arxiv: 2605.01252 · v1 · submitted 2026-05-02 · 🧮 math.FA

Recognition: unknown

L^r- Schwartz spaces on split rank one semisimple symmetric spaces

Iswarya Sitiraju, Sanjoy Pusti

Pith reviewed 2026-05-10 14:53 UTC · model grok-4.3

classification 🧮 math.FA

keywords Schwartz spaceFourier transformkerneldiscrete spectrumLaplace-Beltrami operatorsymmetric spaceharmonic analysis

0 comments

The pith

The kernel of the Fourier transform on left K-invariant L^r-Schwartz spaces is spanned by discrete spectrum eigenfunctions of the Laplace-Beltrami operator.

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

This paper examines the left K-invariant L^r-Schwartz spaces on split rank one semisimple symmetric spaces G/H for 0 < r ≤ 2. The authors define the Fourier transform on these spaces and explicitly determine its kernel. They show that the kernel is spanned by the eigenfunctions associated with the discrete spectrum of the Laplace-Beltrami operator on G/H. This description is important because it pinpoints the functions in the Schwartz space that vanish under the Fourier transform, clarifying the range and injectivity properties in harmonic analysis on symmetric spaces.

Core claim

We explicitly determine the kernel of the Fourier transform on the left K-invariant L^r-Schwartz space and show that it is spanned by eigenfunctions associated with the discrete spectrum of the Laplace-Beltrami operator on G/H.

What carries the argument

The Fourier transform defined on the left K-invariant L^r-Schwartz space, whose kernel is the span of discrete spectrum eigenfunctions of the Laplace-Beltrami operator.

If this is right

The Fourier transform fails to be one-to-one on the L^r-Schwartz space due to the discrete spectrum.
The Schwartz space contains a discrete component that is invisible to the Fourier transform.
Explicit spanning sets for the kernel facilitate the study of the quotient space where the transform becomes injective.

Where Pith is reading between the lines

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

This suggests that the full Plancherel theorem or inversion formula must include a separate discrete part.
The result could guide the definition of modified Schwartz spaces that exclude the discrete spectrum for better invertibility.
Analogous results might hold for other classes of symmetric spaces beyond split rank one.

Load-bearing premise

The Fourier transform is well-defined and continuous on the left K-invariant L^r-Schwartz space.

What would settle it

Computing the Fourier transform of a discrete spectrum eigenfunction and checking if it is zero, or exhibiting a function in the kernel not in the span of those eigenfunctions.

read the original abstract

We study the left $K$-invariant $L^r$-Schwartz space and its Fourier transform on split rank one semisimple symmetric spaces $G/H$ for $0<r\leq 2$. We explicitly determine the kernel of the Fourier transform and show that it is spanned by eigenfunctions associated with the discrete spectrum of the Laplace--Beltrami operator on $G/H$.

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.

Desk Editor's Note private letter to a colleague

The paper explicitly pins down the kernel of the Fourier transform on left K-invariant L^r-Schwartz spaces for split rank-one symmetric spaces as spanned by discrete-spectrum Laplace-Beltrami eigenfunctions.

read the letter

The main point is that the authors give an explicit description of the kernel of the Fourier transform on the left K-invariant L^r-Schwartz space for 0 < r ≤ 2 on split rank one semisimple symmetric spaces G/H, and they show this kernel is spanned by eigenfunctions from the discrete spectrum of the Laplace-Beltrami operator. This is a concrete step in a narrow setting where the low rank makes explicit calculations possible once the Plancherel decomposition is available. The work does well by turning the standard strategy for these spaces into a precise statement about the kernel, which can be useful for anyone needing inversion formulas or multiplier results in this exact class. The approach looks grounded in existing harmonic analysis on symmetric spaces rather than introducing new machinery. The soft spots are limited. The abstract leaves the precise definition of the L^r-Schwartz space and the continuity proof for the transform implicit, so a reader must check whether those steps add new estimates or simply invoke prior results. The spanning claim appears standard for split rank one cases and does not show obvious gaps or circularity in the given outline. No load-bearing assumptions seem hidden or unfalsifiable. This is a paper for specialists already working on Fourier analysis and Schwartz spaces on Lie groups and symmetric spaces. A reader in that subfield gets a usable reference point; outsiders will find it too narrow. The thinking is coherent and engages the literature directly, so it deserves a serious referee even if revisions are needed on the details of the definitions.

Referee Report

0 major / 2 minor

Summary. The paper studies the left K-invariant L^r-Schwartz space and its Fourier transform on split rank one semisimple symmetric spaces G/H for 0 < r ≤ 2. It explicitly determines the kernel of the Fourier transform and shows that this kernel is spanned by eigenfunctions associated with the discrete spectrum of the Laplace-Beltrami operator on G/H.

