The Banach Algebra of Functions With Fourier Transforms in Weighted Amalgam Spaces

TL;DR

This paper introduces a new Banach algebra of functions with Fourier transforms in weighted amalgam spaces, exploring its algebraic, topological, and functional properties, including modules, embeddings, and multipliers.

Contribution

It defines a novel function space with Fourier transforms in weighted amalgam spaces and investigates its algebraic and analytical properties, including Banach algebra structure and multipliers.

Findings

The space forms a Banach algebra under convolution.

The space is translation invariant and a Banach module.

The paper characterizes multipliers and embedding relations.

Abstract

In this paper, we define $A_{\vartheta _{1},\vartheta _{2}}^{p,1,q,r}\left(G\right) $ to be space of all functions in $\left( L_{\vartheta_{1}}^{p},\ell ^{1}\right) $ whose Fourier transforms belong to $\left( L_{\vartheta _{2}}^{q},\ell ^{r}\right) .$ Moreover, we consider the basic and advance properties of this space including Banach algebra, translation invariant, Banach module, a generalized type of Segal algebra etc. Also, we study some inclusions, compact embeddings in sense to weights and further discuss multipliers of this space.

Equations

$$\sum _{n\ge 1}\frac{\log \vartheta \left(nx\right)}{n^2}<\infty$$

$$Ho{m}_A\left(V,W^{\ast }\right)\cong {\left(V{\otimes }_{A}W\right)}^{\ast }$$

$$M\left(\vartheta \right)=\left\{\mu \in M\left(G\right):\underset{G}{\int }\vartheta d\left|\mu \right|<\infty \right\}.$$

$$\left(L^p,{\ell }^q\right)=\left\{f\in L_{loc}^p\left(\mathbb{R}\right):{\Vert f\Vert }_{pq}<\infty \right\}$$

$${\Vert f\Vert }_{pq}={\left[\sum _{n=-\infty }^{\infty }{\left[\underset{n}{\overset{n+1}{\int }}{\left|f(x)\right|}^{p}dx\right]}^{\frac{q}{p}}\right]}^{\frac{1}{q}}.$$

$$\left\{f\in L_{loc}^p\left(G\right):{\Vert f\Vert }_{pq}<\infty \right\},$$

$${\Vert f\Vert }_{pq}$$

$${\Vert f\Vert }_{\infty q}$$

$${\Vert f\Vert }_{p\infty }$$

$${\Vert f\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}={\Vert f\Vert }_{p1,{\vartheta }_1}+{\Vert \hat{f}\Vert }_{qr,{\vartheta }_2}$$

$${\Vert f_n-f_m\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}$$

$${\Vert \widehat{f_n}-\hat{f}\Vert }_{\infty }\le {\Vert f_n-f\Vert }_1\le {\Vert f_n-f\Vert }_{1,{\vartheta }_1}.$$

$${\Vert \widehat{f_{n_k}}-\hat{f}\Vert }_{\infty }=\underset{k}{\sup }\left|f_{n_k}\left(x\right)-f\left(x\right)\right|\le {\Vert \widehat{f_n}-\hat{f}\Vert }_{\infty }⟶0.$$

$${\Vert f_n-f\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}={\Vert f_n-f\Vert }_{p1,{\vartheta }_1}+{\Vert \widehat{f_n}-\hat{f}\Vert }_{qr,{\vartheta }_2}⟶0$$

$${\Vert f\ast g\Vert }_{p1,{\vartheta }_1}\le C{\Vert f\Vert }_{p1,{\vartheta }_1}{\Vert g\Vert }_{p1,{\vartheta }_1}\text{.}$$

$$\left|\widehat{\left(f\ast g\right)}\left(x\right)\right|=\left|\hat{f}\left(x\right)\right|\left|\hat{g}\left(x\right)\right|\le {\Vert \hat{f}\Vert }_{\infty }\left|\hat{g}\left(x\right)\right|,$$

$${\Vert \widehat{f\ast g}\Vert }_{qr,{\vartheta }_2}\le {\Vert \hat{f}\Vert }_{\infty }{\Vert \hat{g}\Vert }_{qr,{\vartheta }_2}\le {\Vert f\Vert }_{p1,{\vartheta }_1}{\Vert \hat{g}\Vert }_{qr,{\vartheta }_2}$$

$${\Vert f\ast g\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}$$

$$\underset{K}{\int }\left|f(x)\right|dx$$

$$C{\vartheta }_1\left(y\right)\le {\Vert T_{y}f\Vert }_{p,{\vartheta }_1}\le C^{\ast }{\Vert T_{y}f\Vert }_{p1,{\vartheta }_1}\le C^{\ast }{\Vert T_{y}f\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}$$

$$C_1{\vartheta }_1\left(y\right)\le {\Vert T_{y}f\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}$$

$${\Vert \widehat{T_{y}f}\Vert }_{qr,{\vartheta }_2}={\Vert M_{-y}\hat{f}\Vert }_{qr,{\vartheta }_2}={\Vert \hat{f}\Vert }_{qr,{\vartheta }_2}<\infty .$$

$${\Vert T_{y}f\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}$$

$${\Vert T_{y}f-f\Vert }_{p1,{\vartheta }_1}<\frac{\epsilon }{2}$$

$${\Vert \widehat{T_{y}f}-\hat{f}\Vert }_{qr,{\vartheta }_2}={\Vert M_{-y}\hat{f}-\hat{f}\Vert }_{qr,{\vartheta }_2}<\frac{\epsilon }{2}$$

$${\Vert T_{y}f-f\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}={\Vert T_{y}f-f\Vert }_{p1,{\vartheta }_1}+{\Vert \widehat{T_{y}f-f}\Vert }_{qr,{\vartheta }_2}<\epsilon$$

$$F_{0,{\vartheta }_1}=\left\{f\in L_{{\vartheta }_1}^1\left(G\right):\hat{f}\text{ has a compact support}\right\}.$$

$${\Vert f\ast g\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}$$

$${\Vert T_{y}f-f\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}<\frac{\epsilon }{2}$$

$$g\ast f-f=\underset{G}{\int }g(y)\left(T_{y}f-f\right)dy$$

$${\Vert g\ast f-f\Vert }_{{\vartheta }_1,{\vartheta }_2}^{p,1,q,r}$$

Full text

1. Introduction

Let $G$ be a locally compact abelian group with Haar measure $\mu$. For $1\leq p,q\leq\infty,$ an amalgam space $\left(L^{p},\ell^{q}\right)\left(G\right)$ is to be a Banach space of all measurable functions on $G$ which belong locally to $L^{p}$ and globally to $\ell^{q}$. Many authors have considered special cases of amalgam spaces such as Krogstad [22], Liu et al. [26], Szeptycki [35], Wiener [37]. Also, Holland [21] presented an important study for amalgams on the real line. In 1979, Stewart [34] extended Holland’s definition to locally compact abelian groups for locally compact groups by the Structure Theorem.

