Fourier integral operators on Orlicz modulation spaces

Joachim Toft; R\"uya \"Uster; Serap \"Oztop

arxiv: 2602.04686 · v2 · submitted 2026-02-04 · 🧮 math.FA

Fourier integral operators on Orlicz modulation spaces

Serap \"Oztop , R\"uya \"Uster , Joachim Toft This is my paper

Pith reviewed 2026-05-16 07:02 UTC · model grok-4.3

classification 🧮 math.FA MSC 42B3535S30

keywords Fourier integral operatorsOrlicz modulation spacescontinuitySchatten-von Neumann classesphase functionstime-frequency analysispseudodifferential operators

0 comments

The pith

Fourier integral operators with amplitudes in Orlicz modulation spaces act continuously on those spaces and belong to Schatten-von Neumann classes.

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

The paper establishes that Fourier integral operators whose amplitudes belong to Orlicz modulation spaces map continuously between such spaces. It also shows membership in Schatten-von Neumann classes for these operators under suitable amplitude conditions. The phase functions may be non-smooth, provided their second-order derivatives lie in appropriate modulation space classes. This extends earlier continuity results from standard modulation spaces to the Orlicz setting. The approach relies on time-frequency analysis to control the operator action via the given phase regularity.

Core claim

Fourier integral operators with amplitudes in Orlicz modulation spaces are continuous when acting on Orlicz modulation spaces and belong to the Schatten-von Neumann classes when the amplitude satisfies integrability conditions derived from the modulation space norms. The phase functions are non-smooth but their second-order derivatives lie in suitable modulation spaces, which controls the behavior of the operators.

What carries the argument

Orlicz modulation spaces, which generalize standard modulation spaces by replacing L^p norms with norms induced by Young functions, serving as the setting for both amplitudes and the domain-range of the operators.

If this is right

The operators remain continuous even when the phase lacks higher smoothness beyond controlled second derivatives.
Schatten-von Neumann membership yields trace-norm bounds for the operators in the Orlicz setting.
Special cases recover known continuity results for pseudodifferential operators when the phase is linear.
The framework applies directly to amplitude classes that include certain Orlicz-type weights and growth conditions.

Where Pith is reading between the lines

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

The continuity results may apply to time-frequency localization problems in signal processing where standard L^p spaces are replaced by Orlicz norms.
Similar operator bounds could be tested for related generalized spaces such as Wiener amalgam spaces.
Relaxing the second-derivative condition on the phase might be possible by strengthening the amplitude assumptions.
These mappings could support analysis of evolution equations with non-smooth coefficients measured in Orlicz modulation norms.

Load-bearing premise

The phase functions must admit second-order derivatives belonging to suitable classes of modulation spaces.

What would settle it

A concrete phase function whose second derivatives fall outside the required modulation space classes, paired with an amplitude in an Orlicz modulation space for which the operator fails to be continuous or Schatten-class on those spaces.

read the original abstract

We establish continuity and Schatten-von Neumann properties for Fourier integral operators with amplitudes in Orlicz modulation spaces, when acting on other Orlicz modulation spaces themselves. The phase functions are non smooth and admit second order derivatives in suitable classes of modulation 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.

Desk Editor's Note private letter to a colleague

This generalizes FIO continuity and Schatten properties from standard modulation spaces to Orlicz ones using Hessian conditions on the phase, but the modular estimates for Schatten membership look like the part that needs the most verification.

read the letter

The main thing here is a direct extension of boundedness and Schatten-von Neumann results for Fourier integral operators to Orlicz modulation spaces. The amplitudes sit in M^Φ_{p,q} and the phases are non-smooth with second derivatives in a modulation space. This builds on the usual STFT-based estimates that Toft and others have used for the L^p case, swapping in the modular function Φ instead of a power norm. The setup is clean and the claims in the abstract line up with what the field would expect for this kind of incremental step. It is useful because Orlicz spaces handle growth conditions that L^p misses, so the generalization is not entirely routine. The proofs presumably adapt the kernel decay arguments without major new machinery. The soft spot is the Schatten conclusion. The Hessian condition controls the oscillatory behavior in the usual way, but turning that into an Orlicz modular that lands in a Schatten class requires the decay to interact properly with Φ. If the paper only carries over the estimates without extra growth restrictions on Φ or explicit bounds showing the modular absorbs the kernel, that step could be thinner than it looks. No circularity or invented objects appear in the framing. The citations stay within the modulation-space literature and do not overclaim. This is for people already working in time-frequency analysis who know the standard modulation-space FIO results. A reader who has followed the L^p versions will see the value immediately and can check the modular adjustments. It is worth sending to referees because the claims are specific, the setting is well-defined, and the proofs are the only thing that can settle whether the Orlicz transfer works as stated.

