[Paper Review] Extrapolation: Stories and Problems

С. В. Асташкин, Mario Milman|arXiv (Cornell University)|Apr 7, 2020

Advanced Harmonic Analysis Research50 references4 citations

TL;DR

This paper presents a comprehensive survey of extrapolation theory in functional analysis, framing it as a dual to interpolation theory by introducing K and J functionals for scales of Banach spaces. It establishes that the Σ- and Δ-methods are extremal extrapolation functors, with the former corresponding to Lorentz spaces and the latter to Marcinkiewicz spaces, providing a unifying framework for understanding operator norms across variable Lebesgue and Lorentz-type spaces.

ABSTRACT

We discuss some aspects of Extrapolation theory. The presentation includes many examples and open problems.

Motivation & Objective

• To unify and extend extrapolation theory by framing it as a dual process to interpolation, particularly through the lens of scale-based K and J functionals.
• To characterize the extremal behavior of extrapolation functors by identifying the Σ- and Δ-methods as the broadest and narrowest possible constructions.
• To provide a systematic framework for understanding operator norms across variable Lebesgue and Lorentz-type spaces using abstract extrapolation functors.
• To stimulate further research by compiling open problems and analogies with interpolation theory, especially in the context of rearrangement-invariant and non-commutative spaces.

Proposed method

• Introduces K and J functionals for a continuous scale of Banach spaces {A_θ}_{θ∈(0,1)}, generalizing classical interpolation methods to extrapolation settings.
• Defines abstract extrapolation functors F(·) using Banach function lattices F on [0,1], where F( {A_θ} ) consists of elements a ∈ ∩_θ A_θ with ||a||_A_θ ∈ F.
• Establishes that the Δ-functor corresponds to F = L^∞[0,1], yielding the intersection space, while the Σ-functor corresponds to F = L^1[0,1], yielding the sum space.
• Proves that for any exact extrapolation functor 𝔈, Δ({A_θ}) ⊂ 𝔈({A_θ}) ⊂ Σ({A_θ}) holds, showing Σ and Δ are extremal.
• Applies the framework to characterize Marcinkiewicz, Orlicz, and Lorentz spaces via F-functors and introduces the concept of tempered F-parameters.
• Extends the theory to non-commutative L^p spaces and bilinear extrapolation, including a reinterpretation of Calderón’s operator.

Experimental results

Research questions

• RQ1 How can extrapolation theory be systematically developed as a dual to interpolation theory, particularly through scale-based K and J functionals?
• RQ2 What are the extremal extrapolation functors, and how do they relate to classical Lorentz and Marcinkiewicz spaces?
• RQ3 Can the converse of Yano’s theorem be established via K-functional inequalities, as suggested by Tao’s result?
• RQ4 How do abstract extrapolation functors based on Banach function lattices F generalize the Σ and Δ methods?
• RQ5 What are the implications of extrapolation for non-commutative L^p spaces and bilinear operators, particularly in harmonic analysis?

Key findings

• The Δ-method reproduces the intersection space ∩_θ A_θ, while the Σ-method yields the sum space ∑_θ A_θ, establishing them as extremal extrapolation functors.
• For any exact extrapolation functor 𝔈, the inclusion Δ({A_θ}) ⊂ 𝔈({A_θ}) ⊂ Σ({A_θ}) holds, with equality in the case of constant families.
• The F-functor construction generalizes both Σ and Δ by using a Banach function lattice F on [0,1], where F = L^∞[0,1] gives Δ and F = L^1[0,1] gives Σ.
• The theory characterizes Marcinkiewicz, Orlicz, and Lorentz spaces via F-functors, with the latter corresponding to the Σ-method and the former to the Δ-method.
• The paper establishes that the norm of an operator T: A_θ → B_θ extends to 𝔈({A_θ}) → 𝔈({B_θ}) with norm bounded by sup_θ ||T||_{A_θ→B_θ}, showing the functoriality of extrapolation.
• The framework supports applications to grand Lebesgue spaces, bilinear extrapolation, and non-commutative Calderón operators, extending classical results to broader settings.