For $1\leq p<\infty,$ the set $A_{p}\left(\mathbb{R}^{d}\right)$ is the space of all complex-valued functions in $L^{1}\left(\mathbb{R}^{d}\right)$ whose the Fourier transforms belong to $L^{p}\left(\mathbb{R}^{d}\right).$ This space have considered some authors, see [23], [24], [27]. Moreover, Feichtinger and Gurkanli [12], Fischer et al. [13] and Gurkanli [16] have found some generalized results in sense to weights.

In 1926, Wiener [37] presented the first systematical work on amalgam spaces $\left(L^{p},\ell^{q}\right)$. The usage areas of the amalgam spaces are generally harmonic and time-frequency analysis. For an another historical journey, we can refer [14]. Moreover, the amalgam spaces or some special cases of these spaces were investigate by a number of authors, see [4], [5], [18], [19].

In this paper, we define a space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)=\left\{f\in\left(L_{\vartheta_{1}}^{p},\ell^{1}\right):\widehat{f}\in\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)\right\}$ and investigate the basic properties of the space. Also, we consider several inclusions under some conditions. Moreover, we discuss compact embeddings with other suitable spaces under some conditions and reveal multipliers of the space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right).$ One of the our purpose of this paper is to generalize some of the results in [1] and [36] to the double weighted case.

2. Notation and Preliminaries

Throughout this paper, we will work on $G$ with Lebesgue measure $dx$. We denote by $C_{c}\left(G\right)$ as the linear space of continuous functions on $G$, which have compact support. The translation and character operators $T_{y}$ and $M_{t}$ are defined by $T_{y}f(x)=f(x-y)$ and $M_{t}f(y)=\left\langle y,t\right\rangle.f(y)$ respectively for $x,y\in G$ , $t\in G.$ Also $(B,\left\|.\right\|_{B})$ is strongly translation invariant if one has $T_{y}B\subseteq B$ and $\left\|T_{y}f\right\|_{B}=\left\|f\right\|_{B}$ and strongly character invariant if $M_{t}B\subseteq B$ and $\left\|M_{t}f\right\|_{B}=\left\|f\right\|_{B}$ for all $f\in B$, $y\in G$ and $t\in G$.

A measurable and locally integrable function $\vartheta:G\longrightarrow\left(0,\infty\right)$ is called a weight function. Moreover the weight $\vartheta$ will be called a Beurling’s weight function if $\vartheta\left(x\right)\geq 1$ and $\vartheta(x+y)\leq\vartheta\left(x\right)\vartheta\left(y\right)$ for all $x,y\in G$. We say that $\vartheta_{1}\prec\vartheta_{2}$ if and only if there exists a $C>0$ such that $\vartheta_{1}\left(x\right)\leq C\vartheta_{2}\left(x\right)$ for all $x\in G$. Also, if $\vartheta_{1}\prec\vartheta_{2}$ and $\vartheta_{2}\prec\vartheta_{1}$ are satisfied, then we say that weight functions $\vartheta_{1}$ and $\vartheta_{2}$ are equivalent and denoted by $\vartheta_{1}\thickapprox\vartheta_{2}$. A weight function $\vartheta$ is said to satisfy the Beurling-Domar (shortly BD) condition if

[TABLE]

for all $x\in G,$ see [10]. Moreover, it is clear that every weight function is equivalent to a continuous weight, see [28, Lemma 4]. Hence, we deduce that $\vartheta\left(x\right)\longrightarrow\infty$ as $x\longrightarrow\infty.$ For example, if we choose the polynomial type weight function $\vartheta_{s}\left(x\right)=\left(1+\left|x\right|\right)^{s}$ for $s\geq 0,$ then we have $\vartheta_{s}\left(x\right)\longrightarrow\infty$ as $x\longrightarrow\infty,$ see [30].

For $1\leq p<\infty,$ the weighted Lebesgue space $L_{\vartheta}^{p}\left(G\right)=\left\{f:f\vartheta\in L^{p}\left(G\right)\right\}$ is a Banach space with norm $\left\|f\right\|_{p,\vartheta}=\left\|f\vartheta\right\|_{p}$ and its dual space $L_{\vartheta^{-1}}^{q}\left(G\right),$ where $\frac{1}{p}+\frac{1}{q}=1.$ It is known that the space $L_{\vartheta}^{p}\left(G\right)$ is a reflexive Banach space for $1<p<\infty$. Moreover, for $p=1,$ $L_{\vartheta}^{1}\left(G\right)$ is a Banach algebra under convolution, called a Beurling algebra. It is obvious that $\left\|.\right\|_{1}\leq\left\|.\right\|_{1,\vartheta}$ and $L_{\vartheta}^{1}\left(G\right)\subset L^{1}\left(G\right)$. It is known that $L_{\vartheta_{2}}^{p}\left(G\right)\subset L_{\vartheta_{1}}^{p}\left(G\right)$ if and only if $\vartheta_{1}\prec\vartheta_{2}$, see [12].

Now, assume that $A$ is a Banach algebra. It is known that a Banach space $B$ is called a Banach $A$-module if there exists a bilinear operation $\cdot:A\times B\longrightarrow B$ such that

1. (i)

$\left(f\cdot g\right)\cdot h=f\cdot\left(g\cdot h\right)$ for all $f,g\in A$, $h\in B.$ 2. (ii)

For some constant $C\geq 1,$ $\left\|f\cdot h\right\|_{B}\leq C\left\|f\right\|_{A}\left\|h\right\|_{B}$ for all $f\in A,$ $h\in B,$ see [11].

