New properties of grand amalgam spaces

TL;DR

This paper explores new properties of grand amalgam spaces, including an equivalent discrete definition and conditions under which these spaces form an algebra under convolution on locally compact Abelian groups.

Contribution

It introduces a discrete definition of grand amalgam spaces and characterizes when these spaces are closed under convolution on Abelian groups.

Findings

Established an equivalent discrete definition of grand amalgam spaces.

Identified necessary and sufficient conditions for these spaces to be convolution algebras.

Extended the understanding of algebraic structures within grand amalgam spaces.

Abstract

In $ \left[14\right]$, a new family called grand amalgam space $W( L^{p),\theta},L^{q),\theta })$ of amalgam spaces was defined and investigated properties of these spaces. The present paper is a sequel to my work $[14].$ In this paper, notations are included in Section 1. In Section 2, we introduce another equivalent but discrete definition of grand amalgam space and study properties of these spaces. In Section 3, we determine necessary and sufficient conditions on a locally compact Abelian group $G$ for the grand amalgam spaces $W(L^{p,)\theta},L^ {q,\theta})$ to be an algebra under convolution.

Equations

$${\Vert f\Vert }_{p)}=\underset{0<\epsilon \le p-1}{\sup }{\left(\epsilon {\int }_{\Omega }{\left|f\right|}^{p-\epsilon }d\mu \right)}^{\frac{1}{p-\epsilon }}$$

$$L^p\left(\Omega \right)\subset L^{p),\theta }\left(\Omega \right)\subset L^{p-\epsilon }\left(\Omega \right).$$

$${\Vert f\Vert }_{p),\theta ,\Omega }={\Vert f\Vert }_{p),\theta }=\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\left({\int }_{\Omega }{\left|f\right|}^{p-\epsilon }d\mu \right)}^{\frac{1}{p-\epsilon }}=\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\Vert f\Vert }_{p-\epsilon };$$

$$\underset{\epsilon \to 0}{\lim }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\Vert f\Vert }_{p-\epsilon }=0,\left[12\right].$$

$${\Vert u\Vert }_{{\ell }^{p),\theta }\left(J\right)}=\underset{0<\epsilon \le p-1}{\sup }{\left({\epsilon }^{\theta }\sum _{k\in J}{\left|u_k\right|}^{p-\epsilon }\right)}^{\frac{1}{p-\epsilon }}=\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\Vert u\Vert }_{{\ell }^{p-\epsilon }\left(J\right)}.$$

$${\Vert g\Vert }_{(p^{{}^{\prime }},\theta }=\underset{g=\sum _{k=1}^{\infty }{g}_k}{\inf }\left\{\sum _{k=1}^{\infty }\underset{0<\epsilon \le p-1}{\inf }{\epsilon }^{-\frac{\theta }{p-\epsilon }}{\left({\int }_{\Omega }{\left|g_k\right|}^{{\left(p-\epsilon \right)}^{{}^{\prime }}}dx\right)}^{\frac{1}{{\left(p-\epsilon \right)}^{{}^{\prime }}}}\right\}$$

$$L^{p{)}^{{}^{\prime }},\theta }\left(\Omega \right)=\left\{g\in {\mathcal{M}}_0:{\Vert g\Vert }_{p{)}^{{}^{\prime }},\theta }<+\infty \right\},$$

$${\Vert g\Vert }_{p{)}^{{}^{\prime },\theta }}=\underset{\begin{array}{c}0\le \psi \le |g|\\ \psi \in L^{(p^{\prime },\theta }\end{array}}{\sup }{\Vert \psi \Vert }_{(p^{{}^{\prime }},\theta }.$$

$$F_f^{p),\theta }\left(x\right)=F_f^{p),\theta ,Q}\left(x\right)={\Vert f{\chi }_{Q+x}\Vert }_{p),\theta }=\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\Vert f{\chi }_{Q+x}\Vert }_{p-\epsilon }$$

$$\Vert f{\Vert }_{W(L^{p),\theta },L^{q),\theta })}=\Vert F_f^{p),\theta }{\Vert }_{q),\theta }=\Vert \Vert f{\chi }_{Q+x}{\Vert }_{p),\theta }{\Vert }_{q),\theta }.$$

