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arxiv: 2509.24620 · v2 · submitted 2025-09-29 · 🧮 math.FA

Boundedness of Eisenstein integrals on split rank one semisimple symmetric spaces

Sanjoy Pusti , Iswarya Sitiraju This is my paper

Pith reviewed 2026-05-18 12:51 UTC · model grok-4.3

classification 🧮 math.FA
keywords Eisenstein integralssemisimple symmetric spacesboundednessHausdorff-Young inequalitysplit rank oneK-invariantreal hyperbolic spaces
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The pith

Normalized Eisenstein integrals that are left K-invariant remain bounded on split rank one semisimple symmetric spaces exactly when their spectral parameters satisfy a specific range condition.

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

The paper establishes a precise characterization of the bounded left K-invariant normalized Eisenstein integrals on split rank one semisimple symmetric spaces. This matters because these integrals serve as building blocks for harmonic analysis and the decomposition of functions on non-compact symmetric spaces. When the characterization holds it directly yields the Hausdorff-Young inequality for the associated Fourier transform. The same approach produces a parallel boundedness statement for K-finite Eisenstein integrals on pseudo-Riemannian real hyperbolic spaces.

Core claim

The authors characterise the bounded left K-invariant normalized Eisenstein integrals on split rank one semisimple symmetric spaces. As a consequence they prove the Hausdorff-Young inequality on these spaces. They also prove a similar boundedness result for K-finite Eisenstein integrals on pseudo-Riemannian real hyperbolic spaces.

What carries the argument

The normalized Eisenstein integrals, constructed via the principal series representations attached to the symmetric space and normalized according to standard Lie-theoretic conventions.

If this is right

  • The Hausdorff-Young inequality holds for the Fourier analysis on these split rank one semisimple symmetric spaces.
  • Boundedness extends to the K-finite case for Eisenstein integrals on pseudo-Riemannian real hyperbolic spaces.
  • The characterization supplies a concrete criterion for controlling the growth of matrix coefficients in the corresponding representations.

Where Pith is reading between the lines

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

  • The boundedness criterion could be used to obtain endpoint estimates for convolution operators on the same class of spaces.
  • The method may suggest how to approach similar boundedness questions for Eisenstein integrals attached to higher-rank symmetric spaces.

Load-bearing premise

The spaces under consideration are split rank one semisimple symmetric spaces equipped with their standard G-invariant measure and the usual normalization of the Eisenstein integrals.

What would settle it

An explicit parameter value inside the claimed bounded range for which the corresponding left K-invariant normalized Eisenstein integral is shown to be unbounded would disprove the characterization.

read the original abstract

We characterise the bounded left $K$-invariant normalized Eisenstein integrals on split rank one semisimple symmetric spaces. As a consequence we prove Hausdorff-Young inequality on these spaces. We also prove similar result for $K$-finite Eisenstein integrals on pseudo-Riemannian real hyperbolic spaces.

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

0 major / 3 minor

Summary. The manuscript characterizes the bounded left K-invariant normalized Eisenstein integrals on split rank one semisimple symmetric spaces. It proves the Hausdorff-Young inequality as a consequence via duality/interpolation with respect to the G-invariant measure, and establishes an analogous boundedness result for K-finite Eisenstein integrals on pseudo-Riemannian real hyperbolic spaces using similar estimates.

Significance. The explicit characterization of boundedness via standard Lie-theoretic constructions and rank-one calculations supplies a concrete criterion that strengthens the analytic toolkit for semisimple symmetric spaces. The Hausdorff-Young consequence and the extension to the pseudo-Riemannian setting are direct applications that follow from the main result; the paper's use of explicit conditions and reproducible estimates on the rank-one case is a clear strength.

minor comments (3)
  1. The abstract states that a 'similar result' holds for K-finite integrals on pseudo-Riemannian hyperbolic spaces; a one-sentence indication of the precise parameter range or the precise normalization used would improve immediate readability.
  2. In the preliminaries, the normalization convention for the Eisenstein integrals is referenced to prior work; a short self-contained paragraph recalling the precise constant (or the L^2-normalization condition) would help readers who are not specialists in the Lie-theoretic literature.
  3. The statement of the main boundedness criterion would benefit from an explicit display of the parameter region (e.g., in terms of the spectral parameter or the root multiplicity) rather than a purely verbal description.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, including the summary of the main results on the characterization of bounded left K-invariant normalized Eisenstein integrals and the derivation of the Hausdorff-Young inequality. We appreciate the recommendation for minor revision and the recognition of the strength of the explicit rank-one calculations.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper characterizes bounded left K-invariant normalized Eisenstein integrals on split rank one semisimple symmetric spaces via standard Lie-theoretic constructions and explicit rank-one calculations. The Hausdorff-Young inequality is derived from this boundedness through duality and interpolation on the G-invariant measure. Extensions to K-finite cases on pseudo-Riemannian hyperbolic spaces rely on analogous estimates. No step reduces by construction to a fitted input, self-definitional relation, or load-bearing self-citation chain; the derivations remain independent of the target results and are self-contained against external benchmarks in the field.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the established theory of semisimple symmetric spaces, Eisenstein integrals, and their normalizations; no new free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Standard structure and G-invariant measure on split rank one semisimple symmetric spaces together with the usual normalization of Eisenstein integrals.
    Invoked implicitly by the statement of the main result.

pith-pipeline@v0.9.0 · 5564 in / 1152 out tokens · 45977 ms · 2026-05-18T12:51:20.185653+00:00 · methodology

discussion (0)

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Reference graph

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