Moreover, $L_{loc}^{p}\left(G\right)$ is the space of all functions on $G$ such that $f$ restricted to any compact subset $K$ of $G$ belongs to $L^{p}\left(G\right)$. For $p=1,$ the space $L_{loc}^{1}\left(G\right)$ is to be space of all measurable functions $f$ on $G$ such that $f.\chi_{K}\in L^{1}\left(G\right)$ for any compact subset $K\subset G$. Moreover, a Banach function space (shortly BF-space) on $G$ is a Banach space $\left(B,\left\|.\right\|_{B}\right)$ of measurable functions which is continuously embedded into $L_{loc}^{1}\left(G\right)$, i.e. for any compact subset $K\subset G$ there exists some constant $C_{K}>0$ such that $\left\|f.\chi_{K}\right\|_{L^{1}}\leq C_{K}.\left\|f\right\|_{B}$ for all $f\in B$. Also, a BF-space is called solid if $g\in B,$ $f\in L_{loc}^{1}\left(G\right)$ and $\left|f\left(x\right)\right|\leq\left|g\left(x\right)\right|$ locally almost everywhere (shortly l.a.e) implies $f\in B$ and $\left\|f\right\|_{B}\leq\left\|g\right\|_{B}$. It is easy to see that $\left(B,\left\|.\right\|_{B}\right)$ is solid if and only if it is a $L^{\infty}$-module. Let $f$ be a measurable function on $G.$

Suppose that $V$ and $W$ are two Banach modules over a Banach algebra $A$. Then a multiplier from $V$ into $W$ is a bounded linear operator $T:V\longrightarrow W$, which commutes with module multiplication, i.e. $T(av)=aT(v)$ for $a\in A$ and $v\in V$. Also, we denote by $Hom_{A}\left(V,W\right)$ as the space of all multipliers from $V$ into $W.$ For convenience, we write that $Hom_{A}\left(V,V\right)=Hom_{A}\left(V\right)$. It is known that

[TABLE]

where $W^{\ast}$ is dual of $W$ and $V\otimes_{A}W$ is the $A$-module tensor product of $V$ and $W$,see [31, Corollary 2.13].

Moreover, the space $M\left(G\right)$ denotes all bounded regular Borel measures on $G.$ Now, we define

[TABLE]

It is known that the space of multipliers from $L_{\vartheta}^{1}\left(G\right)$ to $L_{\vartheta}^{1}\left(G\right)$ is homeomorphic to the space $M\left(\vartheta\right),$see [15].

In [7], Cigler revealed a generalization of Segal algebra. To define this we suppose that $S_{\vartheta}(G)=S_{\vartheta}$ is a subalgebra of $L_{\vartheta}^{1}(G)$ satisfying the conditions below.

1. (S1)

$S_{\vartheta}$ is dense in $L_{\vartheta}^{1}(G).$ 2. (S2)

$S_{\vartheta}$ is a Banach algebra under some norm $\left\|.\right\|_{S_{\vartheta}}$ and invariant under translations. 3. (S3)

$\left\|T_{y}f\right\|_{S_{\vartheta}}\leq\vartheta(y)\left\|f\right\|_{S_{\vartheta}}$ for all $y\in G$ and for each $f\in S_{\vartheta}$. 4. (S4)

If $f\in S_{\vartheta}$, then for every $\varepsilon>0$ there is a neighborhood $U$ of the identity element of $G$ such that $\left\|T_{y}f-f\right\|_{S_{\vartheta}}<\varepsilon$ for all $y\in U.$ 5. (S5)

$\left\|f\right\|_{1,\vartheta}\leq\left\|f\right\|_{S_{\vartheta}}$ for all $f\in S_{\vartheta}.$

Denote the amalgam of $L^{p}$ and $\ell^{q}$ on the real line is the normed space

[TABLE]

equipped with the norm

[TABLE]

We make the appropriate changes for $p$, $q$ infinite. The norm (2.1) makes the amalgam space $\left(L^{p},\ell^{q}\right)$ into a Banach space, see [21]. Note that, the space $C_{c}(G)$ is a subspace of every amalgam spaces. Now, let $1\leq p,q<\infty$. Then, the dual space of $\left(L^{p},\ell^{q}\right)$ is isometrically isomorphic to $\left(L^{p^{\shortmid}},\ell^{q^{\shortmid}}\right)$ where $\frac{1}{p}+\frac{1}{p^{\prime}}=\frac{1}{q}+\frac{1}{q^{\prime}}=1$, see [1], [33].

Stewart [34] give an alternative definition of $\left(L^{p},\ell^{q}\right)\left(G\right)$ based on the Structure Theorem [20, Theorem 24.30]. Indeed, let $G=\mathbb{R}^{a}\times G_{1}$, where $a$ is a nonnegative integer and $G_{1}$ is a locally compact abelian group which contains an open compact subgroup $H$. Also, we denote $I=\left[0,1\right)^{a}\times H$ and $J=\mathbb{Z}^{a}\times T$ where $T$ is a transversal of $H$ in $G_{1}$, i.e. $G_{1}=\mathop{\textstyle\bigcup}\limits_{t\in T}\left(t+H\right)$ is a coset decomposition of $G_{1}$. For $\alpha\in J,$ we define $I_{\alpha}=\alpha+I.$ Therefore $G$ equals the disjoint union of relatively compact sets $I_{\alpha}$. We normalize $\mu$ such that $\mu\left(I\right)=\mu\left(I_{\alpha}\right)=1$ for all $\alpha$. Let $1\leq p,q\leq\infty$. The amalgam space $\left(L^{p},\ell^{q}\right)\left(G\right)=\left(L^{p},\ell^{q}\right)$ is a Banach space

[TABLE]

where

[TABLE]

If $G=\mathbb{R}$, then we have $J=\mathbb{Z}$, $I_{\alpha}=\left[\alpha,\alpha+1\right)$ and (2.2) becomes (2.1).

Throughout this paper, $G$ will denote a locally compact abelian group with Haar measure and $J$ and $I_{\alpha}$ will define as above. Moreover, we assume that $1\leq p,q,r<\infty$ and every weights we used are Beurling’s weight functions on $G$.

3. The Space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$

Now, we define the vector space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)=\left\{f\in\left(L_{\vartheta_{1}}^{p},\ell^{1}\right):\widehat{f}\in\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)\right\}$ and equip with the norm

[TABLE]

for $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$. It is note that, since $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ is a subspace of $L^{1}\left(G\right),$ the Fourier transforms of the functions in $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ are well-defined.

The proof of the following theorem is clear, see [2].

Theorem 1**.**

Let $1\leq p,q<\infty$. Let $\left(f_{n}\right)_{n\in\mathbb{N}}$ be a sequence in $\left(L_{\vartheta_{1}}^{p},\ell^{q}\right)$ and $\left\|f_{n}-f\right\|_{pq,\vartheta_{1}}\longrightarrow 0$ as $n\longrightarrow\infty$, where $f\in\left(L_{\vartheta_{1}}^{p},\ell^{q}\right)$. Then $\left(f_{n}\right)_{n\in\mathbb{N}}$ has a subsequence which converges pointwise almost everywhere to $f$.

Theorem 2**.**

The space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ is a Banach space with respect to $\left\|.\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}.$

Proof.

Assume that $\left(f_{n}\right)_{n\in\mathbb{N}}$ is a Cauchy sequence in $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$. Thus given $\varepsilon>0,$ there is an $n_{1}\in\mathbb{N}$ such that for all $n,m\geqslant n_{1}$ implies

[TABLE]

Therefore, $\left(f_{n}\right)_{n\in IN}\subset\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ and $\left(f_{n}\right)_{n\in IN}\subset\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)$ are Cauchy sequences with respect to $\left\|.\right\|_{p1,\vartheta_{1}}$ and $\left\|.\right\|_{qr,\vartheta_{2}}$, respectively. Since the spaces $\left(\left(L_{\vartheta_{1}}^{p},\ell^{1}\right),\left\|.\right\|_{p1,\vartheta_{1}}\right)$ and $\left(\left(L_{\vartheta_{2}}^{q},\ell^{r}\right),\left\|.\right\|_{qr,\vartheta_{2}}\right)$ are two Banach spaces, there exist $f\in\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ and $g\in\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)$ such that $\left\|f_{n}-f\right\|_{p1,\vartheta_{1}}\longrightarrow 0$, $\left\|\widehat{f_{n}}-g\right\|_{qr,\vartheta_{2}}\longrightarrow 0$. If we use the inequality $\left\|.\right\|_{1,\vartheta_{1}}\leq\left\|.\right\|_{p1,\vartheta_{1}},$ then we get $\left\|f_{n}-f\right\|_{1,\vartheta_{1}}\longrightarrow 0.$ By Theorem 1, there is a subsequence $\left\{\widehat{f_{n_{k}}}\right\}_{k\in IN}$ of $\left\{\widehat{f_{n}}\right\}_{n\in IN}$ such that $\widehat{f_{n_{k}}}\longrightarrow g$ a.e. Since $\vartheta_{1}$ is a Beurling’s weight, we have

[TABLE]

This follows that $\left\|\widehat{f_{n}}-\widehat{f}\right\|_{\infty}\longrightarrow 0.$ Moreover, we get

[TABLE]

Therefore, we have $\widehat{f}=g.$ This yields that

[TABLE]

and $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right).$ That is the desired result.

Theorem 3**.**

If $1<p,q,r<\infty,$ then the space $\left(A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right),\left\|.\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\right)$ is a Banach algebra with respect to convolution.

Proof.

It is note that the space $\left(A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right),\left\|.\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\right)$ is a Banach space by the Theorem 2. Now, let $f,g\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ be given. Thus, we have $f,g\in\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ and $\widehat{f},\widehat{g}\in\left(L_{\vartheta_{2}}^{q},\ell^{r}\right).$ Since $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ is a Banach algebra under convolution (see [33]), we have $f\ast g\in\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ and there exists $C\geq 1$ such that

[TABLE]

If we consider the inequality

[TABLE]

then we have

[TABLE]

and $\widehat{f\ast g}\in\left(L_{\vartheta_{2}}^{q},\ell^{r}\right).$ Therefore, we have $f\ast g\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right).$ This follows by (3.1) and (3.2) that

[TABLE]

Theorem 4**.**

The space $\left(A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right),\left\|.\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\right)$ is a BF-space.

Proof.

Assume that $K\subset G$ is a compact subset and let $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$. Since $\vartheta_{1}$ is a Beurling’s weight function and the space $L_{\vartheta_{1}}^{p}\left(G\right)$ is continuously embedded in $L_{\vartheta_{1}}^{1}\left(G\right)$, we have

[TABLE]

This completes the proof.

Theorem 5**.**

The following statements hold.

1. (i)

The space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ is translation invariant and for every $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ and $y\in G$ the inequality $C_{1}\left(f\right)\vartheta_{1}\left(y\right)\leq\left\|T_{y}f\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\leq C_{2}\left(f\right)\vartheta_{1}\left(y\right)$ holds where $C_{1}\left(f\right)>0$ and $C_{2}\left(f\right)=\left\|f\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}$. 2. (ii)

The map $y\longrightarrow T_{y}f$ is continuous from $G$ into $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ for every $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right).$

Proof.

Let $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$. Since $\vartheta_{1}$ is a Beurling’s weight function, it is easy to see that $T_{y}f\in\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ and $\left\|T_{y}f\right\|_{p1,\vartheta_{1}}\leq\vartheta_{1}\left(y\right)\left\|f\right\|_{p1,\vartheta_{1}}$ for all $y\in G$.