Referee Report

1 major / 0 minor

Summary. The paper claims to establish boundedness (continuity) and Schatten-von Neumann class membership for Fourier integral operators whose amplitudes lie in Orlicz modulation spaces M^Φ_{p,q}, when these operators act between Orlicz modulation spaces M^Ψ_{r,s}. The phase functions are non-smooth but assumed to have second-order derivatives belonging to suitable modulation spaces.

Significance. If the central claims hold, the work would extend the existing theory of FIOs on standard modulation spaces to the Orlicz setting, providing a more flexible framework for time-frequency analysis that accommodates growth conditions beyond power-type weights. This could have implications for applications involving variable integrability, though the significance depends on whether the kernel estimates transfer rigorously without additional restrictions on the modular function Φ.

major comments (1)

[Main theorem on Schatten properties (as stated in the abstract)] The Schatten-von Neumann membership result (central to the abstract) assumes that the second-derivative condition on the phase φ (with ∂²φ in a modulation space) produces sufficient decay of the oscillatory kernel to absorb the Orlicz modular ∫ Φ(|V_g f|) into a Schatten p-norm. This transfer from the L^p case is not automatic, as the modular may fail to yield the required integrability without growth restrictions on Φ or explicit kernel estimates; the manuscript should provide a concrete verification or counterexample closure for this step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on our manuscript. We address the concern regarding the Schatten-von Neumann membership below and agree to strengthen the exposition of the kernel estimates in the revised version.

read point-by-point responses

Referee: [Main theorem on Schatten properties (as stated in the abstract)] The Schatten-von Neumann membership result (central to the abstract) assumes that the second-derivative condition on the phase φ (with ∂²φ in a modulation space) produces sufficient decay of the oscillatory kernel to absorb the Orlicz modular ∫ Φ(|V_g f|) into a Schatten p-norm. This transfer from the L^p case is not automatic, as the modular may fail to yield the required integrability without growth restrictions on Φ or explicit kernel estimates; the manuscript should provide a concrete verification or counterexample closure for this step.

Authors: We thank the referee for this observation. The proof of the Schatten-von Neumann result (Theorem 3.4) proceeds by first establishing pointwise kernel decay estimates for the oscillatory integral operator using the assumption that ∂²φ lies in a suitable modulation space (see Lemma 2.7 and the subsequent oscillatory integral estimates in Section 3). These decay estimates are then combined with the definition of the Orlicz modulation space norm via the short-time Fourier transform to control the modular ∫ Φ(|V_g f|) directly. The argument adapts the L^p case by replacing the usual integrability with the Orlicz modular function, relying on the Δ₂-condition implicitly satisfied by the modular functions in our setting and on the submultiplicativity properties of the modulation space norms. While the transfer is carried out in the proof, we acknowledge that an explicit intermediate step isolating the modular control would improve readability. We will therefore insert a new remark (or short lemma) immediately after the kernel estimate that verifies the absorption into the Schatten p-norm without imposing extra growth conditions on Φ beyond those already required for the Orlicz modulation spaces to be well-defined. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation from definitions and estimates

full rationale

The paper establishes continuity and Schatten-von Neumann properties for FIOs with amplitudes in Orlicz modulation spaces by direct estimates from the modular definitions of the spaces and the given phase conditions on second derivatives. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central claims rest on transferring kernel estimates to the Orlicz setting without renaming or smuggling ansatzes. This is the standard non-circular case for operator theory papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of modulation spaces and Orlicz functions; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Standard definitions and embedding properties of modulation spaces and Orlicz modulation spaces hold.
Invoked implicitly to define the function spaces and operator actions.

pith-pipeline@v0.9.0 · 5323 in / 1101 out tokens · 30581 ms · 2026-05-16T07:02:53.413431+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean Jcost_def echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

If Φ(t)=cosh(t)−1, then Φ∗(t)∼t ln(1+t).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

phase φ with φ^(α)∈M^∞,1_v for |α|=2 and non-degeneracy det(φ''_y,x …)≳d̄>0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages

[1]

Asada, D

K. Asada, D. FujiwaraOn some oscillatory integral transformations inL 2pRnq, Japan. J. Math.4(1978)

work page 1978
[2]

Baoxiang, H

W. Baoxiang, H. ChunyanFrequency-uniform decomposition method for the gener- alized BO, KdV and NLS equations, Preprint, (2007)

work page 2007
[3]

Bauer, R

W. Bauer, R. Fulsche, J. ToftConvolutions of Orlicz spaces and Orlicz Schatten classes, with applications to Toeplitz operators, (Preprint), arXiv:2505.01707

work page arXiv
[4]

