma(m, w)| \; \left|V_{g_1}f(m,w) \right|\; \left|V_{g_2}h(m,w) \right| \; dw \\ & \leq \left\Vert \sigma \right\Vert{}_{L^\infty({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \left\Vert V_{g_1}f \right\Vert{}_{L^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \left\Vert V_{g_2}h \right\Vert{}_{L^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align*} $$

Applying Plancherel’s formula (2), we obtain

$$ \begin{align*} \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f, h \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right| \leq \left\Vert \sigma \right\Vert{}_{L^\infty({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \Vert f \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert g_1 \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert h \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{\ell^2({\mathbb{Z}}^n)}. \end{align*} $$

Since $M^{\Phi }({\mathbb {Z}}^n) \subset \ell ^2({\mathbb {Z}}^n)$ , we get

$$\begin{align*}\Vert g_1 \Vert_{\ell^2({\mathbb{Z}}^n)} \leq \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \quad \text{and} \quad \Vert g_2 \Vert_{\ell^2({\mathbb{Z}}^n)} \leq \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Therefore,

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{L^\infty({\mathbb{Z}}^n \times {\mathbb{T}}^n)}\; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}.\\[-34pt] \end{align*}$$

Proposition 4.3 Let $\Phi $ be a Young function, $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $g_1, g_2 \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, the operator $\mathfrak {L}^{g_1, g_2}_{\sigma } \in \mathcal {B}(\ell ^2({\mathbb {Z}}^n))$ , and

$$ \begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}.\end{align*} $$

Proof Let $f , h \in \ell ^2({\mathbb {Z}}^n)$ . Using the duality between the modulation spaces $M^\infty ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , we obtain

(15) $$ \begin{align} \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f, h \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right| & \leq \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\sigma(m, w)| \; \left|V_{g_1}f(m, w) \; \overline{V_{g_2}h(m, w)} \right| \; dw \nonumber \\ & \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \left\Vert V_{g_1}f \cdot \overline{V_{g_2}h} \right\Vert{}_{M^\infty({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align} $$

Since $L^2({\mathbb {Z}}^n \times {\mathbb {T}}^n) \subset M^\infty ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $M^{\Phi }({\mathbb {Z}}^n) \subset \ell ^2({\mathbb {Z}}^n)$ , applying Plancherel’s formula (2), we get

(16) $$ \begin{align} & \left\Vert V_{g_1}f \cdot \overline{V_{g_2}h} \right\Vert{}_{M^\infty({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \nonumber \\ & \leq \left\Vert V_{g_1}f \cdot \overline{V_{g_2}h} \right\Vert{}_{L^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \nonumber \\ & \leq \left\Vert V_{g_1}f \right\Vert{}_{L^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \left\Vert V_{g_2}h \right\Vert{}_{L^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \nonumber \\ & = \Vert f \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert g_1 \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert h \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{\ell^2({\mathbb{Z}}^n)} \nonumber \\ & \leq \Vert f \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert h \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align} $$

From (15) and (16), we have

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}.\\[-34pt] \end{align*}$$

Proposition 4.4 Let $\Phi $ be a Young function, $\sigma \in M^2({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $g_1, g_2 \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, the operator $\mathfrak {L}^{g_1, g_2}_{\sigma } \in \mathcal {B}(\ell ^2({\mathbb {Z}}^n))$ , and

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Proof Let $f, h \in \ell ^2({\mathbb {Z}}^n)$ . Applying Hölder’s inequality, we get

$$ \begin{align*} \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f, h \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right| & \leq \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\sigma(m, w)| \; \left|V_{g_1}f(m, w) \; \overline{V_{g_2}h(m, w)} \right| \; dw \\ & \leq \left\Vert \sigma \right\Vert{}_{L^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \left\Vert V_{g_1}f \cdot \overline{V_{g_2}h} \right\Vert{}_{L^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align*} $$

Since $L^2({\mathbb {Z}}^n \times {\mathbb {T}}^n)= M^2({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , using (16), we obtain

$$ \begin{align*} & \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f, h \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right| \\ & \leq \left\Vert \sigma \right\Vert{}_{M^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \Vert f \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert h \Vert_{\ell^2({\mathbb{Z}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*} $$

