[Paper Review] Uniform openness of multiplication in Banach spaces $L_p$

Marek Balcerzak, Adam Majchrzycki+1 more|arXiv (Cornell University)|Sep 13, 2013

Advanced Banach Space Theory2 references3 citations

TL;DR

This paper establishes the uniform openness of multiplication maps in $L_p$ and $σ$-finite measure spaces, proving that for $p \in [1,\infty)$ and $q$ conjugate to $p$, the bilinear map $L_p \times L_q \to L_1$ is uniformly open with a quantitative lower bound $\delta = \varepsilon^2/4$. It further extends this result to the sequence spaces $\ell_1 \times c_0 \to \ell_1$, showing uniform openness with $\delta = \varepsilon^2/16$, strengthening prior results on openness of multiplication in Banach function spaces.

ABSTRACT

We show that multiplication from $L_p imes L_q$ to $L_1$ (for $p,q\in [1,\infty]$, $1/p+1/q=1$) is a uniformly open mapping. We also prove the uniform openness of the multiplication from $\ell_1 imes c_0$ to $\ell_1$. This strengthens the former results obtained by M. Balcerzak, A. Majchrzycki and A. Wachowicz.

Motivation & Objective

• To establish the uniform openness of the multiplication map $L_p \times L_q \to L_1$ for conjugate exponents $p, q$ with $1/p + 1/q = 1$.
• To extend the uniform openness result to the sequence space setting, specifically $\ell_1 \times c_0 \to \ell_1$.
• To provide quantitative estimates for the uniform openness modulus, improving upon previous qualitative openness results.
• To generalize the real-line uniform openness lemma (Lemma 1) to function and sequence spaces via measure-theoretic and summability techniques.
• To demonstrate that the uniform openness property holds even in non-reflexive and non-locally convex settings, such as $\ell_1 \times c_0$.

Proposed method

• Utilizes a key lemma (Lemma 1) on uniform openness of multiplication in $\mathbb{R}$, which provides a quantitative lower bound $\delta = rR/4$ for intervals.
• Applies Lemma 3 to approximate $L_1$ functions by simple functions with finite measure support and uniform boundedness.
• Employs countably valued functions in $L_p$ and $L_q$ to reduce the problem to finite-dimensional or simple function cases.
• Constructs perturbations $u$ and $v$ of $f$ and $g$ such that $uv$ approximates any $h$ in a small $L_1$-ball around $fg$, using case analysis based on the support of the difference $h - fg$.
• Applies Lemma 6 on series with non-increasing tails to control the $\ell_Σ$-norm of perturbations in the $\ell_1 \times c_0$ case.
• Uses case distinction based on whether the set of points where $h_n \neq f_n g_n$ is finite or infinite, applying Lemma 1 with tailored parameters $r_k, R_k$ in each case.

Experimental results

Research questions

• RQ1 Is the multiplication map $L_p \times L_q \to L_1$ uniformly open for conjugate exponents $p, q$ with $1/p + 1/q = 1$?
• RQ2 Can the uniform openness of multiplication in $\ell_1 \times c_0 \to \ell_1$ be established with an explicit quantitative modulus?
• RQ3 Does the uniform openness property extend from $\mathbb{R}$ to $L_p$ and sequence spaces using measure-theoretic and summability arguments?
• RQ4 Can the uniform openness modulus be bounded below by a function of $\varepsilon$, such as $\varepsilon^2/4$ or $\varepsilon^2/16$?
• RQ5 What is the role of the $c_0$-structure in ensuring uniform openness in the sequence space case?

Key findings

• The multiplication map $L_p \times L_q \to L_1$ is uniformly open for $p \in [1,\infty)$ and $q$ conjugate to $p$, with the uniform openness modulus satisfying $\operatorname{B}_1(fg, \varepsilon^2/4) \subseteq \operatorname{B}_p(f,\varepsilon) \cdot \operatorname{B}_q(g,\varearepsilon)$.
• The uniform openness of multiplication from $\ell_1 \times c_0$ to $\ell_1$ is established with the modulus $\delta = \varepsilon^2/16$, i.e., $\operatorname{B}_1(xy, \varepsilon^2/16) \subseteq \operatorname{B}_1(x,\varepsilon) \cdot (\operatorname{B}_\infty(y,\varepsilon) \cap c_0)$.
• The proof relies on decomposing the difference $h - fg$ into finite and infinite support parts, applying Lemma 1 with carefully chosen parameters $r_k, R_k$ in each case.
• For $h$ close to $fg$ in $L_1$, the construction yields $u \in \operatorname{B}_p(f,\varepsilon)$ and $v \in \operatorname{B}_q(g,\varepsilon)$ such that $uv = h$, with $v$ in $c_0$ when $g \in c_0$.
• The use of Lemma 6 ensures summability control in the infinite support case, guaranteeing $\|v - y\|_\infty < \varepsilon$ and $\|u - f\|_1 < \varepsilon$.
• The result strengthens earlier openness results by providing a uniform modulus independent of the point $(f,g)$, resolving a gap in the literature on bilinear maps in Banach spaces.