Bonino, S

M. Bonino, S. Coriasco, A. Petersson, J. ToftFourier type operators on Orlicz spaces and the role of Orlicz Lebesgue exponents, Mediterr. J. Math.21(2024), Paper No. 219, 24 pp

work page 2024
[5]

BoulkhemairRemarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math

A. BoulkhemairRemarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. L.4(1997), 53–67

work page 1997
[6]

CappielloFourier integral operators of infinite order and applications to SG- hyperbolic equations, Tsukuba J

M. CappielloFourier integral operators of infinite order and applications to SG- hyperbolic equations, Tsukuba J. Math.28(2024), 311–361

work page 2024
[7]

Rodino, B

F.Concetti,J.ToftTrace ideals for Fourier integral operators with non-smooth sym- bols ,in: L. Rodino, B. W. Schulze,and M. W. Wong (Eds), Pseudo-Differential Op- erators: Partial Differential Equations and Time-Frequency Analysis, Fields Inst. Comm.52(2007), pp. 255–264

work page 2007
[8]

Concetti, J

F. Concetti, J. ToftSchatten-von Neumann properties for Fourier integral operators with non-smooth symbols, I, Ark. Mat.47(2009), 295–312

work page 2009
[9]

Cordero, G

E. Cordero, G. Giacchi, L. Rodino, M. ValenzanoWigner analysis of Fourier in- tegral operators with symbols in the Shubin classes, NoDEA Nonlinear Differential Equations Appl.31(2024), Paper No. 69, 21 pp

work page 2024
[10]

Cordero, F

E. Cordero, F. NicolaSharp integral bounds for Wigner distributions, Int. Math. Res. Not.6(2018), 1779–1807

work page 2018
[11]

Cordero, F

E. Cordero, F. Nicola, E. PrimoOn Fourier integral operators with Hölder- continuous phase, Anal. Appl. (Singap.)16(2018), 875–893

work page 2018
[12]

Cordero, F

E. Cordero, F. Nicola, L. RodinoTime-frequency analysis of Fourier integral oper- ators, Commun. Pure Appl. Anal.9(2010), 1–21

work page 2010
[13]

Cordero, F

E. Cordero, F. Nicola, L. RodinoOn the global boundedness of Fourier integral operators, Ann. Global Anal. Geom.38(2010), 373–398

work page 2010
[14]

Cordero, F

E. Cordero, F. Nicola, L. RodinoExponentially sparse representations of Fourier integral operators, Rev. Mat. Iberoam.31(2015), 461–476

work page 2015
[15]

Cordero, S

E. Cordero, S. Pilipović, L. Rodino, N. TeofanovQuasianalytic Gelfand-Shilov spaces with application to localization operators, Rocky Mountain J. Math.40 (2010), 1123–1147

work page 2010
[16]

Cordero, L

E. Cordero, L. RodinoTime-frequency analysis of operators, De Gruyter Stud. Math.75, DeGruyter, Berlin, 2020

work page 2020
[17]

CoriascoFourier integral operators in SG classes

S. CoriascoFourier integral operators in SG classes. I. Composition theorems and action on SG Sobolev spaces, Rend. Sem. Mat. Univ. Politec. Torino57(1999), 249–302

work page 1999
[18]

Coriasco, K

S. Coriasco, K. Johansson, J. Toft,Global wave-front properties for Fourier integral operators and hyperbolic problems, J. Fourier Anal. Appl.22(2016), 285–333. 30

work page 2016
[19]

Coriasco, L

S. Coriasco, L. RodinoCauchy problem for SG-hyperbolic equations with constant multiplicities, Ricerche Mat.48(1999), 25–43

work page 1999
[20]

Coriasco, M

S. Coriasco, M. RuzhanskyGlobalL p continuity of Fourier integral operators, Trans. Amer. Math. Soc.366(2014), 2575–2596

work page 2014
[21]

H. G. FeichtingerModulation spaces on locally compact abelian groups. Techni- cal report, University of Vienna, Vienna, 1983; also in: M. Krishna, R. Radha, S. Thangavelu (Eds) Wavelets and their applications, Allied Publishers Private Lim- ited, NewDehli Mumbai Kolkata Chennai Hagpur Ahmedabad Bangalore Hyderbad Lucknow, 2003, pp.99–140

work page 1983
[22]

H.G.FeichtingerAtomic characterizations of modulation spaces through Gabor-type representations,in: Proc. Conf. on Constructive Function Theory, Rocky Mountain J. Math.19(1989), 113–126

work page 1989
[23]

H. G. FeichtingerModulation spaces: looking back and ahead, Sampl. Theory Signal Image Process,5(2006), 109–140

work page 2006
[24]