Significance. If the result holds, the explicit description of the kernel provides a concrete link between the L^r-Schwartz space and the discrete spectrum, which strengthens the spectral theory for these spaces beyond the L^2 case. The parameter-free nature of the spanning set (once the discrete eigenvalues are known) is a clear strength for applications in harmonic analysis.

minor comments (2)

[Section 1] The notation for the L^r-Schwartz space (e.g., seminorms or topology) is introduced without an explicit comparison to the classical Schwartz space when r=2; adding a short remark would improve readability.
[Theorem 4.1] In the statement of the main theorem, the precise range of r for which the Fourier transform extends continuously should be restated for emphasis, as the abstract only gives 0<r≤2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript on the L^r-Schwartz spaces and their Fourier transforms on split rank one semisimple symmetric spaces. The report accurately reflects our main result that the kernel is spanned by eigenfunctions associated with the discrete spectrum of the Laplace-Beltrami operator. We note the recommendation for minor revision and will incorporate any such changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via standard spectral theory

full rationale

The paper's central result explicitly determines the kernel of the Fourier transform on the left K-invariant L^r-Schwartz space and shows it is spanned by discrete-spectrum eigenfunctions of the Laplace-Beltrami operator. This follows directly from the Plancherel decomposition and the definition of the Schwartz space on split rank-one symmetric spaces G/H, without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations for uniqueness. The derivation uses established facts from harmonic analysis on these spaces and remains independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions from Lie theory and harmonic analysis on symmetric spaces, with no free parameters or invented entities mentioned.

axioms (2)

domain assumption Standard structure and properties of split rank one semisimple symmetric spaces G/H and their Laplace-Beltrami operator
Invoked implicitly to define the spaces and the discrete spectrum.
domain assumption The Fourier transform is defined and acts on the left K-invariant L^r-Schwartz space
Required to study its kernel.

pith-pipeline@v0.9.0 · 5349 in / 1244 out tokens · 46531 ms · 2026-05-10T14:53:45.969344+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

58 extracted references

[1]

Andersen

N. Andersen. Paley–Wiener theorems for hyperbolic spaces.J. Funct. Anal., 179(1):66–119, 2001

2001
[2]

J. P. Anker and B. Ørsted, editors.Lie theory, volume 230 ofProgress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 2005. Harmonic analysis on symmetric spaces—general Plancherel theorems

2005
[3]

The spherical Fourier transform of rapidly decreasing functions

Jean-Philippe Anker. The spherical Fourier transform of rapidly decreasing functions. A simple proof of a character- ization due to Harish-Chandra, Helgason, Trombi, and Varadarajan.J. Funct. Anal., 96(2):331–349, 1991

1991
[4]

A. A. Artemov and V. F. Molchanov. The Laplace–Beltrami operator on rank one semisimple symmetric spaces in polar coordinates.Sib. Math. J., 31(6):900–910, 1990

1990
[5]

E. P. van den Ban. Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula.Ark. Mat., 25(2):175–187, 1987

1987
[6]

E. P. van den Ban. The principal series for a reductive symmetric space. I.H-fixed distribution vectors.Ann. Sci. ´Ecole Norm. Sup. (4), 21(3):359–412, 1988

1988
[7]

E. P. van den Ban. The principal series for a reductive symmetric space. II. Eisenstein integrals.J. Funct. Anal., 109(2):331–441, 1992

1992
[8]

E. P. van den Ban and H. Schlichtkrull. Expansions for Eisenstein integrals on semisimple symmetric spaces.Ark. Mat., 35(1):59–86, 1997

1997
[9]

E. P. van den Ban and H. Schlichtkrull. The Most Continuous Part of the Plancherel Decomposition for a Reductive Symmetric Space.Ann. of Math., 145(2):267–364, 1997

1997
[10]

E. P. van den Ban and H. Schlichtkrull. Fourier inversion on a reductive symmetric space.Acta Math., 182(1):25–85, 1999

1999
[11]

E. P. van den Ban and H. Schlichtkrull. The Plancherel Decomposition for a Reductive Symmetric Space i: Spherical Functions.Invent. Math., 161(3):453–566, 2005

2005
[12]

E. P. van den Ban and H. Schlichtkrull. The Plancherel decomposition for a reductive symmetric space. ii. Represen- tation theory.Invent. Math., 161:567–628, 2005

2005
[13]

E. P. van den Ban and H. Schlichtkrull. A Paley–Wiener theorem for reductive symmetric spaces.Ann. of Math., 163(3):1041–1131, 2006

2006
[14]

van den Ban and H

E. van den Ban and H. Schlichtkrull. Fourier transform on a semisimple symmetric space.Invent. Math., 130(3):517– 574, 1997

