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Math-2366 · mathematics · Analysis

Uniform Rectifiability from Bounded Riesz Transforms

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $n,d$ be integers with $1\le n<d$, and let $\mu$ be an $n$-dimensional AD-regular measure in $\mathbb R^d$. Is $\mu$ uniformly $n$-rectifiable if and only if its maximal $n$-Riesz transform $R_\mu^*$ is bounded on $L^2(\mu)$?

Formulation references: [S1], [S2].

Short English statement

Let $n,d$ be integers with $1\le n<d$, and let $\mu$ be an $n$-dimensional AD-regular measure in $\mathbb R^d$. Is $\mu$ uniformly $n$-rectifiable if and only if its maximal $n$-Riesz transform $R_\mu^*$ is bounded on $L^2(\mu)$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2366-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2366

The claimed theorem treats the intermediate dimensions $2\le n\le d-2$, while the paper cites established one-dimensional and codimension-one implications. A maximal-transform bound controls every hard truncation, so these implications give uniform rectifiability throughout the recorded dimensional range. Conversely, the standard singular-integral bounds for uniformly rectifiable measures control the maximal Riesz transform. In full ambient dimension the rectifiability statement has the direct Euclidean parametrization. The claimed new theorem therefore completes the missing cases of this equivalence, with its additional constants specified in the mathematical statement.

Uniform Rectifiability from Bounded Riesz Transforms

OpenAI's claim

Higher-codimension Riesz rectifiability: The claimed result turns uniform $L^2$ control of all positive hard truncations of the Riesz transform into quantitative uniform rectifiability. It concerns $n$-regular measures in $\mathbb R^d$ with $d\ge4$ and $2\le n\le d-2$.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $d\ge4$ and $2\le n\le d-2$. Suppose a Radon measure $\mu$ on $\mathbb R^d$ satisfies $C^{-1}r^n\le\mu(B(x,r))\le Cr^n$ for $x\in\operatorname{supp}\mu$ and $0<r\le\operatorname{diam}(\operatorname{supp}\mu)$. If \[R_{\mu,\epsilon}f(x)=\int_{|x-y|>\epsilon}\frac{x-y}{|x-y|^{n+1}}f(y)\,d\mu(y),\qquad\sup_{\epsilon>0}\|R_{\mu,\epsilon}\|_{L^2(\mu)\to L^2(\mu;\mathbb R^d)}\le C_R,\] then there are $\theta>0$ and $M<\infty$, depending only on $d,n,C,C_R$, such that each admissible $B(x,r)$ contains at least $\theta r^n$ of $\mu$ on an $M$-Lipschitz image of the radius-$r$ ball in $\mathbb R^n$. This is uniform $n$-rectifiability; pointwise principal values are not assumed.

Outline of the argument

Higher-codimension Riesz rectifiability: After a flat blowup, the normal height is renormalized by the initial excess so that relative improvement can be detected. A fractional-energy test yields compactness and a limiting equation whose integrability forces the height to be affine. Dyadic packing then counts failures of excess improvement, leading to large Lipschitz pieces at every support scale.

Formal verification

Family 081 describes Lean coverage for the quantitative Lipschitz-piece conclusion in the stated higher-codimension Riesz setting.

Higher-codimension Riesz rectifiability: The manuscript links RieszQuantitative.lean (declaration OAI.RieszRectifiability.quantitative_higher_codimension_riesz_rectifiability).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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