we-solve (beta)

Math-438-A2 · revision r1 · on Math-438

OpenAI claim summary for Math-438

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Hilbert transforms in Lipschitz directions: The manuscript’s result gives one absolute short scale and one absolute L^2 bound for every direction field with Lipschitz constant <=1; dilation supplies all positive Lipschitz constants at a fixed sufficiently small multiple of their inverse, and constant fields are ordinary directional Hilbert transforms. This covers the selected variable-constant local boundedness question and proves uniformity beyond it.

The claimed result implies the stated question and also supplies the additional strength described above.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Stein's Conjecture on the Hilbert Transform along Lipschitz Vector Fields
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Stein's Conjecture on the Hilbert Transform along Lipschitz Vector Fields

OpenAI's claim

Hilbert transforms in Lipschitz directions: The claimed $L^2$ bound is uniform for inner truncations of a Hilbert transform following a planar Lipschitz unit field over a short outer interval. Its length is an absolute multiple of the reciprocal Lipschitz constant, and the field may vary in both coordinates.

Mathematical statement of the claimed result

Mathematical claim 1.
For a Lipschitz $v:\mathbb R^2\to S^1$ with $\operatorname{Lip}(v)\le1$, put $H_{v,a}^{\epsilon}f(x)=\int_{\epsilon<|t|<a}f(x-tv(x))\,dt/t$. The claim gives absolute $0<a_*<1/2$ and $C_*<\infty$ with \[\sup_{0<\epsilon<a_*}\|H_{v,a_*}^{\epsilon}f\|_2\le C_*\|f\|_2\quad(f\in\mathcal S(\mathbb R^2)).\] Thus the symmetric principal value has the same $L^2$ bound and $|\{|H_{v,a_*}f|>\lambda\}|\le C_*^2\lambda^{-2}\|f\|_2^2$. The supremum above is outside the norm; it is not a maximal-truncation norm assertion.

Outline of the argument

Hilbert transforms in Lipschitz directions: Time-frequency tiles produce a popular forest, followed by concentration estimates that count significant directions simultaneously. A finite Fourier inequality and an angular hierarchy organize predicted factors and residuals along paths. Integration of those products, grid averaging, and interpolation assemble the short-scale strong bound without dimension-dependent bin losses.

Formal verification

Family 083 describes Lean coverage for the short-scale Lipschitz-field Hilbert bounds and their stated extension and principal-value consequences.

Hilbert transforms in Lipschitz directions: The manuscript links LipschitzHilbert.lean.

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