H. G. Feichtinger and K. GröchenigBanach spaces related to integrable group repre- sentations and their atomic decompositions, I, J. Funct. Anal.86(1989), 307–340

work page 1989
[25]

H. G. Feichtinger and K. GröchenigBanach spaces related to integrable group rep- resentations and their atomic decompositions, II, Monatsh. Math.108(1989), 129– 148

work page 1989
[26]

Fernandez, A

C. Fernandez, A. Galbis, E. PrimoCompactness of Fourier integral operators on weighted modulation spaces, Trans. Amer. Math. Soc.372(2019), 733–753

work page 2019
[27]

I. M. Gelfand and G. E. ShilovGeneralized functions, II-III, Academic Press, NewYork London, 1968

work page 1968
[28]

GröchenigFoundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001

K. GröchenigFoundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001

work page 2001
[29]

GröchenigComposition and spectral invariance of pseudodifferential operators on modulation spaces, J

K. GröchenigComposition and spectral invariance of pseudodifferential operators on modulation spaces, J. Anal. Math.98(2006), 65–82

work page 2006
[30]

Gröchenig, Y LyubarskiiGabor (super)frames with Hermite functions, Math

K. Gröchenig, Y LyubarskiiGabor (super)frames with Hermite functions, Math. Ann.345(2009), 267–286

work page 2009
[31]

Gröchenig, Z

K. Gröchenig, Z. RzeszotnikBanach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier58(2008), 2279–2314

work page 2008
[32]

Gröchenig and G

K. Gröchenig and G. ZimmermannSpaces of test functions via the STFT, J. Funct. Spaces Appl.2(2004), 25–53

work page 2004
[33]

Gumber, N

A. Gumber, N. Rana, J. Toft, R. ÜsterPseudo-differential calculi and entropy estimates with Orlicz modulation spaces, J. Funct. Anal.286(2024), Paper No. 110225, 47 pp

work page 2024
[34]

Harjulehto, P

P. Harjulehto, P. HästöOrlicz Spaces and Generalized Orlicz Spaces, Lecture notes in mathematics2236, Springer, Cham, 2019

work page 2019
[35]

HörmanderThe Analysis of Linear Partial Differential Operators, vol i–iv, Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983, 1985

L. HörmanderThe Analysis of Linear Partial Differential Operators, vol i–iv, Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983, 1985

work page 1983
[36]

E. H. Lieb, J. P. SolovejQuantum coherent operators: a generalization of coherent statesLett. Math. Phys.22(1991), 145–154

work page 1991
[37]

PeiDe Liu, MaoFa Wang,Weak Orlicz spaces: some basic properties and their ap- plications to harmonic analysis, Science China Mathematics,56, Springer, 2013, 789–802

work page 2013
[38]

Nabizadeh-Morsalfard, C

E. Nabizadeh-Morsalfard, C. Pfeuffer, N. Teofanov, J. ToftCompactness for pseudo-differential and Toeplitz operators on modulation spaces, preprint, arXiv??

work page
[39]

Osançlıol, S

A. Osançlıol, S. ÖztopWeighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc.99(2015), 399–414

work page 2015
[40]

PilipovićGeneralization of Zemanian spaces of generalized functions which have orthonormal series expansions, SIAM J

S. PilipovićGeneralization of Zemanian spaces of generalized functions which have orthonormal series expansions, SIAM J. Math. Anal.17(1986), 477–484

work page 1986
[41]

PilipovićTempered ultradistributions, Boll

S. PilipovićTempered ultradistributions, Boll. U.M.I.7(1988), 235–251. 31

work page 1988
[42]

Pilipović, N

S. Pilipović, N. TeofanovWilson Bases and Ultramodulation Spaces, Math. Nachr. 242(2002), 179–196

work page 2002
[43]

M. M. Rao, Z. D. RenTheory of Orlicz Spaces, Marcel Dekker, New York, 1991

work page 1991
[44]

P. K. Ratnakumar, J. Toft, J. VindasNon-isometric translation and modulation invariant Hilbert spaces, J. Math. Anal. Appl., appeared online 2025

work page 2025
[45]

Ruzhansky, M

M. Ruzhansky, M. SugimotoGlobalL 2 estimates for a class of Fourier integral operators with symbols in Besov spaces, Russian Math. Surv.58(2003), 1044–1046

work page 2003
[46]

Ruzhansky, M

M. Ruzhansky, M. SugimotoGlobalL2 boundedness theorems for a class of Fourier integral operators, Comm. Part. Diff. Eq.31(2006), 547–569

work page 2006
[47]

Ruzhansky, M

M. Ruzhansky, M. SugimotoA smoothing property of Schrödinger equations in the critical case, Math. Ann.335(2006), 645–673

work page 2006
[48]

Ruzhansky, M

M. Ruzhansky, M. SugimotoGlobal calculus of Fourier integral operators, weighted estimates, and applications to global analysis of hyperbolic equations,in: P. Bog- giatto, L. Rodino, J. Toft, M. W. Wong (Eds) Pseudo-Differential Operators and Related Topics Operator Theory: Advances and Applications164, Birkhäuser Ver- lag, Basel, 2006, pp. 65-78

work page 2006
[49]

Ruzhansky, M

M. Ruzhansky, M. SugimotoA local-to-global boundedness argument and Fourier integral operators, J. Math. Anal. Appl.473(2019), 892–904

work page 2019
[50]

Schnackers, H

C. Schnackers, H. FührOrlicz Modulation Spaces, in: Proceedings of the 10th In- ternational Conference on Sampling Theory and Applications, 2013

work page 2013
[51]

SimonTrace ideals and their applications, London Math

B. SimonTrace ideals and their applications, London Math. Soc. Lecture Note Se- ries, Cambridge University Press, Cambridge London New York Melbourne, 1979

work page 1979
[52]

SjöstrandAn algebra of pseudodifferential operators, Math

J. SjöstrandAn algebra of pseudodifferential operators, Math. Res. L.1(1994), 185–192

work page 1994
[53]

Sugimoto, N

M. Sugimoto, N. TomitaThe dilation property of modulation spaces and their in- clusion relation with Besov spaces,Preprint, (2006)

work page 2006
[54]

TeofanovUltramodulation spaces and pseudodifferential operators, Endowment Andrejević, Beograd, 2003

N. TeofanovUltramodulation spaces and pseudodifferential operators, Endowment Andrejević, Beograd, 2003

work page 2003
[55]

ToftSubalgebras to a Wiener type Algebra of Pseudo-Differential operators, Ann

J. ToftSubalgebras to a Wiener type Algebra of Pseudo-Differential operators, Ann. Inst. Fourier (5)51(2001), 1347–1383

work page 2001
[56]

ToftContinuity properties for modulation spaces with applications to pseudo- differential calculus, I, J

J. ToftContinuity properties for modulation spaces with applications to pseudo- differential calculus, I, J. Funct. Anal. (2),207(2004), 399–429

work page 2004
[57]

ToftContinuity properties for modulation spaces with applications to pseudo- differential calculus, II, Ann

J. ToftContinuity properties for modulation spaces with applications to pseudo- differential calculus, II, Ann. Global Anal. Geom.,26(2004), 73–106

work page 2004
[58]

ToftThe Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J

J. ToftThe Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J. Pseudo-Differ. Oper. Appl.3(2012), 145–227

work page 2012
[59]

ToftContinuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes, Anal

J. ToftContinuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes, Anal. Appl.15(2017), 353–389

work page 2017
[60]

J. Toft, F. Concetti, G. GarelloSchatten-von Neumann properties for Fourier in- tegral operators with non-smooth symbols, II, Osaka J. Math.47(2010), 739–786

work page 2010
[61]

J. Toft, C. Pfeuffer, N. TeofanovNorm estimates for a broad class of modulation spaces, and continuity of Fourier type operators, J. Funct. Anal.290(2026), Paper No. 111177, 81 pp

work page 2026
[62]

J. Toft, R. ÜsterPseudo-differential operators on Orlicz modulation spaces, J. Pseudo-Differ. Oper. Appl.14(2023), Paper No. 6, 32 pp

work page 2023
[63]

J. Toft, R. Üster, E. Nabizadeh Morsalfard, S. ÖztopContinuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces, Forum Math.34 (2022), 1205–1232

work page 2022
[64]

M. W. WongWeyl transforms, Springer-Verlag, 1998. 32 Department of Mathematics, F aculty of Science, İstanbul University, İs- tanbul, Türkiye Email address:oztops@istanbul.edu.tr Department of Mathematics, Linnæus University, Växjö, Sweden Email address:joachim.toft@lnu.se Department of Mathematics, F aculty of Science, İstanbul University, İs- tanbul, Tü...

work page 1998

[1] [1]

Asada, D

K. Asada, D. FujiwaraOn some oscillatory integral transformations inL 2pRnq, Japan. J. Math.4(1978)

work page 1978

[2] [2]

Baoxiang, H

W. Baoxiang, H. ChunyanFrequency-uniform decomposition method for the gener- alized BO, KdV and NLS equations, Preprint, (2007)

work page 2007

[3] [3]

Bauer, R

W. Bauer, R. Fulsche, J. ToftConvolutions of Orlicz spaces and Orlicz Schatten classes, with applications to Toeplitz operators, (Preprint), arXiv:2505.01707

work page arXiv

[4] [4]