Therefore,

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}.\\[-34pt] \end{align*}$$

Theorem 4.5 Let $1 < p < 2$ , $\sigma \in M^p({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , $\Phi $ be a Young function, and $g_1, g_2 \in M^{\Phi }({\mathbb {Z}}^n)$ . For fixed $\sigma \in M^p({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ can be uniquely extended to a bounded linear operator on $\ell ^2({\mathbb {Z}}^n)$ , for which

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^p({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Proof Let $1< p < 2$ and $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n) \cap M^2({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . The modulation spaces $M^{p}$ interpolate similar to the corresponding mixed-norm spaces $L^{p}$ . From Proposition 4.3, for $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , we have

$$ \begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}.\end{align*} $$

Also, from Proposition 4.4, for $\sigma \in M^2({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , we have

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^2({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Now, for $1< p < 2$ , using the Riesz–Thorin interpolation theorem (see [Reference Stein26]), we get

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^p({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Let $\sigma \in M^p({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $\{ \sigma _n \}_{n \geq 1}$ be a sequence of functions in $M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n) \cap M^2({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ such that $\sigma _n \to \sigma $ in $M^p({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ as $ n \to \infty $ . Therefore, for any $n, k \in \mathbb {N}$ , we get

$$\begin{align*}\left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma_n} - \mathfrak{L}^{g_1, g_2}_{\sigma_k} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma_n - \sigma_k \Vert_{M^p({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Hence, $\{ \mathfrak {L}^{g_1, g_2}_{\sigma _n} \}_{n \geq 1}$ is a Cauchy sequence in $\mathcal {B}(\ell ^2({\mathbb {Z}}^n))$ . Let $ \mathfrak {L}^{g_1, g_2}_{\sigma _n} \to \mathfrak {L}^{g_1, g_2}_{\sigma }$ as $ n \to \infty $ . Then, the limit $ \mathfrak {L}^{g_1, g_2}_{\sigma }$ remains unaffected by the selection of the sequence $\{ \sigma _n \}_{n \geq 1}$ , and we have

$$ \begin{align*} \left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} & = \lim_{n \to \infty} \left\Vert \mathfrak{L}^{g_1, g_2}_{\sigma_n} \right\Vert{}_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \\ & \leq \lim_{n \to \infty} \Vert \sigma_n \Vert_{M^p({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \\ & = \Vert \sigma \Vert_{M^p({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}.\\[-34pt] \end{align*} $$

Theorem 4.6 Let $1 \leq p \leq 2$ , $\sigma \in M^p({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , $\Phi $ be a Young function, and $g_1, g_2 \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }: \ell ^2({\mathbb {Z}}^n) \to \ell ^2({\mathbb {Z}}^n) $ is compact.

Proof Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $\{ v_n \}_n$ be an orthonormal basis for $\ell ^2({\mathbb {Z}}^n)$ . Since $M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n) \subset L^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , applying Parseval’s identity, we get

$$ \begin{align*} & \sum_{n=1}^\infty \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma} v_n, v_n \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right| \\ & \leq \sum_{n=1}^\infty \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \left| \sigma (m,w) \right| \; \left| \langle v_n, M_w T_m g_1 \rangle_{\ell^2({\mathbb{Z}}^n)} \right| \; \left| \langle M_w T_m g_2 , v_n \rangle_{\ell^2({\mathbb{Z}}^n)} \right| \; dw\\ & = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \left| \sigma (m,w) \right| \left( \sum_{n=1}^\infty \left| \langle v_n, M_w T_m g_1 \rangle_{\ell^2({\mathbb{Z}}^n)} \right|\; \left| \langle M_w T_m g_2 , v_n \rangle_{\ell^2({\mathbb{Z}}^n)} \right| \right) dw \\ & \leq \frac{1}{2} \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \left| \sigma (m,w) \right| \left( \sum_{n=1}^\infty |\langle v_n, M_w T_m g_1 \rangle_{\ell^2({\mathbb{Z}}^n)}|^2 \right. \\ & \qquad \qquad \left. + \sum_{n=1}^\infty |\langle M_w T_m g_2, v_n \rangle_{\ell^2({\mathbb{Z}}^n)}|^2 \right) \; dw\\ & = \frac{1}{2} \Vert \sigma \Vert_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; (\Vert g_1 \Vert^2_{\ell^2({\mathbb{Z}}^n)} + \Vert g_2 \Vert^2_{\ell^2({\mathbb{Z}}^n)}) \\ & \leq \frac{1}{2} \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; (\Vert g_1 \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)} + \Vert g_2 \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)}). \end{align*} $$