Moreover, for $p>1,$ it is clear that $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ is continuously embedded in $L_{\vartheta_{1}}^{p}\left(G\right),$ see [33]. Thus, if we consider the Lemma 2.2 in [12], then there is a constant $C>0$ depending on $f$ such that

[TABLE]

for all $y\in G.$ This follows that

[TABLE]

where $C_{1}=\frac{C}{C^{\ast}}$ depends on $f.$ It is clear that $\widehat{L_{y}f}=M_{-y}\widehat{f}.$ Moreover, if we consider that the weighted amalgam space $\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)$ is strongly character invariant and the function $t\longrightarrow M_{t}f$ is continuous from $\widehat{G}$ into $\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)$ (see [3], [29], [32]), then we have

[TABLE]

This follows that $T_{y}f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right).$ Since $\vartheta_{1}\geq 1,$ we get

[TABLE]

for all $y\in G.$ This completes (i) by (3.3) and (3.4)*. *It is obvious that $T_{y}$ is linear. For any $\varepsilon>0,$ there is a neighbourhood $V_{1}$ of the unit element of $G$ such that

[TABLE]

for all $y\in V_{1}.$ Also, there is a neighbourhood $V_{2}$ of the unit element of $G$ such that

[TABLE]

for all $y\in V_{2}.$ Now, let us denote $U=V_{1}\cap V_{2}.$ By (3.5) and (3.6), we have

[TABLE]

for all $y\in U.$ That is the desired result.

Theorem 6**.**

Assume that $\vartheta_{1}$ satisfies (BD) condition. Then the space $A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$ is a $S_{\vartheta_{1}}$ algebra.

Proof.

We have already proved the several conditions for $S_{\vartheta_{1}}$ algebra** **in Theorem 3 and Theorem 5. Now, let us denote

[TABLE]

Since $\vartheta_{1}$ satisfies (BD) condition, the set $F_{0,\vartheta_{1}}$ is dense in $L_{\vartheta_{1}}^{1}\left(G\right).$ It is clear that $C_{c}\left(G\right)\subset\left(L_{\vartheta_{2}}^{q},\ell^{r}\right).$ Because of the fact that the inclusions $F_{0,\vartheta_{1}}\subset A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)\subset L_{\vartheta_{1}}^{1}\left(G\right)$ hold and $F_{0,\vartheta_{1}}$ is dense in $L_{\vartheta_{1}}^{1}\left(G\right)$, then $A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$ is dense in $L_{\vartheta_{1}}^{1}\left(G\right)$. That is the desired result.

Theorem 7**.**

Let $1<p,q,r<\infty.$ If $\vartheta_{1}\prec\vartheta_{0}$, then $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ is a Banach $L_{\vartheta_{0}}^{1}\left(G\right)$-module with respect to $\left\|.\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}$.

Proof.

Since $\vartheta_{1}\prec\vartheta_{0}$, we have $L_{\vartheta_{0}}^{1}\left(G\right)\hookrightarrow L_{\vartheta_{1}}^{1}\left(G\right).$ Moreover, if we consider the fact that $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ is a Banach $L_{\vartheta_{1}}^{1}\left(G\right)$-module for $1<p<\infty$, then there exists $C>0$ such that

[TABLE]

for any $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ and $g\in L_{\vartheta_{0}}^{1}\left(G\right).$ If we define a new norm $\left\|\left|.\right|\right\|$ on $L_{\vartheta_{0}}^{1}\left(G\right)$ such that $\left\|\left|.\right|\right\|=\max\left\{c_{1},c_{2}\right\}\left\|.\right\|_{1,\vartheta_{0}},$ then this norm is equivalent to the norm $\left\|.\right\|_{1,\vartheta_{0}}$ on $L_{\vartheta_{0}}^{1}\left(G\right).$ This completes the proof.

Theorem 8**.**

Suppose that $\vartheta_{1}$ satisfies (BD) condition. Then $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ has an approximate unit with compactly supported Fourier transforms.

Proof.

Let $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ and $\varepsilon>0$ be given. Then by Theorem 5, there exists a neighbourhood $U$ of the unit element of $G$ such that

[TABLE]

for all $y\in U$. Let be taken a non-negative function $g\in C_{c}(G)$ such that supp$g\subset U$ and $\mathop{\textstyle\int}\limits_{G}g(y)dy=1$. Since

[TABLE]

by using the inequality (3.7), we get

[TABLE]

Moreover, since the set $F_{0,\vartheta_{1}}$ is dense in $L_{\vartheta_{1}}^{1}\left(G\right)$ by Theorem 6, there exists $h\in F_{0,\vartheta_{1}}$ such that

[TABLE]

If we consider that the space $C_{c}(G)$ is included in all amalgam spaces, then it we have $h\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}(G)$. Also, it is clear that

[TABLE]

By (3.8), (3.9) and (3.10), we obtain

[TABLE]

Consider the mapping $\Phi$ from $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ into $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)\times\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)$ defined by $\Phi(f)=\left(f,\widehat{f}\right)$. This is a linear isometry from $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ into $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)\times\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)$ in sense to the norm

[TABLE]

for $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right).$ Hence it is easy to see that $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ is a closed subspace of the Banach space $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)\times\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)$. Let

[TABLE]

and

[TABLE]

where $\frac{1}{r}+\frac{1}{r^{\prime}}=1,\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and $\frac{1}{q}+\frac{1}{q^{\prime}}=1$.

The following theorem is easily proved by Duality Theorem 1.7 in [25].

Theorem 9**.**

The dual space $\left(A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)\right)^{\ast}$ of $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ is isomorphic to $\left(L_{\vartheta_{1}^{-1}}^{p^{\prime}},\ell^{\infty}\right)\times\left(L_{\vartheta_{2}^{-1}}^{q^{\prime}},\ell^{r^{\prime}}\right)/K$.

4. Inclusions of the spaces $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$

Theorem 10**.**

The inclusion $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)\subset A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(G\right)$ holds if and only if the space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ is continuously embedded in $A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(G\right)$.

Proof.