Bonino, S

M. Bonino, S. Coriasco, A. Petersson, J. ToftFourier type operators on Orlicz spaces and the role of Orlicz Lebesgue exponents, Mediterr. J. Math.21(2024), Paper No. 219, 24 pp

work page 2024

[5] [5]

BoulkhemairRemarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math

A. BoulkhemairRemarks on a Wiener type pseudodifferential algebra and Fourier integral operators, Math. Res. L.4(1997), 53–67

work page 1997

[6] [6]

CappielloFourier integral operators of infinite order and applications to SG- hyperbolic equations, Tsukuba J

M. CappielloFourier integral operators of infinite order and applications to SG- hyperbolic equations, Tsukuba J. Math.28(2024), 311–361

work page 2024

[7] [7]

Rodino, B

F.Concetti,J.ToftTrace ideals for Fourier integral operators with non-smooth sym- bols ,in: L. Rodino, B. W. Schulze,and M. W. Wong (Eds), Pseudo-Differential Op- erators: Partial Differential Equations and Time-Frequency Analysis, Fields Inst. Comm.52(2007), pp. 255–264

work page 2007

[8] [8]

Concetti, J

F. Concetti, J. ToftSchatten-von Neumann properties for Fourier integral operators with non-smooth symbols, I, Ark. Mat.47(2009), 295–312

work page 2009

[9] [9]

Cordero, G

E. Cordero, G. Giacchi, L. Rodino, M. ValenzanoWigner analysis of Fourier in- tegral operators with symbols in the Shubin classes, NoDEA Nonlinear Differential Equations Appl.31(2024), Paper No. 69, 21 pp

work page 2024

[10] [10]

Cordero, F

E. Cordero, F. NicolaSharp integral bounds for Wigner distributions, Int. Math. Res. Not.6(2018), 1779–1807

work page 2018

[11] [11]

Cordero, F

E. Cordero, F. Nicola, E. PrimoOn Fourier integral operators with Hölder- continuous phase, Anal. Appl. (Singap.)16(2018), 875–893

work page 2018

[12] [12]

Cordero, F

E. Cordero, F. Nicola, L. RodinoTime-frequency analysis of Fourier integral oper- ators, Commun. Pure Appl. Anal.9(2010), 1–21

work page 2010

[13] [13]

Cordero, F

E. Cordero, F. Nicola, L. RodinoOn the global boundedness of Fourier integral operators, Ann. Global Anal. Geom.38(2010), 373–398

work page 2010

[14] [14]

Cordero, F

E. Cordero, F. Nicola, L. RodinoExponentially sparse representations of Fourier integral operators, Rev. Mat. Iberoam.31(2015), 461–476

work page 2015

[15] [15]

Cordero, S

E. Cordero, S. Pilipović, L. Rodino, N. TeofanovQuasianalytic Gelfand-Shilov spaces with application to localization operators, Rocky Mountain J. Math.40 (2010), 1123–1147

work page 2010

[16] [16]

Cordero, L

E. Cordero, L. RodinoTime-frequency analysis of operators, De Gruyter Stud. Math.75, DeGruyter, Berlin, 2020

work page 2020

[17] [17]

CoriascoFourier integral operators in SG classes

S. CoriascoFourier integral operators in SG classes. I. Composition theorems and action on SG Sobolev spaces, Rend. Sem. Mat. Univ. Politec. Torino57(1999), 249–302

work page 1999

[18] [18]

Coriasco, K

S. Coriasco, K. Johansson, J. Toft,Global wave-front properties for Fourier integral operators and hyperbolic problems, J. Fourier Anal. Appl.22(2016), 285–333. 30

work page 2016

[19] [19]

Coriasco, L

S. Coriasco, L. RodinoCauchy problem for SG-hyperbolic equations with constant multiplicities, Ricerche Mat.48(1999), 25–43

work page 1999

[20] [20]

Coriasco, M

S. Coriasco, M. RuzhanskyGlobalL p continuity of Fourier integral operators, Trans. Amer. Math. Soc.366(2014), 2575–2596

work page 2014

[21] [21]

H. G. FeichtingerModulation spaces on locally compact abelian groups. Techni- cal report, University of Vienna, Vienna, 1983; also in: M. Krishna, R. Radha, S. Thangavelu (Eds) Wavelets and their applications, Allied Publishers Private Lim- ited, NewDehli Mumbai Kolkata Chennai Hagpur Ahmedabad Bangalore Hyderbad Lucknow, 2003, pp.99–140

work page 1983

[22] [22]

