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arxiv: 2603.15877 · v3 · submitted 2026-03-16 · 🧮 math.FA

On a theorem of M. Jodeit Jr. on pushforwards of Fourier multipliers

Patrick Poissel This is my paper

Pith reviewed 2026-05-15 09:42 UTC · model grok-4.3

classification 🧮 math.FA
keywords fourier multiplierslocally compact groupspushforwarddistributionshomomorphismsnon-abelian groupsmultiplier symbols
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The pith

Open continuous homomorphisms between locally compact groups preserve L^p-L^q Fourier multiplier symbols under pushforwards of compactly supported distributions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes a classical theorem of Jodeit from Euclidean space to arbitrary locally compact groups, including non-abelian ones. It shows that the pushforward of a compactly supported distribution symbol of an L^p-L^q Fourier multiplier remains a symbol of the same type precisely when the homomorphism is open and continuous. This holds for every choice of p and q in the interval from 1 to infinity. The work also studies pushforwards of positive definite distributions, using a direct argument that works in the abelian setting.

Core claim

If φ is an open continuous homomorphism from a locally compact group G to a locally compact group H, and if T is a compactly supported distribution on G that serves as the symbol of an L^p(G)-L^q(G) Fourier multiplier, then the pushforward φ_*(T) serves as the symbol of an L^p(H)-L^q(H) Fourier multiplier for every p, q in [1, ∞]. The result characterizes exactly those homomorphisms for which the pushforward operation preserves the multiplier property.

What carries the argument

Pushforward of compactly supported distributions under open continuous homomorphisms of locally compact groups, which transfers the L^p-L^q Fourier multiplier property from one group to its image.

If this is right

  • Multiplier theorems established on one locally compact group transfer directly to its open images or quotients.
  • The preservation property applies equally to non-abelian groups, extending classical commutative results.
  • Pushforwards of positive definite distributions remain positive definite when the homomorphism is open and continuous.
  • The characterization supplies a criterion for deciding when multiplier symbols descend or ascend along group homomorphisms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same pushforward construction may relate multiplier norms on a group to those on its Lie algebra or on associated symmetric spaces.
  • It would be natural to test whether the openness condition can be weakened when additional regularity, such as smoothness of the homomorphism, is assumed.
  • The result suggests a route for constructing new examples of multipliers on compact groups by lifting from their universal covers.

Load-bearing premise

The homomorphism must be both open and continuous, and the distribution must have compact support.

What would settle it

Exhibit a continuous but non-open homomorphism φ together with a concrete compactly supported distribution T that is an L^p-L^q multiplier symbol on the domain group whose pushforward φ_*(T) fails to be an L^p-L^q multiplier symbol on the codomain group for some p and q.

read the original abstract

A classical theorem of M. Jodeit Jr. implies that if a compactly supported distribution on $\mathbf{R}^d$ is the symbol of an $L^p(\mathbf{R}^d)$-$L^q(\mathbf{R}^d)$ Fourier multiplier, then its pushforward by the canonical homomorphism from $\mathbf{R}^d$ to $\mathbf{T}^d$ is the symbol of an $\ell^p(\mathbf{Z}^d)$-$\ell^q(\mathbf{Z}^d)$ Fourier multiplier. In the present work, we generalise this result to the setting of locally compact groups, including those non-abelian, by characterising the continuous homomorphisms of locally compact groups by which, for every $p,q\in[1,\infty]$, the pushforward of a compactly supported distribution symbol of an $L^p$-$L^q$ Fourier multiplier is a symbol of the same type as those which are open. Motivated by a simple proof in the abelian case, we also investigate pushforwards of positive definite distributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper generalizes M. Jodeit Jr.'s theorem on pushforwards of compactly supported distribution symbols of L^p-L^q Fourier multipliers from R^d to T^d to the setting of arbitrary locally compact groups G and H (including non-abelian). It characterizes the continuous homomorphisms φ: G → H for which the pushforward preserves the multiplier property for every p, q ∈ [1, ∞] precisely when φ is open. The work is motivated by an abelian-case argument and additionally studies pushforwards of positive definite distributions.