Hence, the operator $\mathfrak {L}^{g_1, g_2}_{\sigma } \in S_1$ . Next, let $\sigma \in M^p({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . We consider $\{ \sigma _n \}_{n \geq 1}$ in $M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n) \cap M^2({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ such that $\sigma _n \to \sigma $ in $M^p({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ as $n \to \infty $ . Then, applying Theorem 4.5, we obtain

$$\begin{align*}\Vert \mathfrak{L}^{g_1, g_2}_{\sigma_n} - \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(\ell^2({\mathbb{Z}}^n))} \leq \Vert \sigma_n - \sigma \Vert_{M^p({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \to 0, \end{align*}$$

as $n \to \infty $ . Therefore, $\mathfrak {L}^{g_1, g_2}_{\sigma _n} \to \mathfrak {L}^{g_1, g_2}_{\sigma } $ in $\mathcal {B}(\ell ^2({\mathbb {Z}}^n))$ as $n \to \infty $ . From the above, we get that $ \{ \mathfrak {L}^{g_1, g_2}_{\sigma _n} \}_{n \geq 1}$ is a sequence of linear operators in $S_1$ and hence compact, so the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ is compact.

Now, we calculate the adjoint $(\mathfrak {L}^{g_1, g_2}_{\sigma })^*$ of the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ on $\ell ^2({\mathbb {Z}}^n)$ , which determined by the relation

$$\begin{align*}\left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f , h \right\rangle_{\ell^2({\mathbb{Z}}^n)}= \left\langle f , (\mathfrak{L}^{g_1, g_2}_{\sigma})^*h \right\rangle_{\ell^2({\mathbb{Z}}^n)}. \end{align*}$$

We obtain

$$ \begin{align*} & \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f , h \right\rangle_{\ell^2({\mathbb{Z}}^n)} \\ &= \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; V_{g_1}f(m,w) \; \overline{V_{g_2}h(m,w)} \; dw \\ &= \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; \langle f, M_w T_m g_1 \rangle_{\ell^2({\mathbb{Z}}^n)} \; \overline{V_{g_2}h(m,w)} \; dw \end{align*} $$

$$ \begin{align*} &= \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; \langle f, V_{g_2}h(m,w) \; M_w T_m g_1 \rangle_{\ell^2({\mathbb{Z}}^n)} \; dw \\ &= \left\langle f, \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \overline{\sigma(m, w)} \; V_{g_2}h(m,w) \; M_w T_m g_1 \; dw \right\rangle_{\ell^2({\mathbb{Z}}^n)} \\ &= \left\langle f, \mathfrak{L}^{g_2, g_1}_{\overline{\sigma}}h \right\rangle_{\ell^2({\mathbb{Z}}^n)}. \end{align*} $$

Hence, we get

$$\begin{align*}(\mathfrak{L}^{g_1, g_2}_{\sigma})^* = \mathfrak{L}^{g_2, g_1}_{\overline{\sigma}}. \end{align*}$$

Therefore, the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ is a self-adjoint operator if $g_1=g_2$ and $\sigma $ is a real-valued function.

4.2 Localization operators in $S_p$

Here, we show that the operator $\mathfrak {L}^{g, g}_{\sigma } \in S_p$ and give an upper bound of $\Vert \mathfrak {L}^{g, g}_{\sigma } \Vert _{S_p}$ .

Proposition 4.7 Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , $\Phi $ be a Young function, and $g \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, the operator $\mathfrak {L}^{g, g}_{\sigma }: \ell ^2({\mathbb {Z}}^n) \to \ell ^2({\mathbb {Z}}^n) $ is in $S_1$ and

$$\begin{align*}\Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_1} \leq 4 \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Proof Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . Then, from Theorem 4.6, the operator $\mathfrak {L}^{g, g}_{\sigma } \in S_1$ . Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ be nonnegative real-valued. Then, $(({\mathfrak {L}^{g, g}_{\sigma }})^* \mathfrak {L}^{g, g}_{\sigma })^{1/2}=\mathfrak {L}^{g, g}_{\sigma }$ . Let $\{v_n\}_n$ be an orthonormal basis for $\ell ^2({\mathbb {Z}}^n)$ consisting of eigenvalues of the operator $(({\mathfrak {L}^{g, g}_{\sigma }})^* \mathfrak {L}^{g, g}_{\sigma })^{1/2}: \ell ^2({\mathbb {Z}}^n) \to \ell ^2({\mathbb {Z}}^n)$ . If $\sigma $ is nonnegative real-valued function, then by using a similar method as in the proof of Theorem 4.6, we get the following estimate:

(17) $$ \begin{align} \Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_1} & = \sum_{n=1}^\infty \left\langle (({\mathfrak{L}^{g, g}_{\sigma}})^* \mathfrak{L}^{g, g}_{\sigma})^{1/2}v_n, v_n \right\rangle_{\ell^2({\mathbb{Z}}^n)} \nonumber \\ &= \sum_{n=1}^\infty \left\langle \mathfrak{L}^{g, g}_{\sigma} v_n, v_n \right\rangle_{\ell^2({\mathbb{Z}}^n)} \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align} $$

Next, let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ be an arbitrary real-valued function. We write $\sigma = \sigma _+ - \sigma _-$ , where $\sigma _+=\max (\sigma , 0)$ and $\sigma _-=-\min (\sigma , 0)$ . Then, applying the relation (17), we get

(18) $$ \begin{align} \Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_1} & = \Vert \mathfrak{L}^{g, g}_{\sigma_+} -\mathfrak{L}^{g, g}_{\sigma_-} \Vert_{S_1} \nonumber \\ & \leq \Vert \mathfrak{L}^{g, g}_{\sigma_+} \Vert_{S_1} +\Vert \mathfrak{L}^{g, g}_{\sigma_-} \Vert_{S_1} \nonumber \\ & \leq \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)} (\Vert \sigma_+ \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} + \Vert \sigma_- \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}) \nonumber \\ & \leq 2 \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)} \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align} $$

Finally, let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ be a complex-valued function. Then, we write $\sigma =\sigma _1+ i \sigma _2$ , where $\sigma _1, \sigma _2$ are the real and imaginary parts of $\sigma ,$ respectively. Applying the relation (18), we get

$$ \begin{align*} \Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_1} & = \Vert \mathfrak{L}^{g, g}_{\sigma_1}+i \; \mathfrak{L}^{g, g}_{\sigma_2} \Vert_{S_1} \\ & \leq \Vert \mathfrak{L}^{g, g}_{\sigma_1} \Vert_{S_1} + \Vert \mathfrak{L}^{g, g}_{\sigma_2} \Vert_{S_1} \\ &\leq 2 \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)} (\Vert \sigma_1 \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} + \Vert \sigma_2 \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}) \nonumber \\ & \leq 4 \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)}.\\[-34pt] \end{align*} $$

Theorem 4.8 Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , $\Phi $ be a Young function, and $g \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, for $1 \leq p \leq \infty $ , the operator $\mathfrak {L}^{g, g}_{\sigma }: \ell ^2({\mathbb {Z}}^n) \to \ell ^2({\mathbb {Z}}^n)$ is in $S_p$ , and

$$\begin{align*}\Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_p} \leq 2^{2/p} \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g \Vert^{2}_{M^{\Phi}({\mathbb{Z}}^n)}.\end{align*}$$

Proof Using interpolation theorems (see [Reference Wong30, Theorems 2.10 and 2.11]), Proposition 4.4, and Proposition 4.7, we obtain the proof of the theorem.

In the following, we obtain a lower bound of $\Vert \mathfrak {L}^{g, g}_{\sigma } \Vert _{S_1}$ , and improve the constant given in Proposition 4.7.

