Weight Extrapolation in Analysis

• Weight extrapolation is a class of techniques that extend boundedness and norm inequalities from seed weight parameters to a broader range of exponents and weight classes.
• It generalizes Rubio de Francia's one-weight theorem to multilinear, two-weight, off-diagonal, and matrix-valued settings, advancing theoretical and practical applications.
• The method also informs data-driven models and neural network architectures by enabling controlled parameter extrapolation, ensuring robust performance across varied regimes.

Weight extrapolation refers to a broad class of theoretical and algorithmic techniques for extending boundedness results and norm inequalities for operators from a fixed set of "seed" weight parameters (often corresponding to particular integrability exponents or weight classes) to a larger, often full, scale of exponents and (possibly multi-parameter or matrix-valued) weights. The paradigm originated with Rubio de Francia's one-weight extrapolation theorem and has been generalized to multilinear, two-weight, off-diagonal, limited-range, and quasi-Banach settings. It plays a central role in harmonic analysis, operator theory, data-driven statistical models, and even parameter-controlled neural network generation via direct manipulation of "weight space" coefficients.

1. Classical Foundations: Rubio de Francia's Extrapolation

The classical setup begins with the Muckenhoupt $A_p$ class: $$[w]_{A_p} = \sup_Q \left( \frac{1}{|Q|} \int_Q w \right) \left( \frac{1}{|Q|} \int_Q w^{-1/(p-1)} \right)^{p-1} < \infty,$$ where the supremum is over cubes $Q \subset \mathbb{R}^n$. The Hardy–Littlewood maximal operator $M$ is bounded on $L^p(w)$ iff $w \in A_p$.

Rubio de Francia's theorem asserts: If an operator $T$ satisfies

$$\|Tf\|_{L^{p_0}(w)} \leq C([w]_{A_{p_0}}) \|f\|_{L^{p_0}(w)}$$

for all $w \in A_{p_0}$ for some fixed $p_0 > 1$, then for every $p \in (1, \infty)$,

$$\|Tf\|_{L^p(w)} \leq C'([w]_{A_{p}}) \|f\|_{L^p(w)}$$

for all $w \in A_{p}$, transferring the bound across all Lebesgue exponents (Cao et al., 2021).

Key to the extrapolation proof is the construction of majorizing weights via the Rubio de Francia algorithm: $$Rh(x) = \sum_{k=0}^{\infty} \frac{M^k h(x)}{2^k \|M\|_{X \to X}^k},$$ with $X$ a Banach function space on which $M$ is bounded. This yields an $A_1$ majorant of $h$ with controlled norm, allowing reduction of the desired estimate to $p_0$ via duality and factorization (Carro et al., 2010).

2. Multilinear and Two-Weight Extensions

Modern extensions operate on tuples of functions and weights. The genuinely multilinear Muckenhoupt class $A_{\vec{p}}$ for $\vec{p} = (p_1, \ldots, p_n)$ is

$$[\vec{w}]_{A_{\vec{p}}} := \sup_{R=I^1 \times I^2} \langle w^p \rangle_R^{1/p} \prod_{i=1}^n \langle w_i^{-p_i'} \rangle_R^{1/p_i'} < \infty,$$

with $w = \prod_i w_i$, $p$ given by $1/p = \sum_i 1/p_i$, and averages over rectangles $R$.

Two-weight extrapolation, as formalized in (Airta et al., 2021), states that if for some index $j$ and initial weights $(w_1, \ldots, \lambda_j, \ldots, w_n)$,

$$\left\| f \lambda_j \prod_{i \neq j} w_i \right\|_{L^p} \lesssim \prod_{i=1}^n \| f_i w_i \|_{L^{p_i}},$$

holds with weights in $A_{\vec{p}}$ and the ratio $\nu = w_j/\lambda_j \in A_\infty$, then for any other exponents $\vec{q}$,

$$\left\| f \lambda_j \prod_{i \neq j} w_i \right\|_{L^q} \lesssim \| f_j w_j \|_{L^{p_j}} \prod_{i \neq j} \| f_i w_i \|_{L^{q_i}},$$

with corresponding weight conditions. The proof adapts the Rubio iteration to multilinear/product spaces and also covers the quasi-Banach regime ($q < 1$).

For compactness, an operator compact at a seed exponent and weight, and bounded across the $A_p$ scale, is compact for all $p$ and $A_p$ weights (Liu et al., 2021).