1997
[15]

W. H. Barker.L p harmonic analysis on SLp2,Rq.Mem. Amer. Math. Soc., 76(393):iv+110, 1988

1988
[16]

Brylinski and P

J. Brylinski and P. Delorme. Vecteurs distributionsH-invariants pour les s´ eries principales g´ en´ eralis´ ees d’espaces sym´ etriques r´ eductifs et prolongement m´ eromorphe d’int´ egrales d’eisenstein.Invent. Math., 109(3):619–664, 1992

1992
[17]

Carmona and P

J. Carmona and P. Delorme. Base m´ eromorphe de vecteurs de distributionsH-invariants pour les s´ eries principales g´ en´ eralis´ ees d’espaces sym´ etriques r´ eductifs:´Equation fonctionnelle.J. Funct. Anal., 122(1):152–221, 1994

1994
[18]

Carmona and P

J. Carmona and P. Delorme. Transformation de Fourier sur l’espace de Schwartz d’un espace sym´ etrique r´ eductif. Invent. Math., 134(1):59–99, 1998

1998
[19]

P. Delorme. Int´ egrales d’eisenstein pour les espaces sym´ etriques r´ eductifs: temp´ erance, majorations. petite matriceb. J. Funct. Anal., 136(2):422–509, 1996

1996
[20]

P. Delorme. Troncature pour les espaces sym´ etriques r´ eductifs.Acta Math., 179:41–77, 1997

1997
[21]

P. Delorme. Formule de Plancherel pour les espaces sym´ etriques r´ eductifs.Ann. of Math., 147(2):417–452, 1998

1998
[22]

van Dijk

G. van Dijk. Harmonic analysis on rank one symmetric spaces. In A. O. Barut and H. D. Doebner, editors,Conformal Groups and Related Symmetries Physical Results and Mathematical Background, pages 244–252, Berlin, Heidelberg,
[23]

20 SANJOY PUSTI AND ISW ARYA SITIRAJU

Springer Berlin Heidelberg. 20 SANJOY PUSTI AND ISW ARYA SITIRAJU
[24]

van Dijk.Introduction to Harmonic Analysis and Generalized Gelfand Pairs, volume 36 ofDe Gruyter Studies in Mathematics

G. van Dijk.Introduction to Harmonic Analysis and Generalized Gelfand Pairs, volume 36 ofDe Gruyter Studies in Mathematics. Walter de Gruyter, Berlin, 2009

2009
[25]

van Dijk and M

G. van Dijk and M. Poel. The Plancherel formula for the pseudo-Riemannian spaceSLpn,Rq{GLpn´1,Rq. Preprint, University of Leiden (1984), to appear in Compositio Mathematica

1984
[26]

Eguchi and A

M. Eguchi and A. Kowata. On the Fourier transform of rapidly decreasing functions ofL p type on a symmetric space. Hiroshima Math. J., 7(1):143–158, 1977

1977
[27]

Erd´ elyi, W

A. Erd´ elyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi.Higher transcendental functions. Vol. I. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1981. Based on notes left by Harry Bateman, With a preface by Mina Rees, With a foreword by E. C. Watson, Reprint of the 1953 original

1981
[28]

J. Faraut. Distributions sph´ eriques sur les espaces hyperboliques.J. Math. Pures Appl., 58, 1979

1979
[29]

Flensted-Jensen

M. Flensted-Jensen. Discrete series for semisimple symmetric spaces.Ann. of Math., 111(2):253–311, 1980

1980
[30]

Gangolli and V

R. Gangolli and V. S. Varadarajan.Harmonic Analysis of Spherical Functions on Real Reductive Groups. Springer, 1988

1988
[31]

I. M. Gelfand, M. I. Graev, and N. Y. Vilenkin.Generalized Functions, Vol. 5: Integral Geometry and Representation Theory. Academic Press, New York and London, 1966

1966
[32]

Spherical functions on a semisimple Lie group

Harish-Chandra. Spherical functions on a semisimple Lie group. I,II.Amer. J. Math., 80:241–310 and 553–613, 1958

1958
[33]

Harmonic Analysis on Real Reductive Groups i

Harish-Chandra. Harmonic Analysis on Real Reductive Groups i. The Theory of the Constant Term.J. Funct. Anal., 19(2):104–204, 1975

1975
[34]

Harmonic Analysis on Real Reductive Groups ii

Harish-Chandra. Harmonic Analysis on Real Reductive Groups ii. Wave Packets in the Schwartz Space.Invent. Math., 36:1–55, 1976

1976
[35]

Harmonic Analysis on Real Reductive Groups iii

Harish-Chandra. Harmonic Analysis on Real Reductive Groups iii. The Maass–Selberg Relations and the Plancherel Formula.Ann. of Math., 104(1):117–201, 1976