Theorem 4.9 Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ be a nonnegative real-valued function, $\Phi $ be a Young function, and $g \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, the operator $\mathfrak {L}^{g, g}_{\sigma }$ is in $S_1$ , and

$$ \begin{align*} \frac{1}{\Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)}}\; \Vert \tilde{\sigma} \Vert_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \leq \Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_1} \leq \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}, \end{align*} $$

where $\tilde {\sigma }$ is given by $\tilde {\sigma }(m, w)=\left \langle \mathfrak {L}^{g, g}_{\sigma }(M_w T_m g), M_w T_m g \right \rangle _{\ell ^2({\mathbb {Z}}^n)}$ .

Proof Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . Then, from Proposition 4.7, the operator $\mathfrak {L}^{g, g}_{\sigma } \in S_1$ . If $\{s_n(\mathfrak {L}^{g, g}_{\sigma }) \}_n$ are the positive singular values of $\mathfrak {L}^{g, g}_{\sigma }$ , $\{ v_n \}_n$ is an orthonormal basis for the orthogonal complement of the null space of $\mathfrak {L}^{g, g}_{\sigma }$ consisting of eigenvectors of $|\mathfrak {L}^{g, g}_{\sigma }|$ and $\{ u_n \}_n$ is an orthonormal set in $\ell ^2({\mathbb {Z}}^n)$ , then using the canonical form of compact operators (see [Reference Wong30, Theorem 2.2]), we get

(19) $$ \begin{align} \mathfrak{L}^{g, g}_{\sigma}f =\sum_{n=1}^{\infty} s_n(\mathfrak{L}^{g, g}_{\sigma}) \langle f, v_n \rangle_{\ell^2({\mathbb{Z}}^n)} u_n. \end{align} $$

Therefore, we obtain

$$\begin{align*}\sum_{n=1}^{\infty} \langle \mathfrak{L}^{g, g}_{\sigma} v_n, u_n \rangle_{\ell^2({\mathbb{Z}}^n)} = \sum_{n=1}^{\infty} s_n(\mathfrak{L}^{g, g}_{\sigma})= \Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_1}. \end{align*}$$

Applying Bessel’s inequality and Cauchy–Schwarz’s inequality, we obtain

$$ \begin{align*} \Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_1} & = \sum_{n=1}^{\infty} \langle \mathfrak{L}^{g, g}_{\sigma} v_n, u_n \rangle_{\ell^2({\mathbb{Z}}^n)} \\ & = \sum_{n=1}^{\infty} \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; V_{g} v_n (m,w) \; \overline{V_{g} u_n (m,w)} \; dw \\ & \leq \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\sigma(m, w)| \left(\sum_{n=1}^{\infty} |V_{g} v_n (m,w)|^2 \right)^{1/2} \\ & \qquad \qquad \left(\sum_{n=1}^{\infty} |V_{g} u_n (m,w)|^2 \right)^{1/2} dw \\ & \leq \Vert \sigma \Vert_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \;\Vert g \Vert^2_{\ell^2({\mathbb{Z}}^n)} \\ & \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}\; \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*} $$

To prove that $\tilde {\sigma } \in L^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ , applying formula (19), we get

$$ \begin{align*} & |\tilde{\sigma}(m, w)| \\ & = \left|\left\langle \mathfrak{L}^{g, g}_{\sigma}( M_w T_m g), M_w T_m g \right\rangle_{\ell^2({\mathbb{Z}}^n)}\right| \\ & = \left| \sum_{n=1}^{\infty} s_n(\mathfrak{L}^{g, g}_{\sigma}) \left\langle M_w T_m g, v_n \right\rangle_{\ell^2({\mathbb{Z}}^n)} \left\langle u_n, M_w T_m g \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right| \\ & \leq \frac{1}{2} \sum_{n=1}^{\infty} s_n(\mathfrak{L}^{g, g}_{\sigma}) \left( \left| \left\langle M_w T_m g, v_n \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right|^2 + \left| \left\langle M_w T_m g, u_n \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right|^2 \right). \end{align*} $$