$${\cup }_{i\in I}(x_i+U)=R.$$

$$\underset{x\in X}{\sup }♮\left\{i:x\in K+y_i\right\}=\sup ♮\left\{j\in I:K+y_i\cap K+y_j\ne \varphi \right\}<\infty .$$

$${\left(L^{p),\theta }\right)}_d=\left\{\Lambda :\Lambda ={\left({\lambda }_i\right)}_{i\in I}\text{ with }\sum _{i\in I}\left|\lambda \right|{\chi }_{x_i+U}\in L^{p),\theta }\left(\Omega \right)\right\}$$

$${\Vert \Lambda \Vert }_{{\left(L^{p),\theta }\right)}_d}={\Vert \sum _{i\in I}\left|\lambda \right|{\chi }_{x_i+U}\Vert }_{p),\theta }$$

$$\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\left\{{\int }_{\Omega }\mid \sum _{i\in I}\mid {\lambda }_i\mid {\chi }_{x_i+U}{\mid }^{p-\epsilon }dx\right\}}^{\frac{1}{p-\epsilon }}$$

$$=\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\left\{\sum _{i\in I}{\int }_{x_i+U}\mid \sum _{i\in I}\mid {\lambda }_i\mid {\chi }_{x_i+U}{\mid }^{p-\epsilon }dx\right\}}^{\frac{1}{p-\epsilon }}$$

$$=\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\left\{\sum _{i\in I}\mid {{\lambda }_i\mid }^{p-\epsilon }{\int }_{x_i+U}{\chi }_{x_i+U}dx\right\}}^{\frac{1}{p-\epsilon }}$$

$$=\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}\mu {\left(U\right)}^{\frac{1}{p-\epsilon }}{\left\{\sum _{i\in I}\mid {{\lambda }_i\mid }^{p-\epsilon }\right\}}^{\frac{1}{p-\epsilon }}$$

$$=\mu \left(U\right)\underset{0<\epsilon \le p-1}{\sup }{\epsilon }^{\frac{\theta }{p-\epsilon }}{\left\{\sum _{i\in I}\mid {{\lambda }_i\mid }^{p-\epsilon }\right\}}^{\frac{1}{p-\epsilon }}$$

$$=\mu \left(U\right)\Vert {\left\{{\lambda }_i\right\}}_{i\in I}{\Vert }_{{\ell }^{p),\theta }}.$$

$${\left(L^{p),\theta }\right)}_d\subset {\ell }^{p),\theta }\left(I\right).$$

$${\left(L^{p),\theta }\right)}_d\supset {\ell }^{p),\theta }\left(I\right).$$

$${\left(L^{p),\theta }\right)}_d={\ell }^{p),\theta }\left(I\right).$$

$${\Vert f\Vert }_{W\left(L^{p),\theta },L^{p),\theta }\right)}\asymp {\Vert \sum _{i\in I}{\Vert f{\Psi }_i\Vert }_{p),\theta }{\chi }_{V+y_i}\Vert }_{q),\theta }.$$

$${\Vert f\Vert }_{W\left(L^{p),\theta },L^{q),\theta }\right)}\asymp {\Vert \sum _{i\in I}{\Vert f{\psi }_i\Vert }_{p),\theta }{\chi }_{U+y_i}\Vert }_{q),\theta }={\Vert {\left\{{\Vert f{\psi }_i\Vert }_{p),\theta }\right\}}_{i\in I}\Vert }_{{\ell }^{q),\theta }}.$$

$${\Vert f\Vert }_{W\left(L^{p),\theta },L^{q),\theta }\right)}$$

$$=\underset{0<\eta \le q-1}{\sup }{\eta }^{\frac{\theta }{q-\eta }}{\Vert \sum _{i\in I}{\Vert f{\psi }_i\Vert }_{p),\theta }{\chi }_{U+y_i}\Vert }_{q-\eta }$$

$$=\underset{0<\eta \le q-1}{\sup }{\eta }^{\frac{\theta }{q-\eta }}(|{\int }_{\Omega }\sum _{i\in I}\Vert f{\psi }_i{\Vert }_{p),\theta }{\chi }_{U+y_i}{|}^{q-\eta }dx{)}^{\frac{1}{q-\eta }}$$