3. Off-Diagonal, Limited-Range, and Mixed-Norm Extrapolation

Extrapolation has evolved to accommodate off-diagonal scenarios and limited-range conditions. For exponents $(p_0, q_0)$ and weights in $A_{p_0, q_0}$,

$$\left\| T f \right\|_{L^{q_0}(w)} \leq C \left\| f \right\|_{L^{p_0}(w)}$$

transfers to all $(p, q)$ with $1/p - 1/q = 1/p_0 - 1/q_0$ (and related weight conditions), yielding

$$\left\| T f \right\|_{L^q(w)} \leq C' \left\| f \right\|_{L^p(w)}.$$

This principle generalizes to multilinear/mixed-norm settings, where the extrapolated norm and the weight characteristic decouple from the initial exponent, controlled via a consistency index $\gamma$ (Sauer, 4 Nov 2025). The extrapolation covers the full range $(0,\infty]$, includes end-point cases (weak-type, $L^1 \rightarrow L^\infty$), quasi-Banach target spaces (e.g., $q < 1$), and product spaces.

4. Weight Extrapolation in General Function Spaces

The extension to general (quasi-)Banach function spaces requires recasting maximal operator bounds, duality, and convexity/concavity properties. For spaces $X$ with $p$-convexity, $q$-concavity, if $T$ satisfies a weighted norm inequality for some $p_0$ and $A_{p_0}$ weights, then analogous bounds hold for $X$ and corresponding weighted quasi-norm spaces, with constants controlled by the maximal operator bounds and factorization theorems (Lozanovskiĭ, Jones) (Nieraeth, 2022, Cao et al., 2021).

Representatives include weighted Lorentz, Morrey, variable Lebesgue, and modular/Orlicz spaces, where Boyd indices or rearrangement invariance dictate applicability. The generalized theory subsumes vector-valued inequalities, multilinear extensions, and endpoints.

5. Matrix Weights and Convex-Set-Valued Operators

For matrix-valued weights $W(x)$ (e.g., mappings into positive semidefinite $d \times d$ matrices), the matrix $\mathcal A_p$ class is

$$[W]_{\mathcal A_p} =\sup_Q \left( \frac{1}{|Q|} \int_Q \left( \frac{1}{|Q|} \int_Q | W(x) W(y)^{-1} |_{\text{op}}^{p'} dy \right)^{p/p'} dx \right)^{1/p},$$

and extrapolation (and Jones factorization) require embedding into convex-set-valued function spaces, defining convex-set Hardy–Littlewood maximal operators via Aumann integrals, and iterated Rubio de Francia algorithms for ball-valued functions (Bownik et al., 2022). Matrix weights necessitate geometric mean constructions and non-commutative factorization, with accompanying technical details for duality and norm control.

6. Extrapolation in Data-Driven and Neural Weight Spaces

Outside harmonic analysis, weight extrapolation governs parameter-borrowing in composite likelihoods. The extrapolation coefficient (or weight) $\alpha$, typically restricted to $[a, b] \subset (0,1)$, specifies information transfer in composite likelihood: $$L_c(\theta) = [L_r(\theta)]^\alpha \cdot [L_t(\theta)]^{1-\alpha},$$ providing a controlled interpolation between full pooling and segregation in multi-cohort estimation. Similar ideas manifest in Bayesian power priors (Gamalo et al., 2022).

In neural architectures for controlled 3D generation, e.g., LAMP ("Linear Affine Mixing of Parametric shapes"), weights $\mathbf{w}_i$ for MLP decoders are aligned by shared initialization, then new weights synthesized via affine mixing: $$\mathbf{w}_d = \sum_{i=1}^N \alpha_i \mathbf{w}_i, \quad \sum_{i=1}^N \alpha_i = 1,$$ subject to parameter matching constraints for controllable extrapolation. Safe extrapolation is certified by linearity-mismatch metrics to ensure the mixing remains valid within the regime of the decoder function (Nehme et al., 26 Oct 2025). Empirically, LAMP achieves robust extrapolation up to $100\%$ parameter range extension with only $\mathcal{O}(10^2)$ exemplars.

7. Applications and Open Problems

Weight extrapolation underpins modern boundedness and compactness results for Calderón–Zygmund operators, commutators, fractional integrals, Fourier multipliers, Littlewood–Paley operators, and vector-valued harmonic analysis. Two-weight and multi-parameter settings are crucial for robust PDE estimates, commutator compactness, and optimal norm dependencies.

Open questions persist:

• The necessity of auxiliary $A_\infty$ (e.g., "Bloom weight") conditions in two-weight extrapolation (Airta et al., 2021).
• Extension of sparse domination techniques to two-weight/multilinear settings.
• Precise characterizations of function space structure (BMO, product spaces, capacitary spaces) yielding compactness and sharp bounds.
• Algebraic and algorithmic generalizations in matrix/convex-set regimes.
• Quantitative extrapolation constants in quasi-Banach and limited-range settings.

The interplay of extrapolation, operator theory, and geometric/functional analysis continues to be a foundational theme for analysis, applied mathematics, and data-driven modeling (Cao et al., 2021, Nieraeth, 2018, Nieraeth, 2022, Sauer, 4 Nov 2025, Nehme et al., 26 Oct 2025).