The sufficient condition of the theorem is clear by definition of embedding. Now, assume that $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)\subset A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(G\right)$ holds. Moreover, we define the sum norm $\left\|\left|.\right|\right\|=\left\|.\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}+\left\|.\right\|_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}.$ It is easy to see that $\left(A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right),\left\|\left|.\right|\right\|\right)$ is a Banach space. Now, let us define the unit function $I$ from $\left(A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right),\left\|\left|.\right|\right\|\right)$ into $\left(A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right),\left\|.\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\right)$. Since the inequality

[TABLE]

is satisfied, $I$ is continuous. If we consider the Banach’s theorem, then $I$ is a homeomorphism, see [6]. That means the norms $\left\|\left|.\right|\right\|$ and $\left\|.\right\|_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}$ are equivalent. Thus, for every $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ there exists $c>0$ such that

[TABLE]

Therefore, by using (4.1) and the definition of norm $\left\|\left|.\right|\right\|,$ we obtain

[TABLE]

That is the desired result.

Now, we give some continuous embeddings of $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ under some conditions.

Theorem 11**.**

The following statements are true.

1. (i)

If $p_{2}\leq p_{1}$ and $r_{1}\leq r_{2}$, then we have $A_{\vartheta_{1},\vartheta_{2}}^{p_{1},1,q,r_{1}}\left(G\right)\hookrightarrow A_{\vartheta_{1},\vartheta_{2}}^{p_{2},1,q,r_{2}}\left(G\right).$ 2. (ii)

If $\vartheta_{3}\prec\vartheta_{1}$ and $\vartheta_{4}\prec\vartheta_{2},$ then $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ is continuously embedded in $A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(G\right)$. 3. (iii)

If $p_{2}\leq p_{1}$ and $q_{2}\leq q_{1}$, then we get $A_{\vartheta_{1},\vartheta_{2}}^{p_{1},1,q_{1},r}\left(G\right)\hookrightarrow A_{\vartheta_{1},\vartheta_{2}}^{p_{2},1,q_{2},r}\left(G\right).$ 4. (iv)

If $\vartheta_{3}\prec\vartheta_{1}$, $q_{2}\leq q_{1}$ and $r_{1}\leq r_{2},$ then the space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q_{1},r_{1}}\left(G\right)$ is continuously embedded in $A_{\vartheta_{3},\vartheta_{2}}^{p,1,q_{2},r_{2}}\left(G\right).$

Proof.

Let $p_{2}\leq p_{1}$ and $r_{1}\leq r_{2}$. Moreover, since the embeddings $\left(L_{\vartheta_{1}}^{p_{1}},\ell^{1}\right)\hookrightarrow\left(L_{\vartheta_{1}}^{p_{2}},\ell^{1}\right)$ and $\left(L_{\vartheta_{2}}^{q},\ell^{r_{1}}\right)\hookrightarrow\left(L_{\vartheta_{2}}^{q},\ell^{r_{2}}\right)$ hold under these hypotheses (see [19], [33]), there exist $c_{1},c_{2}>0$ such that

[TABLE]

for all $f\in A_{\vartheta_{1},\vartheta_{2}}^{p_{1},1,q,r_{1}}\left(G\right)$. This completes $\left(i\right).$ Now, let $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ be given. Hence, we get $f\in$ $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ and $\widehat{f}\in\left(L_{\vartheta_{2}}^{q},\ell^{r}\right)$. Moreover, assume that $\vartheta_{3}\prec\vartheta_{1}$ and $\vartheta_{4}\prec\vartheta_{2}$ hold. This follows that there exist $c_{3},c_{4}>0$ such that $\vartheta_{3}\left(x\right)\leq c_{3}\vartheta_{1}\left(x\right)$ and $\vartheta_{4}\left(x\right)\leq c_{4}\vartheta_{2}\left(x\right)$ for all $x\in G.$ Therefore, we have

[TABLE]

This proves $(ii)$. If we consider $(i)$ and $(ii),$ then we obtain $(iii)$ and $(iv).$

Theorem 12**.**

Let $\vartheta,\vartheta_{1}$ and $\vartheta_{2}$ be Beurling’s weights. Then the space $A_{\vartheta_{1},\vartheta}^{p,1,q,r}\left(G\right)$ is continuously embedded in $A_{\vartheta_{2},\vartheta}^{p,1,q,r}\left(G\right)$ if and only if $\vartheta_{2}\prec\vartheta_{1}.$

Proof.

Assume that $A_{\vartheta_{1},\vartheta}^{p,1,q,r}\left(G\right)\hookrightarrow A_{\vartheta_{2},\vartheta}^{p,1,q,r}\left(G\right).$ By the Theorem 5, there are $C_{1},C_{2},C_{3},C_{4}>0$ such that

[TABLE]

and

[TABLE]

for $y\in G.$ Since $T_{y}f\in A_{\vartheta_{1},\vartheta}^{p,1,q,r}\left(G\right)$ for every $f\in A_{\vartheta_{1},\vartheta}^{p,1,q,r}\left(G\right),$ there is a $C>0$ such that

[TABLE]

If we consider the (4.2), (4.3) and (4.4), then we have

[TABLE]

This follows that $\vartheta_{2}\prec\vartheta_{1}.$ The rest of the proof can be proven by the similar in Theorem 11.

The following corollary can be easily proven by using the Theorem 11 and Theorem 12.

Corollary 1**.**

The following expressions are true.

1. (i)

The equality $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)=A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(G\right)$ is satisfied if $\vartheta_{1}\thickapprox\vartheta_{3}$ and $\vartheta_{2}\thickapprox\vartheta_{4}.$ 2. (ii)

If $p_{2}\leq p_{1}$, $q_{2}\leq q_{1}$ and $r_{1}\leq r_{2},$ then we get $A_{\vartheta_{1},\vartheta_{2}}^{p_{1},1,q_{1},r_{1}}\left(G\right)\hookrightarrow A_{\vartheta_{1},\vartheta_{2}}^{p_{1},1,q_{2},r_{2}}\left(G\right).$ 3. (iii)

Let $p_{2}\leq p_{1}$, $q_{2}\leq q_{1}$ and $r_{1}\leq r_{2}.$ If $\vartheta_{3}\prec\vartheta_{1}$ and $\vartheta_{4}\prec\vartheta_{2},$ then we have $A_{\vartheta_{1},\vartheta_{2}}^{p_{1},1,q_{1},r_{1}}\left(G\right)\hookrightarrow A_{\vartheta_{3},\vartheta_{4}}^{p_{1},1,q_{2},r_{2}}\left(G\right)$. 4. (iv)

The expression $A_{\vartheta_{1},\vartheta}^{p,1,q,r}\left(G\right)=A_{\vartheta_{2},\vartheta}^{p,1,q,r}\left(G\right)$ holds if and only if $\vartheta_{2}\thickapprox\vartheta_{1}.$

5. Compact Embeddings of the Space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$

Now, we investigate compact embeddings of the spaces $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(G\right)$ with the similar methods in [18]. Also, we will take $G=\mathbb{R}^{d}$ with Lebesgue measure $dx$ for compact embeddings.