Now, using Plancherel’s formula (2), we obtain

$$ \begin{align*} & \Vert \tilde{\sigma} \Vert_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\tilde{\sigma}(m, w)| \; dw \\ & \leq \frac{1}{2} \sum_{n=1}^{\infty} s_n(\mathfrak{L}^{g, g}_{\sigma}) \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \left( \left| \left\langle M_w T_m g, v_n \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right|^2 \right. \\ & \qquad \qquad + \left. \left| \left\langle M_w T_m g, u_n \right\rangle_{\ell^2({\mathbb{Z}}^n)} \right|^2 \right) dw \\ & \leq \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)} \sum_{n=1}^{\infty} s_n(\mathfrak{L}^{g, g}_{\sigma}) \\ & = \Vert g \Vert^2_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert \mathfrak{L}^{g, g}_{\sigma} \Vert_{S_1}.\\[-34pt] \end{align*} $$

4.3 $M^{\Phi }({\mathbb {Z}}^n)$ boundedness

Here, we obtain that the operators $\mathfrak {L}^{g_1, g_2}_{\sigma } : M^{\Phi }({\mathbb {Z}}^n) \to M^{\Phi }({\mathbb {Z}}^n)$ are bounded. Let us start with the following propositions.

Proposition 4.10 Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . Let $\left (\Phi , \Psi \right )$ be a complementary Young pair, and $\Phi $ satisfies a local $\Delta _2$ -condition and continuous. Let $g_1 \in M^{\Psi }({\mathbb {Z}}^n)$ and $g_2 \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }: M^{\Phi }({\mathbb {Z}}^n) \to M^{\Phi }({\mathbb {Z}}^n)$ is a bounded linear operator, and

$$\begin{align*}\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(M^{\Phi}({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{ M^{\Psi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Proof Let $f \in M^{\Phi }({\mathbb {Z}}^n)$ and $g \in M^{\Psi }({\mathbb {Z}}^n)$ . Since $(M^{\Phi }({\mathbb {Z}}^n))^* \cong M^{\Psi }({\mathbb {Z}}^n)$ , applying Hölder’s inequality, we get

(20) $$ \begin{align} \left| V_{g} f (m, w) \right| \leq \Vert f \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g \Vert_{M^{\Psi}({\mathbb{Z}}^n)}. \end{align} $$

For any $f \in M^{\Phi }({\mathbb {Z}}^n)$ and $h \in M^{\Psi }({\mathbb {Z}}^n)$ , applying relations (14) and (20), we have

$$ \begin{align*} & \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f, h \right\rangle \right| \\ & \leq \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\sigma(m, w)| \; \left|V_{g_1}f(m, w) \right| \left| V_{g_2}h(m, w) \right| \; dw \\ & \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert f \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \; \Vert g_1 \Vert_{M^{\Psi}({\mathbb{Z}}^n)} \; \Vert h \Vert_{M^{\Psi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*} $$

Therefore,

$$\begin{align*}\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(M^{\Phi}({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{ M^{\Psi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}.\\[-34pt] \end{align*}$$

Next, using the Schur technique, we get an $M^{\Phi }({\mathbb {Z}}^n)$ -boundedness of the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ . We obtain a new estimate for $\Vert \mathfrak {L}^{g_1, g_2}_{\sigma } \Vert _{\mathcal {B}(M^{\Phi }({\mathbb {Z}}^n))}$ .

Proposition 4.11 Let $\sigma \in M^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $g_1, g_2 \in M^1({\mathbb {Z}}^n) \cap \ell ^\infty ({\mathbb {Z}}^n)$ . Then, there exists a bounded linear operator $\mathfrak {L}^{g_1, g_2}_{\sigma }: M^{\Phi }({\mathbb {Z}}^n) \to M^{\Phi }({\mathbb {Z}}^n) $ for which

$$ \begin{align*} & \Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(M^{\Phi}({\mathbb{Z}}^n))} \\ & \leq \max(\Vert g_1 \Vert_{M^1({\mathbb{Z}}^n)} \Vert g_2 \Vert_{\ell^\infty({\mathbb{Z}}^n)}, \Vert g_1 \Vert_{\ell^\infty({\mathbb{Z}}^n)} \Vert g_2 \Vert_{M^1({\mathbb{Z}}^n)}) \; \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align*} $$

Proof We define the function $\mathcal {K}$ on ${\mathbb {Z}}^n \times {\mathbb {Z}}^n$ by

(21) $$ \begin{align} \mathcal{K}(k, l) = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w)\; \overline{M_w T_m g_1(l)} \; M_w T_m g_2(k) \; dw. \end{align} $$

