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Off-diagonal matrix extrapolation for Muckenhoupt bases

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Pith's one-line read The authors prove off-diagonal and general-basis Rubio de Francia extrapolation for matrix weights, and show multiparameter bases satisfy the required maximal operator bound.

arxiv 2504.12407 v1 pith:DQSTGOKV submitted 2025-04-16 math.CA

David Cruz-Uribe , Fatih Şirin This is my paper · ORCID
classification math.CA
keywords extrapolationmatrixbasesextendoff-diagonaltheoryassumedauthor
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

Many problems in analysis ask whether an operator T (like a maximal function or singular integral) maps one weighted space into another. The Rubio de Francia extrapolation theorem says that if T satisfies a weighted inequality for one exponent p, then it satisfies weighted inequalities for all exponents p, as long as the weights belong to the Muckenhoupt classes. Recent work by Bownik and Cruz-Uribe extended this theorem to matrix weights, where the weight is a positive definite matrix function and functions are vector-valued. This paper pushes that theory in two directions.

First, it proves off-diagonal extrapolation, where the source and target exponents p and q are different (p < q). Such estimates are typical for fractional integral operators. A surprise in the matrix setting is that the natural weight class A_{p,q} is not sufficient: the hypothesis and conclusion use the stronger condition that W^s belongs to a standard A_r class, with s determined by the off-diagonal exponents. The authors show by example that A_{p,q} is strictly weaker, and they leave it as an open question whether A_{p,q} suffices for the conclusion.

Second, the paper replaces the usual cubes by an arbitrary collection of sets, a basis, and defines a matrix Muckenhoupt basis as one for which a certain maximal operator is bounded for all matrix A_p weights. Under this assumption, extrapolation holds. The paper verifies the assumption for dyadic cubes, ordinary cubes, and all multiparameter bases (rectangles that are products of cubes), extending a result of Vuorinen. The proofs use convex-set valued operators and Rubio de Francia iteration, following the framework of Bownik and Cruz-Uribe.

Extended reading notes

Core claim

Theorem 1.10 (with Theorem 1.2 as the cube case): if a sublinear operator T satisfies a weighted (p0,q0) estimate for all matrix weights with W_0^s in A_{r0,B}, then for all 1<p<=q<infinity with 1/p-1/q = 1/p0-1/q0, T satisfies the corresponding (p,q) estimate for all W with W^s in A_{r,B}. Theorem 1.12 verifies the needed maximal operator bound for every multiparameter basis R_alpha with constant [W]^{j p'}_{A_{p,R_alpha}}. If true, off-diagonal matrix extrapolation holds on all multiparameter bases.

Load-bearing premise

The proof of the multiparameter theorem (Section 6, Lemma 6.1) invokes a matrix version of the reverse Holder/self-improvement property for A_p weights, attributed to [12, Theorem 1.5], an unpublished preprint by the first author and M. Penrod. This is used to place the slice integrals F_{Q_k,v} and G_{Q_k,v} in L^s_loc, s>1, so that the Jessen-Marcinkiewicz-Zygmund differentiation theorem applies. If that unpublished result is false, the verification that multiparameter bases are matrix Muckenhoupt bases is unsupported.

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Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted constants; its constants are all tracked through inequalities. The main extra assumptions are the matrix Muckenhoupt basis property, the unpublished matrix reverse Holder result [12], the imported convex-set valued machinery from [2], and a sketched reduction in Theorem 1.11. No new physical entities or free parameters are introduced; the new notion of matrix Muckenhoupt basis is a definition, not a postulated object with independent evidence requirements.

assumptions (5)
  • domain assumption Matrix Muckenhoupt basis property (Definition 1.5): for every 1<p<infinity and W in A_{p,B}, the Christ-Goldberg maximal operator M_{W,B} is bounded on L^p(R^n) with norm depending on [W]_{A_{p,B}}.
    This is the defining assumption for Theorems 1.10 and 4.1. The paper verifies it for cubes, dyadic cubes, and multiparameter bases, but not for the Zygmund basis, which is left open (Remark 1.13).
  • domain assumption Matrix reverse Holder/self-improvement for A_p weights ([12, Theorem 1.5]).
    Used in Lemma 6.1 to obtain W in A_{sp,Q} and W^{-1} in A_{sp',Q}, hence L^{sp}_loc integrability for the JMZ differentiation theorem. [12] is an unpublished preprint by the first author and Penrod.
  • domain assumption Convex-set valued operator machinery from [2]: iteration operator R_W (Theorem 3.10), exhausters, A^K_1 classes, reverse factorization (Proposition 3.13).
    The paper states these for general bases, saying the proofs are the same as in [2] for cubes. The reader is referred to [2] for details; [2] is an unpublished preprint.
  • domain assumption Uniform boundedness of averaging operators implies the reducing operator condition (2.2) and hence A_p, used at the end of Theorem 1.11.
    The proof is sketched with references [7, Theorem 1.18], [21, Proposition 2.1], [32, Lemma 1.3], not carried out in the paper.
  • standard math Standard real analysis facts: Cordes inequality (Lemma 2.3), measurable diagonalization (Lemma 2.4), the Jessen-Marcinkiewicz-Zygmund theorem for strong differentiation.
    These are background results used without proof.

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Pith. "Pith review of Off-diagonal matrix extrapolation for Muckenhoupt bases." pith.science (2026). https://pith.science/paper/DQSTGOKV

@misc{pith2026250412407,
  author       = {Pith},
  title        = {Pith review of: Off-diagonal matrix extrapolation for Muckenhoupt bases},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DQSTGOKV}},
  note         = {Machine review of arXiv:2504.12407}
}
abstract

In this paper we extend the theory of Rubio de Francia extrapolation for matrix weights, recently introduced by Bownik and the first author, to off-diagonal extrapolation. We also show that the theory of matrix weighted extrapolation can be extended to matrix $\mathcal{A}_p$ classes defined with respect to a general basis, provided that a version of the Christ-Goldberg maximal operator is assumed to be bounded. Finally, we extend a recent result by Vuorinen and show that all of the multiparameter bases have this property.

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