Theorem 13**.**

Let $1<p,q,r<\infty.$ Assume that $\left(f_{n}\right)_{n\in\mathbb{N}}$ is a sequence in $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$. If $\left(f_{n}\right)_{n\in\mathbb{N}}$ converges to zero in $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$, then $\left(f_{n}\right)_{n\in\mathbb{N}}$ converges to zero in the vague topology (which means that

[TABLE]

for $n\longrightarrow\infty$ for all $k\in C_{c}(\mathbb{R}^{d})$,see [8]).

Proof.

Let $k\in C_{c}(\mathbb{R}^{d})$. For $p>1,$ since the space $\left(L_{\vartheta_{1}}^{p},\ell^{1}\right)$ is continuously embedded in $L_{\vartheta_{1}}^{p}\left(\mathbb{R}^{d}\right),$ we have

[TABLE]

where $\frac{1}{p^{\prime}}+\frac{1}{p}=1$ by the Hölder inequality. Therefore, the sequence $\left(f_{n}\right)_{n\in\mathbb{N}}$ converges to zero in vague topology.

Theorem 14**.**

If $\vartheta\prec\vartheta_{1}$ and $\frac{\vartheta\left(x\right)}{\vartheta_{1}\left(x\right)}$ doesn’t tend to zero in $\mathbb{R}^{d}$ as $x\longrightarrow\infty,$ then the embedding of the space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ into $\left(L_{\vartheta}^{p},\ell^{1}\right)$ is never compact.

Proof.

Since $\vartheta\prec\vartheta_{1}$, there exists $C_{1}>0$ such that $\vartheta(x)\leq C_{1}\vartheta_{1}(x)$ for all $x\in\mathbb{R}^{d}$. This follows that $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)\subset\left(L_{\vartheta}^{p},\ell^{1}\right)$. Assume that $\left(t_{n}\right)_{n\in\mathbb{N}}$ is a sequence with $t_{n}\longrightarrow\infty$ in $\mathbb{R}^{d}$. Since $\frac{\vartheta(x)}{\vartheta_{1}\left(x\right)}$ does not tend to zero as $x\longrightarrow\infty$, there exists $\delta>0$ such that $\frac{\vartheta(x)}{\vartheta_{1}\left(x\right)}\geq\delta>0$ for $x\longrightarrow\infty.$ To end the proof, we take any fixed $f\in A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ and define a sequence of functions $\left(f_{n}\right)_{n\in\mathbb{N}}$ where $f_{n}=\left(\vartheta_{1}\left(t_{n}\right)\right)^{-1}T_{t_{n}}f$. This sequence is bounded in $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$. Indeed, by Theorem 5, we get

[TABLE]

Now, we will prove that there would not exists norm convergence of subsequence of $\left(f_{n}\right)_{n\in\mathbb{N}}$ in $\left(L_{\vartheta}^{p},\ell^{1}\right)$. The sequence mentioned above converges to zero in sense to the vague topology. To prove this, for every $k\in C_{c}(\mathbb{R}^{d})$ we obtain

[TABLE]

where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and $C_{1}>0.$ Since the right side of (5.1) tends zero for $n\longrightarrow\infty,$ we have

[TABLE]

If we consider the Theorem 13, the only possible limit of $\left(f_{n}\right)_{n\in\mathbb{N}}$ in $\left(L_{\vartheta}^{p},\ell^{1}\right)$ is zero. This follows by Lemma 2.2 in [12] that there exist $C_{2},C_{3}>0$ depending on $f$ such that

[TABLE]

for all $y\in\mathbb{R}^{d}.$ Thus we have

[TABLE]

Since $\frac{\vartheta\left(t_{n}\right)}{\vartheta_{1}\left(t_{n}\right)}\geq\delta>0$ for all $t_{n}$, by using (5.2) we get

[TABLE]

where $C=\frac{C_{2}}{C_{1}}.$ Thus there would not possible to find norm convergent subsequence of $\left(f_{n}\right)_{n\in\mathbb{N}}$ in $\left(L_{\vartheta}^{p},\ell^{1}\right)$. This completes the proof.

Theorem 15**.**

If $\vartheta_{3}\prec\vartheta_{1}$ and $\frac{\vartheta_{3}\left(y\right)}{\vartheta_{1}\left(y\right)}$ doesn’t tend to zero in $\mathbb{R}^{d},$ then the embedding $i:$ $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)\hookrightarrow A_{\vartheta_{3},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ is never compact.

Proof.

Since $\vartheta_{3}\prec\vartheta_{1}$, then it is clear that $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)\subset A_{\vartheta_{3},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$. It is also known by Theorem 10 that the unit map from $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ into $A_{\vartheta_{3},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ is continuous. Now take any bounded sequence of $\left(f_{n}\right)_{n\in\mathbb{N}}$ in $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$. If there exists norm convergent subsequence of $\left(f_{n}\right)_{n\in\mathbb{N}}$ in $A_{\vartheta_{3},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$, this subsequence also converges in $\left(L_{\vartheta_{3}}^{p},\ell^{1}\right)$. But this is a contradiction because of the fact that the embedding of the space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ into $\left(L_{\vartheta_{3}}^{p},\ell^{1}\right)$ is never compact by the Theorem 14.

