[#P2924] Openness of convolution on l1 of the integers

A mathematical schematic of Openness of convolution on l1 of the integers. — A statement-only illustration of the mathematical objects and operations in this problem.

Problem. Is the convolution map \(\ell_1(\mathbb Z)\times\ell_1(\mathbb Z)\to\ell_1(\mathbb Z)\), \((a,b)\mapsto a*b\), an open map?

1Context

Fourier zero sets, finitely supported approximation lemmas, and quantitative local radii can be reused. The source already separates pointwise openness from the stronger uniform-open-map property.

2Problem setup

Remark 1. For a,b in ell_1(Z), convolution is (a*b)_n=sum_{k in Z} a_k b_{n-k}, which again belongs to ell_1(Z).

Definition 1. A map is open if it sends every open subset of the product Banach space to an open subset of ell_1(Z); equivalently, it is locally open at every pair (a,b).

3What counts as a solution

Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b).
A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.

1Status

Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair. Exact unresolved remainder: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.^[3][2][1]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair. Exact unresolved remainder: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to ab that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.^[3][2][1] Relevance to this problem For Openness of convolution on l1 of the integers, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 3, Reference 2, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair. Exact unresolved remainder: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to ab that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method. Open full record →
	Trace	Reported
attempt · Route 1 Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to ab that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method.^[1] Relevance to this problem For Openness of convolution on l1 of the integers, record l1z-convolution-open-map-next-route-20260801 (“Complete the stated acceptance conditions”) documents a concrete method, search boundary, or failed route. The record states: Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to ab that cannot be factored by pairs converging to (a,b). Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. Prove local openness at every pair (a,b), with a neighborhood estimate that may depend on the pair, or exhibit a pair and an explicit sequence converging to a*b that cannot be factored by pairs converging to (a,b). A negative result must rule out all nearby factorizations, rather than showing failure of a single constructive factorization method. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-27. The thread records local openness when one factor is finitely supported and failure of a uniform openness estimate. It does not establish or refute openness at every pair.

Work on open bilinear mappings, including DOI 10.1215/17358787-3599741, was checked for a theorem applying directly to convolution on ell_1(Z). No such specialization was found.
Norm-controlled inversion results for convolution algebras were checked because invertible factors yield local solution mechanisms. They do not cover arbitrary pairs whose Fourier transforms have common zeros.
A local corpus search for open convolution map, ell_1(Z), and Banach algebra multiplication found no duplicate.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. If one factor is invertible in the convolution algebra, multiplication by that factor is a Banach-space isomorphism and gives local openness at the pair.

2See also

An infinite-dimensional Banach space where every operator attains its normfunctional analysis
Bochner-Riesz conjecture in higher dimensionsharmonic analysis
Sharp fourth-power norm of the cyclic Hilbert transform at order 31harmonic analysis

How to cite

TheoremDB contributors, “Openness of convolution on l1 of the integers,” TheoremDB research memory, snapshot of August 1, 2026. https://theoremdb.org/statements/l1z-convolution-open-map

This page as plain text: l1z-convolution-open-map.md

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1References

Packet source. MathOverflow question 504613, “Openness of convolution on l1 of the integers,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 504613 and every visible answer and comment were checked on 2026-07-27.Also cited at Full question, answers, and visible comments concerning Openness of convolution on l1 of the integers; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Openness of convolution on l1 of the integers, the reviewed source scope is Full question, answers, and visible comments concerning Openness of convolution on l1 of the integers; checked 2026-08-01.. The packet makes no inference beyond that cited scope.Source named by the research packet.
Marek Balcerzak, Ehrhard Behrends, and Filip Strobin, “On certain uniformly open multilinear mappings,” Banach Journal of Mathematical Analysis 10(3) (2016), 482-494. DOI 10.1215/17358787-3599741. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and uniform-openness results for pointwise multiplication and nonzero multilinear functionals.Source used to assess the problem's recorded status.For Openness of convolution on l1 of the integers, this source supplies techniques and examples for different multilinear maps without settling l1(Z) convolution.
Karlheinz Gröchenig and Andreas Klotz, “Norm-controlled inversion in smooth Banach algebras, I,” Journal of the London Mathematical Society 88(1) (2013), 49-64. DOI 10.1112/jlms/jdt004. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at main norm-controlled inversion inequalities for smooth Banach subalgebras.Source used to assess the problem's recorded status.For Openness of convolution on l1 of the integers, this source controls inverse norms in convolution-type Banach algebras without proving openness of l1(Z) convolution.

An original CC0 textbook restatement motivated by the cited MathOverflow functional-analysis question; no page wording was copied.

[#P2924] Openness of convolution on l1 of the integers

1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance