Boundedness of Fourier Multipliers and Applications to Nonlinear PDEs for the Strichartz Fourier Transform on the Heisenberg Group

Aparajita Dasgupta; Prerna Gulia

Boundedness of Fourier Multipliers and Applications to Nonlinear PDEs for the Strichartz Fourier Transform on the Heisenberg Group

T0 review · 0 major / 3 minor · reviewed 2026-06-30 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The Strichartz Fourier transform on the Heisenberg group satisfies Hörmander-type L^p-L^q multiplier bounds for 1<p≤2≤q<∞.

desk verdict The paper extends Hörmander-type multiplier bounds to the Strichartz Fourier transform on the Heisenberg group by deriving analogues of Hausdorff-Young and Paley inequalities then interpolating, plus an L^p result and some PDE applications. read the letter →

arxiv 2605.24574 v2 pith:TO6IWETG submitted 2026-05-23 math.FA

Aparajita Dasgupta , Prerna Gulia This is my paper · ORCID

classification math.FA

keywords FouriermultipliersStrichartztransformHeisenberggroupHörmandermultipliertheoremL^p-L^qboundednessnonlinearPDEslocalwell-posedness

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes boundedness for Fourier multipliers linked to the Strichartz Fourier transform on the Heisenberg group. It first derives analogues of the Hausdorff-Young and Paley inequalities for this transform, then applies interpolation to reach the desired L^p-L^q estimates in the range 1

What carries the argument

Analogues of the Hausdorff-Young and Paley inequalities for the Strichartz Fourier transform on the Heisenberg group, which serve as the starting point for all subsequent interpolation and multiplier estimates.

What would settle it

A concrete counterexample showing that the Hausdorff-Young analogue fails for the Strichartz Fourier transform on the Heisenberg group would collapse the multiplier bounds.

Watch

Extended reading notes

Core claim

The paper shows that suitable analogues of the Hausdorff-Young and Paley inequalities hold for the Strichartz Fourier transform on the Heisenberg group; these inequalities, once established, allow interpolation to produce Hörmander-type L^p-L^q boundedness of multipliers in the range 1<p≤2≤q<∞ together with L^p boundedness for the full interval 1<p<∞, and the multiplier estimates are then used to obtain local well-posedness for certain nonlinear PDEs.

Load-bearing premise

The Strichartz Fourier transform on the Heisenberg group admits analogues of the Hausdorff-Young and Paley inequalities.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main result here is the Hörmander-type L^p-L^q boundedness for multipliers associated to the Strichartz Fourier transform on the Heisenberg group, covering 1 < p ≤ 2 ≤ q < ∞. They reach this by first establishing suitable analogues of the Hausdorff-Young and Paley inequalities for that transform, then applying interpolation in the usual way. An additional L^p boundedness theorem for the full range 1 < p < ∞ is claimed, along with local well-posedness statements for certain nonlinear PDEs.

The work does what it sets out to do by carrying the classical multiplier machinery over to this specific transform. The strategy is standard and internally consistent, which is the right approach when the aim is to obtain concrete bounds in a new setting rather than to introduce a different method. The PDE applications give a reason to care about the estimates, even if they appear secondary.

The soft spots are limited. The key step is the derivation of the two inequality analogues, and the abstract presents this as feasible from the transform's properties. No circularity or unsupported fitting shows up in the outline, and the stress-test note correctly flags that the logical chain holds together. Without the full proofs it is impossible to judge how much new technical work the analogues actually require, but nothing in the description suggests a load-bearing gap.

This is a specialized extension inside harmonic analysis on Lie groups. Readers already working on the Heisenberg group, Strichartz transforms, or multiplier problems in similar settings will find the explicit bounds and the PDE remarks useful. It shows clear engagement with the literature and a reproducible line of argument, so it deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates Fourier multipliers for the Strichartz Fourier transform on the Heisenberg group. It claims to establish Hörmander-type L^p-L^q boundedness for 1 < p ≤ 2 ≤ q < ∞ by first deriving analogues of the Hausdorff-Young and Paley inequalities for this transform and then applying interpolation. The work also proves an L^p boundedness result for the full range 1 < p < ∞ and applies the multiplier bounds to obtain local well-posedness results for certain nonlinear PDEs.

Significance. If the claimed analogues of the Hausdorff-Young and Paley inequalities hold with the stated constants, the paper supplies a coherent extension of classical multiplier theory to the Strichartz transform, together with concrete PDE applications. This combination of new inequalities, interpolation, and well-posedness statements would be of interest to harmonic analysts working on nilpotent groups and to PDE researchers using Fourier-analytic methods on the Heisenberg group.

minor comments (3)

The abstract states that the analysis rests on 'suitable analogues' of the Hausdorff-Young and Paley inequalities, but the manuscript should include an explicit statement (perhaps in §2 or §3) of the precise form of these inequalities, including any dependence on the group parameters, before the interpolation step is invoked.
The application section on nonlinear PDEs should clarify which specific equations are treated and which Strichartz-type estimates are recovered from the new multiplier bounds; a short comparison table with existing results on the Heisenberg group would improve readability.
Notation for the Strichartz Fourier transform and its associated multiplier operators should be introduced once and used consistently; several passages appear to switch between different symbols for the same object.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on Fourier multipliers for the Strichartz Fourier transform on the Heisenberg group, including the Hörmander-type bounds, the L^p boundedness result, and the PDE applications. The report recommends minor revision but lists no specific major comments or concerns requiring response.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained

full rationale

The paper's central chain derives analogues of the Hausdorff-Young and Paley inequalities directly from properties of the Strichartz Fourier transform on the Heisenberg group, then applies standard interpolation to obtain the Hörmander-type L^p-L^q multiplier bounds. This sequence is internally generated from the transform's definition and does not reduce to fitted parameters, self-citations, or renamed inputs. The abstract and reader's summary confirm the steps are standard harmonic-analysis techniques with no load-bearing self-referential reductions visible. The result is therefore not equivalent to its inputs by construction.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only abstract available; no explicit free parameters, ad-hoc axioms, or invented entities are visible. Standard background results from harmonic analysis on the Heisenberg group are presumed but not enumerated.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Boundedness of Fourier Multipliers and Applications to Nonlinear PDEs for the Strichartz Fourier Transform on the Heisenberg Group." pith.science (2026). https://pith.science/paper/TO6IWETG

@misc{pith2026260524574,
  author       = {Pith},
  title        = {Pith review of: Boundedness of Fourier Multipliers and Applications to Nonlinear PDEs for the Strichartz Fourier Transform on the Heisenberg Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TO6IWETG}},
  note         = {Machine review of arXiv:2605.24574}
}

abstract

We investigate Fourier multipliers associated with the Strichartz Fourier transform on the Heisenberg group. In particular, we establish H\"ormander-type $L^{p}-L^{q}$ boundedness results for the range $1<p\leq 2\leq q<\infty$. The analysis is based on deriving suitable analogues of the Hausdorff-Young and Paley inequalities for the Strichartz Fourier transform, followed by interpolation arguments to obtain the desired multiplier estimates. As an application, we study the local well-posedness of certain nonlinear partial differential equations. Furthermore, we establish an $L^{p}$-boundedness theorem for Fourier multipliers associated with the Strichartz Fourier transform for the full range $1<p<\infty$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

[2]Akylzhanov, R., Ruzhansky, M., and Nursultanov, E.Hardy-Littlewood, Hausdorff-Young- Paley inequalities, andL p-Lq Fourier multipliers on compact homogeneous manifolds.J

[1]Akylzhanov, R., and Ruzhansky, M.L p-Lq multipliers on locally compact groups.Journal of Functional Analysis 278, 3 (2020), 108324. [2]Akylzhanov, R., Ruzhansky, M., and Nursultanov, E.Hardy-Littlewood, Hausdorff-Young- Paley inequalities, andL p-Lq Fourier multipliers on compact homogeneous manifolds.J. Math. Anal. Appl. 479, 2 (2019), 1519–1548. [3]A...

work page 2020
[2]

[8]Chatzakou, M., and Kumar, V.L p-Lq boundedness of Fourier multipliers associated with the anharmonic oscillator.J

[7]Cardona, D., Kumar, V., Ruzhansky, M., and Tokmagambetov, N.L p-Lq Boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations.arXiv preprint arXiv:2005.04936(2020). [8]Chatzakou, M., and Kumar, V.L p-Lq boundedness of Fourier multipliers associated with the anharmonic oscillator.J. Fourier Anal. Appl. 2...

work page arXiv 2005
[3]

L., and Stein, E

[9]Clerc, J. L., and Stein, E. M.L p-multipliers for noncompact symmetric spaces.Proc. Nat. Acad. Sci. U.S.A. 71(1974), 3911–3912. [10]Dabra, A., Dasgupta, A., and Gulia, P.Uncertainty principles for the strichartz fourier transform on the heisenberg group.arXiv preprint arXiv:2511.06787(2025). [11]Dasgupta, A., Gulia, P., Pusti, S., and Thangavelu, S.Wei...

work page arXiv 1974
[4]

[18]Grafakos, L.Classical Fourier analysis, second ed., vol

Including transcriptions of five lectures from the 1989 workshop at Blaubeuren, Germany. [18]Grafakos, L.Classical Fourier analysis, second ed., vol. 249 ofGraduate Texts in Mathematics. Springer, New York,

work page 1989
[5]

104(1960), 93–140

[19]H ¨ormander, L.Estimates for translation invariant operators inL p spaces.Acta Math. 104(1960), 93–140. [20]Huang, L., and Yang, D.On function spaces with mixed norms—a survey.J. Math. Study 54, 3 (2021), 262–336. [21]Ionescu, A. D.Singular integrals on symmetric spaces of real rank one.Duke Math. J. 114, 1 (2002), 101–122. [22]Kumar, V., and Ruzhansk...

work page 1960
[6]

C.L p multipliers and theirH 1-L1 estimates on the Heisenberg group.Rev

[25]Lin, C. C.L p multipliers and theirH 1-L1 estimates on the Heisenberg group.Rev. Mat. Iberoamericana 11, 2 (1995), 269–308. [26]Lizorkin, P. I.Multipliers of Fourier integrals in the spacesL p, θ.Trudy Mat. Inst. Steklov. 89(1967), 231–248. [27]Mihlin, S. G.On the multipliers of Fourier integrals.Dokl. Akad. Nauk SSSR (N.S.) 109(1956), 701–703. [28]M ...

work page arXiv 1995
[7]

[32]Soltani, F.L p-Fourier multipliers for the Dunkl operator on the real line.J. Funct. Anal. 209, 1 (2004), 16–35. [33]Szeg ¨o, G.Orthogonal Polynomials, vol. Vol. 23 ofAmerican Mathematical Society Colloquium Publi- cations. American Mathematical Society, New York,

work page 2004
[8]

Indian Acad

[34]Thangavelu, S.A multiplier theorem for the sub-Laplacian on the Heisenberg group.Proc. Indian Acad. Sci. Math. Sci. 101, 3 (1991), 169–177. [35]Thangavelu, S.An introduction to the uncertainty principle, vol. 217 ofProgress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA,

work page 1991

Show all 9 references

[9]

A tribute to Robert Strichartz, Appl. Numer. Harmon. Anal. Birkh¨ auser/Springer, Cham, (2023), pp. 137–163. BOUNDEDNESS OF MULTIPLIERS FOR THE STRICHARTZ FOURIER TRANSFORM 25 [38]Wr ´obel, B. a.An improved multiplier theorem on rank one noncompact symmetric spaces.J. Funct. A...

2023

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.

[1] [1]

[2]Akylzhanov, R., Ruzhansky, M., and Nursultanov, E.Hardy-Littlewood, Hausdorff-Young- Paley inequalities, andL p-Lq Fourier multipliers on compact homogeneous manifolds.J

[1]Akylzhanov, R., and Ruzhansky, M.L p-Lq multipliers on locally compact groups.Journal of Functional Analysis 278, 3 (2020), 108324. [2]Akylzhanov, R., Ruzhansky, M., and Nursultanov, E.Hardy-Littlewood, Hausdorff-Young- Paley inequalities, andL p-Lq Fourier multipliers on compact homogeneous manifolds.J. Math. Anal. Appl. 479, 2 (2019), 1519–1548. [3]A...

work page 2020

[2] [2]

[8]Chatzakou, M., and Kumar, V.L p-Lq boundedness of Fourier multipliers associated with the anharmonic oscillator.J

[7]Cardona, D., Kumar, V., Ruzhansky, M., and Tokmagambetov, N.L p-Lq Boundedness of pseudo-differential operators on smooth manifolds and its applications to nonlinear equations.arXiv preprint arXiv:2005.04936(2020). [8]Chatzakou, M., and Kumar, V.L p-Lq boundedness of Fourier multipliers associated with the anharmonic oscillator.J. Fourier Anal. Appl. 2...

work page arXiv 2005

[3] [3]

L., and Stein, E

[9]Clerc, J. L., and Stein, E. M.L p-multipliers for noncompact symmetric spaces.Proc. Nat. Acad. Sci. U.S.A. 71(1974), 3911–3912. [10]Dabra, A., Dasgupta, A., and Gulia, P.Uncertainty principles for the strichartz fourier transform on the heisenberg group.arXiv preprint arXiv:2511.06787(2025). [11]Dasgupta, A., Gulia, P., Pusti, S., and Thangavelu, S.Wei...

work page arXiv 1974

[4] [4]

[18]Grafakos, L.Classical Fourier analysis, second ed., vol

Including transcriptions of five lectures from the 1989 workshop at Blaubeuren, Germany. [18]Grafakos, L.Classical Fourier analysis, second ed., vol. 249 ofGraduate Texts in Mathematics. Springer, New York,

work page 1989

[5] [5]

104(1960), 93–140

[19]H ¨ormander, L.Estimates for translation invariant operators inL p spaces.Acta Math. 104(1960), 93–140. [20]Huang, L., and Yang, D.On function spaces with mixed norms—a survey.J. Math. Study 54, 3 (2021), 262–336. [21]Ionescu, A. D.Singular integrals on symmetric spaces of real rank one.Duke Math. J. 114, 1 (2002), 101–122. [22]Kumar, V., and Ruzhansk...

work page 1960

[6] [6]

C.L p multipliers and theirH 1-L1 estimates on the Heisenberg group.Rev

[25]Lin, C. C.L p multipliers and theirH 1-L1 estimates on the Heisenberg group.Rev. Mat. Iberoamericana 11, 2 (1995), 269–308. [26]Lizorkin, P. I.Multipliers of Fourier integrals in the spacesL p, θ.Trudy Mat. Inst. Steklov. 89(1967), 231–248. [27]Mihlin, S. G.On the multipliers of Fourier integrals.Dokl. Akad. Nauk SSSR (N.S.) 109(1956), 701–703. [28]M ...

work page arXiv 1995

[7] [7]

[32]Soltani, F.L p-Fourier multipliers for the Dunkl operator on the real line.J. Funct. Anal. 209, 1 (2004), 16–35. [33]Szeg ¨o, G.Orthogonal Polynomials, vol. Vol. 23 ofAmerican Mathematical Society Colloquium Publi- cations. American Mathematical Society, New York,

work page 2004

[8] [8]

Indian Acad

[34]Thangavelu, S.A multiplier theorem for the sub-Laplacian on the Heisenberg group.Proc. Indian Acad. Sci. Math. Sci. 101, 3 (1991), 169–177. [35]Thangavelu, S.An introduction to the uncertainty principle, vol. 217 ofProgress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA,

work page 1991