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Math-2269 · mathematics · Analysis

Lipschitz extension from Hilbert spaces into L1

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.
Does the pair $(L^2,L^1)$ have the Lipschitz extension property? Explicitly, is there a constant $C<\infty$ such that every Lipschitz map from every subset of the Hilbert space $L^2$ into $L^1$ extends to the whole $L^2$ with Lipschitz constant at most $C$ times the original constant?

Formulation references: [S1], [S2].

Short English statement

Does the pair $(L^2,L^1)$ have the Lipschitz extension property? Explicitly, is there a constant $C<\infty$ such that every Lipschitz map from every subset of the Hilbert space $L^2$ into $L^1$ extends to the whole $L^2$ with Lipschitz constant at most $C$ times the original constant?

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2269-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2269

The selected theorem concerns the discrete sequence target $\ell_1$ and gives a universal Lipschitz extension bound for maps from Hilbert-space subsets into that target. The recorded question asks for an arbitrary $L^1$ target, including the nonatomic space. No reduction of that target to $\ell_1$ is supplied. Thus the discrete-target extension result is a genuine restricted case, and the larger extension problem remains partial.

Lipschitz extension from Hilbert spaces into L1

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 332; 2026-10-05), the asserted result is that real $\ell_1$ has metric Markov cotype two, giving uniformly controlled Lipschitz extension from Hilbert-space subsets to this target.

Mathematical statement of the claimed result

Manuscript 1.
For every finite reversible stochastic matrix $A=(a_{ij})$ with stationary distribution $\pi$, every integer $t\ge1$ and every $x_1,\ldots,x_n\in\ell_1(\mathbb R)$, the claim supplies $y_1,\ldots,y_n$ satisfying \[ \sum_i\pi_i\|x_i-y_i\|_1^2+t\sum_{i,j}\pi_i a_{ij}\|y_i-y_j\|_1^2 \le3024\sum_{i,j}\pi_i\left(\frac1t\sum_{s=1}^tA^s\right)_{ij}\|x_i-x_j\|_1^2. \] Thus the metric Markov cotype-two constant is at most $12\sqrt{21}$. The asserted extension consequence is one universal $K<\infty$ such that every Lipschitz $f:S\to\ell_1$ from a subset of any real Hilbert space $\mathcal H$ extends to $F:\mathcal H\to\ell_1$ with $\operatorname{Lip}(F)\le K\operatorname{Lip}(f)$.

Outline of the argument

For manuscript 1 (family 332; 2026-10-05), finite cut representations reconstruct an expected coordinatewise median of three stopped-walk endpoints. A flat-map martingale and quartic potential control the geometric endpoint costs, which a metric comparison converts to the Cesàro energy needed for cotype.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 332; 2026-10-05).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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