[Paper Review] Extrapolation on function and modular spaces, and applications

Mingming Cao, Juan Marín+1 more|arXiv (Cornell University)|Jan 15, 2021

Advanced Harmonic Analysis Research4 citations

TL;DR

This paper extends Rubio de Francia's extrapolation theory to Banach and modular function spaces by characterizing boundedness of operators through weighted estimates of the Hardy-Littlewood maximal function. The key contribution is a general extrapolation framework valid on measure spaces with Muckenhoupt bases, enabling new weighted inequalities for square functions, layer potentials, and Schrödinger operators.

ABSTRACT

We generalize the extrapolation theory of Rubio de Francia to the context of Banach function spaces and modular spaces. Our results are formulated in terms of some natural weighted estimates for the Hardy-Littlewood maximal function and are stated in measure spaces and for general Muckenhoupt bases. Finally, we give several applications in analysis and partial differential equations.

Motivation & Objective

• To generalize Rubio de Francia's extrapolation theory beyond weighted Lebesgue spaces to the broader context of Banach and modular function spaces.
• To establish a unified framework for extrapolation based on weighted boundedness of the Hardy-Littlewood maximal function and its dual.
• To provide a systematic approach to deriving weighted norm inequalities for operators in analysis and PDEs using modular space structures.
• To extend known results on square functions, layer potentials, and Schrödinger operators to rearrangement-invariant and modular function spaces with sharp weight conditions.

Proposed method

• Formulate extrapolation in terms of weighted estimates for the Hardy-Littlewood maximal function and its dual operator $ M_v' $ on general measure spaces.
• Use Muckenhoupt bases instead of the standard dyadic cubes to generalize the theory to non-homogeneous and non-doubling settings.
• Apply the theory to Banach function spaces $ \mathbb{X} $ and modular spaces by requiring $ \|Mh\|_{\mathbb{X}_v} \leq C\|h\|_{\mathbb{X}_v} $ and $ \|M_v'h\|_{\mathbb{X}'_v} \leq C\|h\|_{\mathbb{X}'_v} $.
• Introduce the concept of $ \mathbb{X}^{1/r} $ being a Banach function space to ensure compatibility with extrapolation techniques.
• Derive weighted inequalities for conical square functions associated with heat and Poisson semigroups via extrapolation from known $ A_p $-weighted estimates.
• Use the theory to establish boundedness of layer potential operators and solutions to the Dirichlet problem on uniformly rectifiable domains in modular spaces.

Experimental results

Research questions

• RQ1 How can Rubio de Francia's extrapolation theory be extended from weighted Lebesgue spaces to general Banach and modular function spaces?
• RQ2 What conditions on weights and maximal function behavior ensure extrapolation in modular and rearrangement-invariant function spaces?
• RQ3 Can the extrapolation framework be applied to operators such as conical square functions, layer potentials, and Schrödinger operators in non-standard function spaces?
• RQ4 What are the sharp weight conditions for boundedness of square functions in modular spaces associated with elliptic operators?
• RQ5 How does the theory generalize to non-homogeneous settings and Muckenhoupt bases beyond the standard Euclidean measure?

Key findings

• The paper establishes a general extrapolation principle for Banach function spaces $ \mathbb{X}_v $, showing that if the maximal function is bounded on $ \mathbb{X}_v $ and its dual on $ \mathbb{X}'_v $, then any operator bounded on $ L^{p_0}(w_0) $ for some $ p_0 \in [1,\infty) $ and $ w_0 \in A_{p_0} $ is bounded on $ \mathbb{X}_v $.
• For conical square functions $ S_{m,\mathrm{H}}, G_{m,\mathrm{H}}, \mathcal{G}_{m,\mathrm{H}} $, the paper proves that $ \|(Tf)w\|_{\mathbb{X}} \lesssim \|fw\|_{\mathbb{X}} $ whenever $ w^{p_{\mathbb{X}}} \in A_{p_{\mathbb{X}}/p_{-}(L)} $ and $ w^{q_{\mathbb{X}}} \in A_{q_{\mathbb{X}}/p_{-}(L)} $, for $ p_{-}(L) < p_{\mathbb{X}} \leq q_{\mathbb{X}} < \infty $.
• For Poisson-type square functions $ S_{k,\mathrm{P}}, G_{k,\mathrm{P}}, \mathcal{G}_{k,\mathrm{P}} $, boundedness holds on $ \mathbb{X} $ if $ w^{p_{\mathbb{X}}} \in A_{p_{\mathbb{X}}/p_{-}(L)} \cap RH_{(p_{+}(L)^{k,*}/p_{\mathbb{X}})^\prime} $ and $ w^{q_{\mathbb{X}}} \in A_{q_{\mathbb{X}}/p_{-}(L)} \cap RH_{(p_{+}(L)^{k,*}/q_{\mathbb{X}})^\prime} $, provided $ q_{\mathbb{X}} < p_{+}(L)^{k,*} $.
• The theory yields new weighted estimates for the Dirichlet problem in the upper half-space, showing well-posedness in modular spaces under sharp weight conditions.
• The framework applies to Schrödinger operators with potentials, extending boundedness results to $ L^p $, $ L^{p,q} $, and $ L^p(\log L)^\alpha $ spaces with sharp weights.
• The results generalize known $ A_p $-weighted bounds for layer potential operators on uniformly rectifiable domains to modular and rearrangement-invariant function spaces with precise weight classes.