[Paper Review] Bilateral Small Lebesgue Spaces

E. Ostrovsky, L. Sirota|ArXiv.org|Apr 29, 2009

Advanced Harmonic Analysis Research14 references3 citations

TL;DR

This paper introduces and analyzes Bilateral Small Lebesgue Spaces (BSL) as the associate spaces to Rearrangement-Invariant Grand Lebesgue Spaces (G(ψ)), establishing their fundamental properties including norm estimation, Boyd’s indices, fundamental functions, and convergence behavior. The key contribution is a comprehensive characterization of BSL spaces, including their duality, separability, non-reflexivity, and boundedness of classical operators like the Hardy-Littlewood maximal operator and Hilbert transform under specific parameter conditions.

ABSTRACT

In this article we investigate the so-called Bilateral Small Lebesgue Spaces: prove that they are associated to the Grand Lebesgue spaces, calculate its fundamental functions and Boyd's indices find its dual spaces etc.

Motivation & Objective

• To define and investigate Bilateral Small Lebesgue Spaces (BSL) as the associate spaces to Grand Lebesgue Spaces G(ψ).
• To characterize the fundamental functions and Boyd’s indices of BSL(ψ) spaces.
• To establish norm estimation, convergence, and compactness criteria for BSL(ψ) using a novel functional |||·||| derived from the measure χ.
• To determine conditions under which classical operators (e.g., Hardy-Littlewood maximal, Hilbert transform) are bounded on BSL(ψ).

Proposed method

• Define BSL(ψ) as the associate space G*(ψ) to the Grand Lebesgue Space G(ψ), using the duality pairing ∫f·g dμ.
• Introduce a measure-like functional χ(μ(A)) = ||I(A)||SL(ψ) to define a new norm |||f||| = ∫|f| dχ, which dominates the original BSL norm.
• Use the representation of functions via simple functions in Sim(ψ) to derive norm estimates and establish density in BSL(ψ).
• Apply the theory of rearrangement-invariant (r.i.) spaces to derive Boyd’s indices and fundamental functions of BSL(ψ).
• Analyze boundedness of operators (e.g., maximal, Hilbert) on BSL(ψ) by relating their behavior to the parameters a, b, and the function ψ.
• Prove convergence and compactness results by relating convergence in |||·||| to convergence in the original BSL norm.

Experimental results

Research questions

• RQ1 What is the precise structure of Bilateral Small Lebesgue Spaces as the associate space to Grand Lebesgue Spaces G(ψ)?
• RQ2 How do Boyd’s indices and fundamental functions of BSL(ψ) depend on the function ψ and the interval (a,b)?
• RQ3 Under what conditions is the Hardy-Littlewood maximal operator bounded on BSL(ψ)?
• RQ4 When is the Hilbert transform bounded on BSL(ψ) for X = ℝ^d?
• RQ5 How can the norm of a function in BSL(ψ) be effectively estimated or approximated?

Key findings

• The Bilateral Small Lebesgue space BSL(ψ) is isometrically isomorphic to the associate space G*(ψ) of the Grand Lebesgue space G(ψ).
• The functional |||f||| = ∫|f| dχ provides a dominating norm estimate: ||f||SL(ψ) ≤ |||f|||SL(ψ), with equality in specific cases.
• The Hardy-Littlewood maximal operator M is bounded on SL(ψ) if and only if a > 1.
• The Hilbert transform H is bounded on SL(ψ) if and only if a > 1.
• For X = [0, 2π), Fourier series of functions in SL(ψ) converge in norm if and only if a > 0 and b < ∞.
• The space BSL(ψ) is separable and non-reflexive, and satisfies the Fatou, Lebesgue, and absolute continuous norm properties.