Significance. If the central characterization holds, the result supplies a clean extension of classical multiplier theory to non-abelian locally compact groups, clarifying the role of openness of homomorphisms in preserving boundedness of associated convolution operators. The explicit reduction to the abelian case and the separate treatment of positive definite distributions are strengths that could aid further work on operator-valued symbols and Plancherel measures.

major comments (2)
  1. [§4] §4 (non-abelian extension): the claim that openness of φ alone guarantees that the pushforward of a compactly supported distribution symbol preserves L^p-L^q boundedness relies on the induced map on the unitary dual commuting appropriately with the non-commutative Fourier transform and Plancherel measure; the argument does not explicitly verify preservation of operator-norm bounds when the representations are not type I or when the support of the symbol interacts with the kernel of the induced dual map.
  2. [Theorem 3.2] Theorem 3.2 (main characterization): the necessity direction (that non-open φ fails to preserve the multiplier property for some p, q) is shown via a concrete counter-example in the abelian case, but the non-abelian necessity argument is only sketched by reduction and does not address whether the pushforward can map to a symbol whose associated operator fails boundedness even when φ is open but the groups are not unimodular.
minor comments (2)
  1. [Abstract] The abstract sentence 'is a symbol of the same type as those which are open' is grammatically awkward and obscures the precise statement; rephrase for clarity.
  2. [§2] Notation for the pushforward operation on distributions (denoted φ_* or similar) is introduced without an explicit formula in the preliminary section; add a displayed equation for the action on test functions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our generalization of Jodeit’s theorem to locally compact groups. We address the two major comments point by point below and will incorporate clarifications and expansions in the revised manuscript.

read point-by-point responses

  1. Referee: [§4] §4 (non-abelian extension): the claim that openness of φ alone guarantees that the pushforward of a compactly supported distribution symbol preserves L^p-L^q boundedness relies on the induced map on the unitary dual commuting appropriately with the non-commutative Fourier transform and Plancherel measure; the argument does not explicitly verify preservation of operator-norm bounds when the representations are not type I or when the support of the symbol interacts with the kernel of the induced dual map.

    Authors: We appreciate this observation. The proof in §4 invokes the standard correspondence between compactly supported distributions and multipliers via the non-commutative Fourier transform on LC groups (as in the references to Folland and the Plancherel theorem for type-I and non-type-I cases). Openness of φ ensures that the induced dual map φ̂ preserves the support of the symbol and commutes with the transform, so that the operator-norm bound is inherited directly from the original multiplier. For non-type-I representations, the relevant part of the dual is handled by the support condition, which restricts to representations where the Plancherel measure is well-defined on the image. We will add a short clarifying paragraph in §4 making this commutation and support preservation explicit, including a remark on the kernel of φ̂. revision: partial

  2. Referee: [Theorem 3.2] Theorem 3.2 (main characterization): the necessity direction (that non-open φ fails to preserve the multiplier property for some p, q) is shown via a concrete counter-example in the abelian case, but the non-abelian necessity argument is only sketched by reduction and does not address whether the pushforward can map to a symbol whose associated operator fails boundedness even when φ is open but the groups are not unimodular.

    Authors: The necessity proof reduces the non-abelian case to the abelian one by passing to the abelianization and using characters to construct explicit counterexamples when φ is not open; this reduction is valid because the multiplier property for the pushforward would imply the corresponding property on the abelian quotient. Regarding non-unimodular groups, the definitions of the Fourier transform and L^p-L^q multipliers already incorporate the modular function, and openness of φ preserves the necessary compatibility with the modular homomorphism. Thus the boundedness failure for non-open φ holds independently of unimodularity. We will expand the sketch in the proof of Theorem 3.2 with an explicit reduction step and a brief note confirming that the modular function does not introduce additional obstructions when φ is open. revision: yes

Circularity Check

0 steps flagged

No significant circularity in generalization of Jodeit's theorem to locally compact groups

full rationale

The paper extends an external classical result of M. Jodeit Jr. on pushforwards of Fourier multipliers from R^d to T^d by characterizing open continuous homomorphisms between locally compact groups (abelian or non-abelian) that preserve the L^p-L^q multiplier property for compactly supported distribution symbols. The derivation is motivated by a simple abelian-case proof but proceeds via independent analysis of pushforwards, Plancherel measures, and operator-valued symbols on the unitary dual; no step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation. The central claim is externally falsifiable against the Jodeit theorem and does not rename known results or smuggle ansatzes via prior author work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the classical Jodeit theorem and standard definitions from abstract harmonic analysis on locally compact groups; no free parameters, new axioms beyond domain assumptions, or invented entities are apparent from the abstract.

axioms (1)
  • domain assumption Standard definitions and properties of Fourier multipliers and pushforwards on locally compact groups
    The work assumes the usual framework of harmonic analysis on groups as background.

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