[Paper Review] Multipliers of grand and small Lebesgue spaces

A. Turan Gürkanlı|arXiv (Cornell University)|Mar 15, 2019

Advanced Harmonic Analysis Research12 references4 citations

TL;DR

This paper establishes that the generalized grand Lebesgue space $L^{p), heta}(G)$ on a locally compact abelian group $G$ forms a Banach algebra under convolution if and only if $G$ is compact. It further characterizes the multipliers of $L^{1}(G)$ into the small Lebesgue space $L^{(p', heta}(G)$, showing they are isometrically isomorphic to $L^{p), heta}(G)$, using approximate identities and duality in Banach module theory.

ABSTRACT

Let $G$ a locally compact abelian group with Haar measure $μ$ and let $1

Motivation & Objective

• To determine necessary and sufficient conditions on a locally compact abelian group $G$ for the generalized grand Lebesgue space $L^{p), heta}(G)$ to be a Banach algebra under convolution.
• To characterize the space of multipliers from $L^{1}(G)$ into the small Lebesgue space $L^{(p', heta}(G)$, particularly in relation to duality and module structure.
• To extend multiplier theory to subspaces of small Lebesgue spaces via relative completion and bounded approximate identities.
• To establish isomorphism between multiplier spaces and generalized grand Lebesgue spaces using duality and module factorization.

Proposed method

• Uses the norm definition $\|f\|_{p),\theta} = \sup_{0<\varepsilon\leq p-1} \varepsilon^{\theta/(p-\varepsilon)} \|f\|_{p-\varepsilon}$ to define generalized grand Lebesgue spaces on $G$.
• Applies known results on convolution algebras in $L^{p-\varepsilon}(G)$ for $0<\varepsilon\leq p-1$ to analyze $L^{p),\theta}(G)$ via uniform bounds.
• Employs the dual space identification $(L^{(p',\theta}(G))^* = L^{p),\theta}(G)$ to relate multipliers to grand Lebesgue spaces.
• Utilizes bounded approximate identities in $L^1(G)$ and the module factorization theorem to prove that $L^1(G)*L^{(p',\theta}(G) = L^{(p',\theta}(G)$.
• Introduces the relative completion $A^\sim$ of a subspace $A \subset L^{(p,\theta}(G)$ via $\|f\|_{A^\sim} = \sup_\alpha \|f*e_\alpha\|_A$, where $\{e_\alpha\}$ is a bounded approximate identity.
• Applies duality and Banach module theory to show $M(L^1(G), A) \cong A^\sim$ for subspaces $A$ satisfying specific bounded approximate identity and norm conditions.

Experimental results

Research questions

• RQ1 Under what conditions on a locally compact abelian group $G$ is the generalized grand Lebesgue space $L^{p),\theta}(G)$ a Banach algebra under convolution?
• RQ2 What is the precise characterization of the multiplier space $M(L^1(G), L^{(p',\theta}(G))$?
• RQ3 How do bounded approximate identities in $L^1(G)$ interact with the structure of small Lebesgue spaces $L^{(p',\theta}(G)$?
• RQ4 Can the multiplier space from $L^1(G)$ to a subspace $A \subset L^{(p',\theta}(G)$ be identified with a relative completion $A^\sim$?
• RQ5 What is the relationship between the dual of $L^{(p',\theta}(G)$ and the grand Lebesgue space $L^{p),\theta}(G)$ in the context of multiplier theory?

Key findings

• The generalized grand Lebesgue space $L^{p),\theta}(G)$ is a Banach algebra under convolution if and only if the underlying group $G$ is compact.
• The space of multipliers $M(L^1(G), L^{(p',\theta}(G))$ is isometrically isomorphic to $L^{p),\theta}(G)$, with the isomorphism established via duality and module factorization.
• For any subspace $A \subset L^{(p,\theta}(G)$ satisfying the conditions of Definition 1, the multiplier space $M(L^1(G), A)$ is algebraically and topologically isomorphic to the relative completion $A^\sim$.
• The space $L^{(p',\theta}(G)$ admits a bounded approximate identity in $L^1(G)$, which is essential for proving the density and factorization results.
• The norm $\|f\|_{(p',\theta}$ is defined via an infimal decomposition over sequences $\{g_k\}$, and satisfies $\|g\|_{(p',\theta} = \|g\|_{p)^{\prime},\theta}$, enabling duality arguments.
• The proof relies on the inequality $\|f*g\|_{(p',\theta} \leq \|f\|_1 \|g\|_{(p',\theta}$, which confirms that $L^1(G)$ acts as a multiplier module over $L^{(p',\theta}(G)$.