Then, we have

$$ \begin{align*} \mathfrak{L}^{g_1, g_2}_{\sigma}f(k) & = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; V_{g_1}f(m,w) \; M_w T_m g_2(k) \; dw \\ & = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; \left\langle f, M_w T_m {g_1} \right\rangle \; M_w T_m g_2(k) \; dw \\ & = \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; \sum_{l \in {\mathbb{Z}}^n} f(l) \overline{M_w T_m {g_1} (l)} \; M_w T_m g_2(k) \; dw \\ & = \sum_{l \in {\mathbb{Z}}^n} f(l) \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} \sigma(m, w) \; \overline{M_w T_m {g_1} (l)}\; M_w T_m g_2(k) \; dw \\ & = \sum_{l \in {\mathbb{Z}}^n} \mathcal{K}(k,l) \; f(l). \end{align*} $$

For every $l \in {\mathbb {Z}}^n$ , we get

(22) $$ \begin{align} \sum_{k \in {\mathbb{Z}}^n} | \mathcal{K}(k, l) | &\leq \sum_{k \in {\mathbb{Z}}^n} \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\sigma(m, w)| \left|\overline{ M_w T_m g_1(l)} \right| \; \left|M_w T_m g_2(k) \right| \; dw \nonumber \\ & \leq \Vert g_1 \Vert_{\ell^\infty({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^1({\mathbb{Z}}^n)} \; \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}, \end{align} $$

and for every $k \in {\mathbb {Z}}^n$ , we have

(23) $$ \begin{align} \sum_{l \in {\mathbb{Z}}^n} | \mathcal{K}(k, l) | \leq \Vert g_1 \Vert_{M^1({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{\ell^\infty({\mathbb{Z}}^n)} \; \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align} $$

The kernel function $\mathcal {K}(k, l)$ of the localization operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ satisfies (22) and (23). Hence, by Schur’s lemma (see [Reference Folland17]), the operator $\mathfrak {L}^{g_1, g_2}_{\sigma }$ extends to a bounded linear operator $\mathfrak {L}^{g_1, g_2}_{\sigma }: M^{\Phi }({\mathbb {Z}}^n) \to M^{\Phi }({\mathbb {Z}}^n)$ with the operator norm

$$ \begin{align*} & \Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(M^{\Phi}({\mathbb{Z}}^n))} \\ & \leq \max(\Vert g_1 \Vert_{M^1({\mathbb{Z}}^n)} \Vert g_2 \Vert_{\ell^\infty({\mathbb{Z}}^n)}, \Vert g_1 \Vert_{\ell^\infty({\mathbb{Z}}^n)} \Vert g_2 \Vert_{M^1({\mathbb{Z}}^n)}) \; \Vert \sigma \Vert_{M^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)}.\\[-32pt] \end{align*} $$

Remark 4.12 Based on Proposition 4.11, it is determined that the bounded linear operator on $M^{\Phi }({\mathbb {Z}}^n)$ identified in Proposition 4.10 is, in fact, the discrete integral operator on $M^{\Phi }({\mathbb {Z}}^n)$ characterized by the kernel $\mathcal {K}$ as provided in (21).

Theorem 4.13 Let $\sigma \in L^1({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ . Let $\left (\Phi , \Psi \right )$ be a complementary Young pair, and $\Phi $ satisfies a local $\Delta _2$ -condition and continuous. Let $g_1 \in M^{\Psi }({\mathbb {Z}}^n)$ and $g_2 \in M^{\Phi }({\mathbb {Z}}^n)$ . Then, the operator $ \mathfrak {L}^{g_1, g_2}_{\sigma } \in \mathcal {B}(M^{\Phi }({\mathbb {Z}}^n))$ , and

$$\begin{align*}\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(M^{\Phi}({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Psi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*}$$