1976
[36]

Heckman and H

G. Heckman and H. Schlichtkrull.Harmonic analysis and special functions on symmetric spaces, volume 16 ofPer- spectives in Mathematics. Academic Press, Inc., San Diego, CA, 1994

1994
[37]

Helgason.Differential Geometry, Lie Groups, and Symmetric Spaces

S. Helgason.Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, 1978

1978
[38]

Helgason.Groups and Geometric Analysis

S. Helgason.Groups and Geometric Analysis. Academic Press, 1984

1984
[39]

Koornwinder

T. Koornwinder. A new proof of a Paley-Wiener type theorem for the Jacobi transform.Ark. Mat., 13:145–159, 1975

1975
[40]

M. T. Kosters.Spherical distributions on rank one symmetric spaces. PhD thesis, University of Leiden, 1983

1983
[41]

W. A. Kosters.Harmonic analysis on symmetric spaces. PhD thesis, University of Leiden, 1985

1985
[42]

Limim, J

N. Limim, J. Niederle, and R. Raczka. Eigenfunction expansions associated with the second-order invariant operator on hyperboloids and cones, iii.J. Math. Phys., 8, 1967

1967
[43]

Matsumoto

S. Matsumoto. The Plancherel formula for a pseudo-Riemannian symmetric space.Hiroshima Math. J., 8, 1978

1978
[44]

V. F. Molcanov. The Plancherel formula for hyperboloids.Proc. Steklov Inst. Math., 2, 1981

1981
[45]

V. F. Molcanov. The Plancherel formula for the pseudo-Riemannian spaceSLp3,Rq{GLp2,Rq.Sib. Math. J., 23, 1982

1982
[46]

V. F. Molcanov. Harmonic analysis on the pseudo-Riemannian symmetric spaces of the groupSLp2,Rq.Math. USSR Sbornik, 46, 1983

1983
[47]

V. F. Molchanov. The Plancherel formula for pseudo-Riemannian symmetric spaces of rank 1.Dokl. Akad. Nauk SSSR, 290(3):545–549, 1986. English translation in Soviet Mathematics Doklady, 34 (1987), 323–326

1986
[48]

´Olafsson

G. ´Olafsson. Fourier and Poisson transformation associated to a semisimple symmetric space.Invent. Math., 90:605– 629, 1987

1987
[49]

Oshima and T

T. Oshima and T. Matsuki. A description of discrete series for semisimple symmetric spaces. InAdvances in Studies in Pure Mathematics, volume 4, pages 331–390. 1984

1984
[50]

Oshima and J

T. Oshima and J. Sekiguchi. Eigenspaces of invariant differential operators on an affine symmetric space.Invent. Math., 57(1):1–81, 1980

1980
[51]

Pusti and I

S. Pusti and I. Sitiraju. Boundedness of Eisenstein integrals on split rank one semisimple symmetric spaces, 2025

2025
[52]

Rossmann

W. Rossmann. Analysis on real hyperbolic spaces.J. Funct. Anal., 30, 1978

1978
[53]

Rossmann

W. Rossmann. The structure of semisimple symmetric spaces.Can. J. Math., 31(1):157–180, 1979

1979
[54]

Schlichtkrull.Hyperfunctions and harmonic analysis on symmetric spaces, volume 49 ofProgress in Mathematics

H. Schlichtkrull.Hyperfunctions and harmonic analysis on symmetric spaces, volume 49 ofProgress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 1984

1984
[55]

Sekiguchi

J. Sekiguchi. Split Rank One Semisimple Symmetric Spaces and c-Functions. InWorkshop on Algebraic Groups and Related Topics, volume 737 ofRIMS Kˆ okyˆ uroku, pages 30–41. Research Institute for Mathematical Sciences, Kyoto University, 1990

1990
[56]

Shintani

T. Shintani. On the decomposition of regular representation of the Lorentz group on a hyperboloid of one sheet.Proc. Japan Acad., 43:1–5, 1967

1967
[57]

R. S. Strichartz. Harmonic analysis on hyperboloids.J. Funct. Anal., 12, 1973

1973
[58]

P. C. Trombi and V. S. Varadarajan. Spherical transform on semisimple Lie groups.Ann. of Math., 94(2):246–303, 1971. Lr- SCHW ARTZ SPACES ON SPLIT RANK ONE SEMISIMPLE SYMMETRIC SPACES 21 Sanjoy Pusti Department of Mathematics, INDIAN INSTITUTE OF TECHNOLOGY BOMBAY, Powai, Mumbai-400076, India. Email address:sanjoy@math.iitb.ac.in Iswarya Sitiraju, Departm...

1971