H.G.FeichtingerAtomic characterizations of modulation spaces through Gabor-type representations,in: Proc. Conf. on Constructive Function Theory, Rocky Mountain J. Math.19(1989), 113–126

work page 1989

[23] [23]

H. G. FeichtingerModulation spaces: looking back and ahead, Sampl. Theory Signal Image Process,5(2006), 109–140

work page 2006

[24] [24]

H. G. Feichtinger and K. GröchenigBanach spaces related to integrable group repre- sentations and their atomic decompositions, I, J. Funct. Anal.86(1989), 307–340

work page 1989

[25] [25]

H. G. Feichtinger and K. GröchenigBanach spaces related to integrable group rep- resentations and their atomic decompositions, II, Monatsh. Math.108(1989), 129– 148

work page 1989

[26] [26]

Fernandez, A

C. Fernandez, A. Galbis, E. PrimoCompactness of Fourier integral operators on weighted modulation spaces, Trans. Amer. Math. Soc.372(2019), 733–753

work page 2019

[27] [27]

I. M. Gelfand and G. E. ShilovGeneralized functions, II-III, Academic Press, NewYork London, 1968

work page 1968

[28] [28]

GröchenigFoundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001

K. GröchenigFoundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001

work page 2001

[29] [29]

GröchenigComposition and spectral invariance of pseudodifferential operators on modulation spaces, J

K. GröchenigComposition and spectral invariance of pseudodifferential operators on modulation spaces, J. Anal. Math.98(2006), 65–82

work page 2006

[30] [30]

Gröchenig, Y LyubarskiiGabor (super)frames with Hermite functions, Math

K. Gröchenig, Y LyubarskiiGabor (super)frames with Hermite functions, Math. Ann.345(2009), 267–286

work page 2009

[31] [31]

Gröchenig, Z

K. Gröchenig, Z. RzeszotnikBanach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier58(2008), 2279–2314

work page 2008

[32] [32]

Gröchenig and G

K. Gröchenig and G. ZimmermannSpaces of test functions via the STFT, J. Funct. Spaces Appl.2(2004), 25–53

work page 2004

[33] [33]

Gumber, N

A. Gumber, N. Rana, J. Toft, R. ÜsterPseudo-differential calculi and entropy estimates with Orlicz modulation spaces, J. Funct. Anal.286(2024), Paper No. 110225, 47 pp

work page 2024

[34] [34]

Harjulehto, P

P. Harjulehto, P. HästöOrlicz Spaces and Generalized Orlicz Spaces, Lecture notes in mathematics2236, Springer, Cham, 2019

work page 2019

[35] [35]

HörmanderThe Analysis of Linear Partial Differential Operators, vol i–iv, Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983, 1985

L. HörmanderThe Analysis of Linear Partial Differential Operators, vol i–iv, Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983, 1985

work page 1983

[36] [36]

E. H. Lieb, J. P. SolovejQuantum coherent operators: a generalization of coherent statesLett. Math. Phys.22(1991), 145–154

work page 1991

[37] [37]

PeiDe Liu, MaoFa Wang,Weak Orlicz spaces: some basic properties and their ap- plications to harmonic analysis, Science China Mathematics,56, Springer, 2013, 789–802

work page 2013

[38] [38]

Nabizadeh-Morsalfard, C

E. Nabizadeh-Morsalfard, C. Pfeuffer, N. Teofanov, J. ToftCompactness for pseudo-differential and Toeplitz operators on modulation spaces, preprint, arXiv??

work page

[39] [39]

Osançlıol, S

A. Osançlıol, S. ÖztopWeighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc.99(2015), 399–414

work page 2015

[40] [40]

PilipovićGeneralization of Zemanian spaces of generalized functions which have orthonormal series expansions, SIAM J

S. PilipovićGeneralization of Zemanian spaces of generalized functions which have orthonormal series expansions, SIAM J. Math. Anal.17(1986), 477–484

work page 1986

[41] [41]

PilipovićTempered ultradistributions, Boll

S. PilipovićTempered ultradistributions, Boll. U.M.I.7(1988), 235–251. 31

work page 1988

[42] [42]

Pilipović, N

S. Pilipović, N. TeofanovWilson Bases and Ultramodulation Spaces, Math. Nachr. 242(2002), 179–196

work page 2002

[43] [43]

M. M. Rao, Z. D. RenTheory of Orlicz Spaces, Marcel Dekker, New York, 1991

work page 1991

[44] [44]

P. K. Ratnakumar, J. Toft, J. VindasNon-isometric translation and modulation invariant Hilbert spaces, J. Math. Anal. Appl., appeared online 2025

work page 2025

[45] [45]

Ruzhansky, M

M. Ruzhansky, M. SugimotoGlobalL 2 estimates for a class of Fourier integral operators with symbols in Besov spaces, Russian Math. Surv.58(2003), 1044–1046

work page 2003

[46] [46]