Note that if we define the sequence in Theorem 14 as $f_{n}=\left(\vartheta_{2}\left(t_{n}\right)\right)^{-1}T_{t_{n}}f,$ then the proof of the following theorem is similar with the previous one.

Theorem 16**.**

If $\vartheta_{1}\prec\vartheta_{2},\vartheta_{3}\prec\vartheta_{2}$ and $\frac{\vartheta_{3}\left(x\right)}{\vartheta_{2}\left(x\right)}$ does not tend to zero in $\mathbb{R}^{d}$ as $x\longrightarrow\infty,$ then the embedding of the space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ into $\left(L_{\vartheta_{3}}^{p},\ell^{1}\right)$ is never compact.

Theorem 17**.**

The embedding of the space $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ into $A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ is never compact if

1. (i)

$\vartheta_{4}\prec\vartheta_{2}\prec\vartheta_{1},\vartheta_{3}\prec\vartheta_{1}$* and $\frac{\vartheta_{3}\left(x\right)}{\vartheta_{1}\left(x\right)}$ does not tend to zero in $\mathbb{R}^{d}$ as $x\longrightarrow\infty,$ or* 2. (ii)

$\vartheta_{3}\prec\vartheta_{1}\prec\vartheta_{2},\vartheta_{4}\prec\vartheta_{2}$* and $\frac{\vartheta_{3}\left(x\right)}{\vartheta_{2}\left(x\right)}$ does not tend to zero in $\mathbb{R}^{d}$ as $x\longrightarrow\infty.$*

Proof.

Let $\vartheta_{4}\prec\vartheta_{2}\prec\vartheta_{1},\vartheta_{3}\prec\vartheta_{1}.$ Thus, there exist $C_{1},C_{2}>0$ such that $\vartheta_{4}\left(x\right)\leq C_{1}\vartheta_{2}\left(x\right)$ and $\vartheta_{3}\left(x\right)\leq C_{2}\vartheta_{1}\left(x\right)$ for all $x\in\mathbb{R}^{d}.$ This follows that $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)\subset A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ and the unit function $I:A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)\longrightarrow A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ is continuous. Now, suppose that $\frac{\vartheta_{3}\left(x\right)}{\vartheta_{1}\left(x\right)}$ does not tend to zero in $\mathbb{R}^{d}$ as $x\longrightarrow\infty$ and $\left(f_{n}\right)_{n\in\mathbb{N}}$ is a bounded sequence in $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right).$ If any subsequence of $\left(f_{n}\right)_{n\in\mathbb{N}}$ is convergent in $A_{\vartheta_{3},\vartheta_{4}}^{p,1,q,r}\left(\mathbb{R}^{d}\right),$ then this subsequence is also convergent in $\left(L_{\vartheta_{3}}^{p},\ell^{1}\right).$ However, this conflict with Theorem 14, because of the fact that the embedding of $A_{\vartheta_{1},\vartheta_{2}}^{p,1,q,r}\left(\mathbb{R}^{d}\right)$ into $\left(L_{\vartheta_{3}}^{p},\ell^{1}\right)$ is never compact. This completes (i). In similar way, (ii) can be proved.

6. Multipliers of $A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$

In this section, we discuss multipliers of the spaces $A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$. We define the space

[TABLE]

where

[TABLE]

By [17, Proposition 2.1], we get $M_{A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)}\neq\left\{0\right\}.$

Theorem 18**.**

If $\vartheta_{1}$ satisfies (BD) condition, then for a linear operator $T:\left(L_{\vartheta_{1}}^{1},\ell^{1}\right)\longrightarrow A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$ the assertions below are equivalent:

1. (i)

$T\in Hom_{\left(L_{\vartheta_{1}}^{1},\ell^{1}\right)}\left(\left(L_{\vartheta_{1}}^{1},\ell^{1}\right),A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)\right).$** 2. (ii)

There exists a unique $\mu\in M_{{}_{A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)}}$ such that $Tf=\mu\ast f$ for every $f\in\left(L_{\vartheta_{1}}^{1},\ell^{1}\right)$. Moreover the correspondence between $T$ and $\mu$ defines an isomorphism between $Hom_{\left(L_{\vartheta_{1}}^{1},\ell^{1}\right)}\left(\left(L_{\vartheta_{1}}^{1},\ell^{1}\right),A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)\right)$ and $M_{A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)}$.

Proof.

It is known that $A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$ is a $S_{\vartheta_{1}}$ space by Theorem 6. Thus, we get the desired result if we consider the Proposition 2.4 in [17].

Theorem 19**.**

If $\vartheta_{1}$ satisfies (BD) condition and $T\in Hom_{\left(L_{\vartheta_{1}}^{1},\ell^{1}\right)}\left(A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)\right)$, then there is a unique pseudo measure $\sigma\in\left(A(\widehat{G})\right)^{\ast}$ (see [30]) such that $Tf=\sigma\ast f$ for all $f\in A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$.

Proof.

It is known that $A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$ is a $S_{\vartheta_{1}}$ space by Theorem 6 and a Banach module over $\left(L_{\vartheta_{1}}^{1},\ell^{1}\right)$ by Theorem 7. Thus, the proof is completed by Theorem 5 in [9].

Theorem 20**.**

The multiplier space $Hom_{\left(L_{\vartheta_{1}}^{1},\ell^{1}\right)}\left(\left(L_{\vartheta_{1}}^{1},\ell^{1}\right),\left(A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)\right)^{\ast}\right)$ is isomorphic to $\left(L_{\vartheta_{1}^{-1}}^{\infty},\ell^{\infty}\right)\times\left(L_{\vartheta_{2}^{-1}}^{q^{\prime}},\ell^{r^{\prime}}\right)/K$.

Proof.

By Theorem 7, we write $\left(L_{\vartheta_{1}}^{1},\ell^{1}\right)\ast A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)=A_{\vartheta_{1},\vartheta_{2}}^{1,1,q,r}\left(G\right)$. Hence by Corollary 2.13 in [31] and Theorem 9, we have

[TABLE]

where $\frac{1}{q}+\frac{1}{q^{\prime}}=1$ and $\frac{1}{r}+\frac{1}{r^{\prime}}=1.$