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Math-2367 · mathematics · Analysis

Higher-Dimensional Riesz Transforms and Uniform Rectifiability

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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.
For an integer n with 0<n<d and n not equal to d-1, if E is n-Ahlfors regular and all Riesz transforms of singularity n are bounded on $L^2(E)$, is E necessarily n-uniformly rectifiable?

Formulation references: [S1], [S2].

Short English statement

Let $0<n<d$ be integers with $n\neq d-1$, and let $E\subseteq\mathbb R^d$ be $n$-Ahlfors regular. If its $n$-dimensional Riesz transforms are bounded on $L^2(E)$, must $E$ be uniformly $n$-rectifiable? Ahlfors regularity means that its $n$-dimensional measure in balls is comparable to the $n$th power of their radius; uniform rectifiability requires quantitative large pieces of Lipschitz images.

Sources

Catalogue entry

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

Contributions 1

Math-2367-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2367

The paper claims the Riesz-bound implication in the remaining range $2\le n\le d-2$. Its historical discussion identifies the one-dimensional implication as already established by the cited curvature results. Combining those cases covers every integer $0<n<d$ other than codimension one, exactly the range on this page. The new claim supplies the previously missing intermediate dimensions.

Higher-Dimensional Riesz Transforms and Uniform Rectifiability

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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