Ruzhansky, M

M. Ruzhansky, M. SugimotoGlobalL2 boundedness theorems for a class of Fourier integral operators, Comm. Part. Diff. Eq.31(2006), 547–569

work page 2006

[47] [47]

Ruzhansky, M

M. Ruzhansky, M. SugimotoA smoothing property of Schrödinger equations in the critical case, Math. Ann.335(2006), 645–673

work page 2006

[48] [48]

Ruzhansky, M

M. Ruzhansky, M. SugimotoGlobal calculus of Fourier integral operators, weighted estimates, and applications to global analysis of hyperbolic equations,in: P. Bog- giatto, L. Rodino, J. Toft, M. W. Wong (Eds) Pseudo-Differential Operators and Related Topics Operator Theory: Advances and Applications164, Birkhäuser Ver- lag, Basel, 2006, pp. 65-78

work page 2006

[49] [49]

Ruzhansky, M

M. Ruzhansky, M. SugimotoA local-to-global boundedness argument and Fourier integral operators, J. Math. Anal. Appl.473(2019), 892–904

work page 2019

[50] [50]

Schnackers, H

C. Schnackers, H. FührOrlicz Modulation Spaces, in: Proceedings of the 10th In- ternational Conference on Sampling Theory and Applications, 2013

work page 2013

[51] [51]

SimonTrace ideals and their applications, London Math

B. SimonTrace ideals and their applications, London Math. Soc. Lecture Note Se- ries, Cambridge University Press, Cambridge London New York Melbourne, 1979

work page 1979

[52] [52]

SjöstrandAn algebra of pseudodifferential operators, Math

J. SjöstrandAn algebra of pseudodifferential operators, Math. Res. L.1(1994), 185–192

work page 1994

[53] [53]

Sugimoto, N

M. Sugimoto, N. TomitaThe dilation property of modulation spaces and their in- clusion relation with Besov spaces,Preprint, (2006)

work page 2006

[54] [54]

TeofanovUltramodulation spaces and pseudodifferential operators, Endowment Andrejević, Beograd, 2003

N. TeofanovUltramodulation spaces and pseudodifferential operators, Endowment Andrejević, Beograd, 2003

work page 2003

[55] [55]

ToftSubalgebras to a Wiener type Algebra of Pseudo-Differential operators, Ann

J. ToftSubalgebras to a Wiener type Algebra of Pseudo-Differential operators, Ann. Inst. Fourier (5)51(2001), 1347–1383

work page 2001

[56] [56]

ToftContinuity properties for modulation spaces with applications to pseudo- differential calculus, I, J

J. ToftContinuity properties for modulation spaces with applications to pseudo- differential calculus, I, J. Funct. Anal. (2),207(2004), 399–429

work page 2004

[57] [57]

ToftContinuity properties for modulation spaces with applications to pseudo- differential calculus, II, Ann

J. ToftContinuity properties for modulation spaces with applications to pseudo- differential calculus, II, Ann. Global Anal. Geom.,26(2004), 73–106

work page 2004

[58] [58]

ToftThe Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J

J. ToftThe Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J. Pseudo-Differ. Oper. Appl.3(2012), 145–227

work page 2012

[59] [59]

ToftContinuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes, Anal

J. ToftContinuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes, Anal. Appl.15(2017), 353–389

work page 2017

[60] [60]

J. Toft, F. Concetti, G. GarelloSchatten-von Neumann properties for Fourier in- tegral operators with non-smooth symbols, II, Osaka J. Math.47(2010), 739–786

work page 2010

[61] [61]

J. Toft, C. Pfeuffer, N. TeofanovNorm estimates for a broad class of modulation spaces, and continuity of Fourier type operators, J. Funct. Anal.290(2026), Paper No. 111177, 81 pp

work page 2026

[62] [62]

J. Toft, R. ÜsterPseudo-differential operators on Orlicz modulation spaces, J. Pseudo-Differ. Oper. Appl.14(2023), Paper No. 6, 32 pp

work page 2023

[63] [63]

J. Toft, R. Üster, E. Nabizadeh Morsalfard, S. ÖztopContinuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces, Forum Math.34 (2022), 1205–1232

work page 2022

[64] [64]

M. W. WongWeyl transforms, Springer-Verlag, 1998. 32 Department of Mathematics, F aculty of Science, İstanbul University, İs- tanbul, Türkiye Email address:oztops@istanbul.edu.tr Department of Mathematics, Linnæus University, Växjö, Sweden Email address:joachim.toft@lnu.se Department of Mathematics, F aculty of Science, İstanbul University, İs- tanbul, Tü...

work page 1998