$$=\underset{0<\eta \le q-1}{\sup }{\eta }^{\frac{\theta }{q-\eta }}{\left(\sum _{i\in I}{\Vert f{\psi }_i\Vert }_{p),\theta }^{q-\eta }{\int }_{\Omega }{\left|{\chi }_{U+y_i}\right|}^{q-\eta }dx\right)}^{\frac{1}{q-\eta }}$$

$${\int }_{\Omega }{\left|{\chi }_{U+y_i}\right|}^{q-\eta }dx=\mu \left(U+y_i\right)=\mu \left(U\right).$$

$${\Vert \sum _{i\in I}{\Vert f{\psi }_i\Vert }_{p),\theta }{\chi }_{U+y_i}\Vert }_{q),\theta }$$

$${\Vert f\ast g\Vert }_{p),\theta }\le {\Vert f\Vert }_{p),\theta }{\Vert g\Vert }_{p),\theta }$$

$${\Vert f\ast g\Vert }_{p-\epsilon }\le {\Vert f\Vert }_{p-\epsilon }{\Vert g\Vert }_{p-\epsilon }.$$

$${\Vert f\ast g\Vert }_{p),\theta }$$

Full text

1. Notations

Let $\Omega$ be a bounded subset of $\ R$. The grand Lebesgue space $L^{p)}\left(\Omega\right)$ was introduced by Iwaniec-Sbordone in $\left[16\right].$ These authors, in their studies related with the integrability properties of the Jacobian in a bounded open set $\Omega$, defined the grand lebesgue space. This Banach space is defined by the norm

[TABLE]

where $1<p<\infty.$ For $0<\varepsilon\leq p-1,$ $L^{p}\left(\Omega\right)\subset L^{p)}\left(\Omega\right)\subset\L^{p-\varepsilon}\left(\Omega\right)$ hold. For some properties and applications of $L^{p)}\left(\Omega\right)$ spaces we refer to papers $\left[1\right],\left[3\right],\left[5\right],\left[6\right],$ $\left[12\right]$and $\left[13\right].$ We have for all $1<p<\infty$ and $\varepsilon>0$

[TABLE]

A generalization of the grand Lebesgue spaces are the spaces $L^{p),\theta}\left(\Omega\right),$ $\theta\geq 0,$ defined by the norm

[TABLE]

when $\theta=0$ the space $L^{p),0}\left(\Omega\right)$ reduces to Lebesgue space $L^{p}\left(\Omega\right)$ and when $\theta=1$ the space $L^{p),1}\left(\Omega\right)$ reduces to grand Lebesgue space $L^{p)}\left(\Omega\right)$, $\left(\text{see }\left[1\right],\left[12\right]\right)$ . It is known that the subspace $\overline{C_{0}^{\infty}}$ is not dense in $L^{p)}\left(\Omega\right).$ Its closure consists of functions $f\in L^{p)}\left(\Omega\right)$ such that

[TABLE]

It is also known that the grand Lebesgue space $L^{p),\theta}\left(\Omega\right),$ is not reflexive. Different properties and applications of these spaces were discussed in $\left[1\right],\left[12\right]$ and $\left[13\right].$

Let $1\leq p<\infty,\theta\geq 0,$ and $J$ be the one of the set of $\mathbb{N},$ $\mathbb{Z}$ or $\mathbb{Z}^{n}.$ We define the grand Lebesgue sequence space $\ell^{p),\theta}=\ell^{p),\theta}\left(J\right)$ by the norm

[TABLE]

Let $p^{{}^{\prime}}=\frac{p}{p-1},$ $1<p<\infty.$ First consider an auxiliary space namely $L^{(p^{\prime},\theta}\left(\Omega\right),\theta>0,$ defined by

[TABLE]

where the functions $g_{k},k\in\mathbb{N},$ being in $\mathcal{M}_{0},$ the set of all real valued measurable functions, finite a.e. in $\Omega.$ After this definition the generalized small Lebesgue spaces have been defined by

[TABLE]

where

[TABLE]

For $\theta=0$ it is $\left\|f\right\|_{(p^{{}^{\prime}},0}=\left\|f\right\|_{p)^{{}^{\prime}},\theta},\left[4\right],\left[11\right].$

Let $1\leq p,q\leq\infty.$ The space $(L^{p),\theta})_{loc}$consists of (classes of) measurable functions $f$ $:\Omega\rightarrow\mathbb{C}$ such that $f\chi_{K}\in L^{p),\theta},$ for any compact subset $K\subset\Omega,$ where $\chi_{K}$ is the characteristic function of $f.$ It is a topological vector space with the family of seminorms $f\rightarrow\left\|f\right\|_{p),\theta}.$ Since $L^{p}\subset L^{p),\theta},$ it is easy to show that $\left(L^{p}\right)_{loc}\hookrightarrow\left(L^{p)}\right)_{loc}.$

The grand amalgam space was defined and studied some properties in [14]. This space is defined as follows:

Let $\Omega$ be a finite subset of real numbers $R$. Also assume that $1<p,q<\infty$ and $Q\subset\Omega$ is a fix compact subset with nonemty interior. The grand Wiener amalgam space $W(L^{p),\theta},L^{q),\theta})$ consists of all functions (classes of) $f\in\left(L^{p),\theta}\right)_{loc}$ such that the control function

[TABLE]

lies in $L^{q),\theta}$, where $x\in R$. The norm of $W(L^{p),\theta},L^{q),\theta})$ defined by

[TABLE]

Since generalized grand Lebesgue spaces are not translation invariant, this result reflects to the grand Wiener amalgam spaces. Then the definition of $W\left(L^{p),\theta},L^{q)}\right)\left(\Omega\right)$ depends on the choice of $Q.$

Given any neighbourhood $U$ of [math]  in the set of real numbers $R$, a family $X=(x_{i})_{i\in I}\subset R$ is called $U$-dense if the family $(x_{i}+U)_{i\in I}$ cover $R$. That is

[TABLE]

The family $X$ is called separated if the sets $(x_{i}+U)_{i\in I}$ are pairwise disjoint. The family $X$ is called relatively separated proved that it is finite union of separated sets. We will call a family $X=(x_{i})_{i\in I}\subset R$ is well-spread in $R$ if it is both $U$-dense and relatively separated.

Let $\Omega$ be a finite subset of real numbers $R$. Another equivalent but discrete definition of $W\left(L^{p),\theta},L^{q)}\right)\left(\Omega\right)$ is given by using the bounded uniform partition of unity (for short BUPU), that is a sequence of non-negative functions $\Psi=\left(\psi_{i}\right)_{i\in I}$ on $\Omega$ corresponding to a sequence $\left(y_{i}\right)$ in $\Omega$ such that

• (a)

$\sum\psi_{i}\equiv 1,$

• (b)

$\sup_{i\in I}\left\|\psi_{i}\right\|_{L^{\infty}}<M,$ for some $M>0,$

• (c)

There exists a compact subset $U\subset\Omega$ with non-emty interior and $y_{i}\in\Omega$ such that $\sup\left(\psi_{i}\right)\subset U+y_{i},$

• (d)

For each compact subset $K\subset\Omega,$

[TABLE]

2. Discrete Grand Amalgam Space

In this section we will introduce another equivalent but discrete definition of grand amalgam spaces and study properties of these spaces.

Definition 1**.**

Let $\left(\ x_{i}\right)_{i\in I}\subset\Omega$ be well-spread family in $\Omega.$ For any grand Lebesgue space $L^{p),\theta}\left(\Omega\right),$ $\theta\geq 0$, we define the associate discrete space $\left(L^{p),\theta}\right)$ as

[TABLE]

with the norm

[TABLE]

Since $L^{p),\theta}$ is not translation invariant then $\left(L^{p),\theta}\right)_{d}$ depends on the choice of $U$.

Theorem 1**.**

The discrete of the grand Lebesgue space $L^{p),\theta}\left(\Omega\right)$ is the the grand Lebesgue sequence space $\ell^{p),\theta}\left(I\right).$

Proof.

$\Lambda=\left(\lambda_{i}\right)_{i\in I}\in\left(L^{p),\theta}\right)_{d}.$ Then

[TABLE]

The righ hand side of this equality is finite. This implies

[TABLE]

Conversely let $\Lambda=\left(\lambda_{i}\right)_{i\in I}\in\l^{p),\theta}\left(\ I\right).$ Again from (9) we have

[TABLE]

Finally from (10) and (11) we have

[TABLE]

∎

One can easily prove the following theorem if uses the norm of Grand amalgam apace instead of the norm of classical Wiener amalgam space and the technics in ( $\left[8\right],$ Theorem 2 and $\left[15\right],$ Theorem 11.6.2).