Proof Let $f \in M^{\Phi }({\mathbb {Z}}^n)$ and $h \in M^{\Psi }({\mathbb {Z}}^n)$ . Using the duality between $M^{\Phi }({\mathbb {Z}}^n)$ and $M^{\Psi }({\mathbb {Z}}^n)$ , we get

$$ \begin{align*} & \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f, h \right\rangle \right| \\ & \leq \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\sigma(m, w)| \; \left|V_{g_1}f(m, w) \right| \; \left| V_{g_2}h(m, w) \right| \; dw \\ & \leq \left\Vert \sigma \right\Vert{}_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \Vert f \Vert_{M^{\Phi}({\mathbb{Z}}^n)} \Vert g_1 \Vert_{M^{\Psi}({\mathbb{Z}}^n)} \Vert h \Vert_{M^{\Psi}({\mathbb{Z}}^n)} \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}. \end{align*} $$

Hence,

$$\begin{align*}\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(M^{\Phi}({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{L^1({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \; \Vert g_1 \Vert_{M^{\Psi}({\mathbb{Z}}^n)} \; \Vert g_2 \Vert_{M^{\Phi}({\mathbb{Z}}^n)}.\\[-34pt] \end{align*}$$

Theorem 4.14 Let $\left (\Phi _i, \Psi _i \right )$ be complementary Young pairs which satisfy local $\Delta _2$ -condition, strictly convex and continuous for $i = 1, 2$ . Let $\sigma \in L^{\Phi _1, \Phi _2} ({\mathbb {Z}}^n \times {\mathbb {T}}^n)$ and $g_1, g_2 \in \mathcal {S}({\mathbb {Z}}^n)$ . Also, let there exists a constant $x_0> 0$ such that $\Psi _i(x) \lesssim \Phi _i(x)$ for all $0 \leq x \leq x_0$ . Then, the localization operator $ \mathfrak {L}^{g_1, g_2}_{\sigma }$ is in $\mathcal {B}(M^{\Phi _1, \Phi _2}({\mathbb {Z}}^n))$ , and we have

$$\begin{align*}\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(M^{\Phi_1, \Phi_2}({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align*}$$

Proof Let $f \in M^{\Phi _1, \Phi _2}({\mathbb {Z}}^n)$ and $h \in M^{\Psi _1, \Psi _2}({\mathbb {Z}}^n)$ . From the given condition, $\Phi _i, \Psi _i$ satisfy condition (3) of Theorem 3.7. Hence, we have $M^{\Phi _1, \Phi _2}({\mathbb {Z}}^n) \subseteq M^{\Psi _1, \Psi _2}({\mathbb {Z}}^n)$ . Now, applying Hölder’s inequality (7), we obtain

$$ \begin{align*} \left| \left\langle \mathfrak{L}^{g_1, g_2}_{\sigma}f, h \right\rangle \right| & \leq \sum_{m \in {\mathbb{Z}}^n} \int_{{\mathbb{T}}^n} |\sigma(m, w)| \; \left|V_{g_1}f(m, w) \right| \; \left| V_{g_2}h(m, w) \right| \; dw \\ & \leq \left\Vert \sigma \right\Vert{}_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \Vert V_{g_1}f \Vert_{L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \Vert V_{g_2}h \Vert_{L^{\Psi_1, \Psi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \\ & = \left\Vert \sigma \right\Vert{}_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \Vert f \Vert_{M^{\Psi_1, \Psi_2}({\mathbb{Z}}^n)} \Vert h \Vert_{M^{\Psi_1, \Psi_2}({\mathbb{Z}}^n)} \\ & \leq \left\Vert \sigma \right\Vert{}_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)} \Vert f \Vert_{M^{\Phi_1, \Phi_2}({\mathbb{Z}}^n)} \Vert h \Vert_{M^{\Psi_1, \Psi_2}({\mathbb{Z}}^n)}. \end{align*} $$

Hence, using the duality between $M^{\Phi _1, \Phi _2}({\mathbb {Z}}^n)$ and $M^{\Psi _1, \Psi _2}({\mathbb {Z}}^n)$ , we get

$$\begin{align*}\Vert \mathfrak{L}^{g_1, g_2}_{\sigma} \Vert_{\mathcal{B}(M^{\Phi_1, \Phi_2}({\mathbb{Z}}^n))} \leq \Vert \sigma \Vert_{L^{\Phi_1, \Phi_2} ({\mathbb{Z}}^n \times {\mathbb{T}}^n)}. \end{align*}$$

This completes the proof.