Theorem 2**.**

If $\ \Psi=\left(\psi_{i}\right)_{i\in I}$ is a bounded uniform partition unity and $V$ is a compact set containing $U,$ then

[TABLE]

Theorem 3**.**

Let $1\leq p\leq\infty$ and let $\ \Psi=\left(\psi_{i}\right)_{i\in I}$ be a bounded uniform partition unity. Then

[TABLE]

Proof.

From Theorem 1 and Theorem 2, we have

[TABLE]

In particular,

[TABLE]

Then from $\left(13\right)$ and $\left(14\right),$

[TABLE]

This completes the proof. ∎

3. Which grand amalgam spaces are convolution algebras

Let $G$ be locally compact Abelian group and $\mu$ its Haar measure. Zelazko $\left[18\right]$ has proved that $L^{p}\left(G\right)$ is an algebra if and only if $G$ is compact. Steward and Salem $\left[17\right]$ have determined necessary and sufficient conditions on $G$ for the Wiener amalgam space $W\left(L^{p},L^{q}\right)$ to be an algebra under convolution. In this chapter we have given similar results for the generalized grand Lebesgue space $L^{p),\theta}$ and the grand amalgam spaces.

Theorem 4**.**

Let $G$ be a locally compact Abelian group, and $\mu$ its Haar measure. Then the generalized grand Lebesgue space $L^{p),\theta}$, $1<p<\infty$ is a Banach algebra under convolution if and only if $G$ is compact.

Proof.

Let $G$ be a compact Abelian group. For the proof of the generalized grand Lebesgue space $L^{p),\theta}$, $1<p<\infty$ is a Banach algebra under convolution, it is enough to show that

[TABLE]

for all $f,g\in L^{p),\theta}\left(G\right).$ Since $G$ is compact, then $L^{p}\left(G\right)$ is a Banach convolution algebra for all $p,$ $1\leq p\leq\infty,$ by Zelasko $\left[18\right].$Thus $\left(L^{p-\varepsilon}\left(G\right),\left\|.\right\|_{p-\varepsilon}\right)$ is a Banach algebra under convolution for all $0<\varepsilon\leq p-1.$ Then for all $f,g\in L^{p-\varepsilon}\left(G\right),$ we have

[TABLE]

Thus

[TABLE]

Conversely Assume that $L^{p),\theta}(G)$ is a Banach algebra under convolution for all $\theta\geq 0$ and, $1<p<\infty$. Since $L^{p),\theta}(G)$ reduces to $L^{p}(G)$ when $\theta=0$, then $L^{p}(G)$ is a Banach algebra under convolution. Thus $G$ is compact again from Zelasko $\left[18\right].$ ∎

Theorem 5**.**

The grand Wiener amalgam space $W\left(L^{p),\theta},L^{q),\theta}\right)\left(G\right),p\geq 1,q>1$ is a Banach algebra under convolution if and only if $G$ is compact.

Proof.

It is known by $\left[17\right]$ that $W\left(L^{p},L^{q}\right)\left(G\right)$ is a Banach algebra under convolution if and only if $G$ is compact. Let $G$ be a compact set. For the proof it is enough to show that

[TABLE]

for all $f,g\in W\left(L^{p),\theta},L^{q),\theta}\right)\left(G\right).$ Since $G$ is compact,$W\left(L^{p-\varepsilon},L^{q-\eta}\right)$=$L^{p-\varepsilon}$=$L^{q-\eta}$ and the norms are equvalent. Let $f,g\in W\left(L^{p),\theta},L^{q),\theta}\right)\left(G\right)$. Then by using Theorem 3, a simple calculation shows that

[TABLE]

Conversely assume that $W\left(L^{p),\theta},L^{q),\theta}\right)\left(G\right)$ is a Banach convolution algebra for all $\theta\geq 0.$ Then it is Banach convolution algebra for $\theta=0.$ But if $\theta=0,$ then the grand amalgam space $W\left(L^{p),\theta},L^{q),\theta}\right)\left(G\right)$ reduces to classical Wiener amalgam space $W\left(L^{p},L^{q}\right)\left(G\right).$ So, $W\left(L^{p},L^{q}\right)\left(G\right)$ becomes a Banach convolution algebra. Then by Theorem 1, in $\left[17\right],$ $G$ becomes compact.

∎