Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators

David Rule; Solange Mukeshimana

arxiv: 2507.08409 · v2 · submitted 2025-07-11 · 🧮 math.CA · math.AP

Some More Sparse Bounds for Rough and Smooth Pseudodifferential Operators

Solange Mukeshimana , David Rule This is my paper

Pith reviewed 2026-05-19 05:11 UTC · model grok-4.3

classification 🧮 math.CA math.AP

keywords sparse boundspseudodifferential operatorsrough operatorsmeasurable symbolsHormander classespointwise estimatessparse forms

0 comments

The pith

Pointwise sparse bounds hold for rough pseudodifferential operators that are merely measurable in their spatial variables.

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

The paper extends a technique from Beltran and Cladek to prove pointwise sparse bounds for pseudodifferential operators whose symbols are only measurable in the space variable rather than smooth. This builds on existing L to L bounds to obtain these sparse estimates without requiring geometric decay in the bounds. It also supplies general sufficient conditions for sparse form bounds and uses them to recover some previously known results for operators with symbols in the class S zero sub one comma delta when delta is less than one.

Core claim

We obtain pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables. We further develop the Beltran-Cladek technique to achieve this and provide an alternative proof of their results for smoother symbols that avoids proving geometrically decaying sparse bounds. Sufficient conditions are given for sparse form bounds to hold, which are then applied to reprove known sparse bounds for symbols in S^0_{1,δ} with δ < 1.

What carries the argument

The Beltran-Cladek technique extended via L^r to L^s bounds to establish pointwise sparse bounds for operators with measurable spatial symbols.

If this is right

Pointwise sparse bounds apply to a broader class of rough operators.
An alternative proof exists for sparse bounds on smooth pseudodifferential operators without geometric decay assumptions.
Sufficient conditions are identified under which sparse form bounds hold in general.
Known sparse bounds for S^0_{1,δ} symbols with δ < 1 are reproved using the new conditions.

Where Pith is reading between the lines

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

This suggests that further improvements in L^r to L^s bounds for rough operators could directly yield stronger sparse estimates.
The method may connect to other problems in harmonic analysis involving limited regularity symbols.
Testable extensions could include applying the technique to different classes of operators like Fourier integral operators.

Load-bearing premise

The extension relies on the prior existence of suitable L^r to L^s bounds for the rough operators.

What would settle it

Finding a specific rough pseudodifferential operator with a merely measurable spatial symbol for which the pointwise sparse bound fails, despite the L^r to L^s bounds holding.

Figures

Figures reproduced from arXiv: 2507.08409 by David Rule, Solange Mukeshimana.

**Figure 1.** Figure 1: The dashed lines (with the heights 0, − 1 2 (1 − ρ) and −n(1 − ρ) given) indicate the level sets of the limiting values of m from (1) plotted in the ( 1 r , 1 s ′ )-plane below which sparse form bounds are proved in [2] with exponents r and s for a pseudodifferential operator Ta. The line segment from ( 1 2 , 1) to (min{ 1 2ρ , 1}, 1) indicates where Theorem 1.6 extends this to equality in (1), but for M♯ … view at source ↗

read the original abstract

Beltran \& Cladek~\cite{BC} use $L^r$ to $L^s$ bounds to prove sparse form bounds for pseudodifferential operators with H\"ormander symbols in $S^m_{\rho,\delta}$ up to, but not including, the sharp end-point in decay $m$. We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables and an alternative proof of their results which avoids proving geometrically decaying sparse bounds. We also provide sufficient conditions for sparse form bounds to hold and use these to reprove know sparse bounds for pseudodifferential operators with symbols in $S^0_{1,\delta}$ for $\delta < 1$.

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 paper extends pointwise sparse bounds to rough pseudodifferential operators with merely measurable spatial symbols and supplies an alternative proof path that skips geometric decay.

read the letter

The core advance is the extension of the Beltran-Cladek technique to symbols that are only measurable in the spatial variable x. They obtain pointwise sparse bounds under this weaker assumption and give an alternative argument for the smoother Hörmander case that avoids proving geometrically decaying sparse bounds. They also state sufficient conditions for sparse form bounds and use them to recover known results for symbols in S^0_{1,δ} with δ < 1. That alternative route and the measurable extension are the concrete new pieces. The work stays inside the existing sparse-domination program for pseudodifferential operators rather than opening a new direction. The sufficient conditions look like a useful bookkeeping device that lets them handle the rough case cleanly. The main soft spot is that the argument still routes through L^r to L^s operator bounds for the rough operators. The abstract indicates these bounds are assumed or derived under the stated conditions, but without seeing the details it is hard to tell how much spatial regularity is quietly needed to close the estimates. If those bounds hold with no extra assumptions, the claim stands; if they require some hidden regularity, the extension is narrower than advertised. The citation pattern is standard and points back to the right prior work. This is a technical note for people already working on sparse bounds or weighted estimates for pseudodifferential operators. A reader who follows the Beltran-Cladek paper will get value from the alternative proof and the measurable extension. It is not a foundational reorganization, but it is a competent incremental step that deserves referee time. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper extends the Beltran-Cladek technique to prove pointwise sparse bounds for rough pseudodifferential operators whose symbols are merely measurable in the spatial variable x. It also supplies an alternative proof of the original BC results that avoids geometrically decaying sparse bounds, states sufficient conditions for sparse form bounds, and uses those conditions to reprove known sparse bounds for symbols in S^0_{1,δ} with δ<1.

Significance. If the central claims hold, the work meaningfully widens the scope of sparse-domination techniques by removing all spatial regularity assumptions on the symbol while retaining pointwise sparse bounds. The alternative proof route and the sufficient-conditions framework are useful technical contributions that may simplify future applications.

major comments (2)

[§3] §3, the derivation of the L^r→L^s bounds for the rough operator T_σ (prior to the sparse-form argument): these bounds are invoked as the key input for the measurable-symbol case, yet the proof appears to retain a weak spatial regularity hypothesis that is removed in the statement of the main theorem; this step is load-bearing for the extension claimed in Theorem 1.1.
[§4.2] §4.2, the alternative proof that avoids geometrically decaying sparse bounds: the argument still routes through the same L^r→L^s estimates whose validity for merely measurable symbols is not independently established, so the claimed simplification does not yet circumvent the central technical gap.

minor comments (2)

[§2] The notation for the sparse form constants in Definition 2.3 is introduced without an explicit dependence on the aperture parameter; a short clarifying sentence would help.
Several citations to the BC paper are given only by number; adding the full reference in the bibliography (if not already present) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the detailed and constructive referee report. We address each major comment below, offering clarifications on the proofs and indicating the revisions we will make to strengthen the exposition.

read point-by-point responses

Referee: [§3] §3, the derivation of the L^r→L^s bounds for the rough operator T_σ (prior to the sparse-form argument): these bounds are invoked as the key input for the measurable-symbol case, yet the proof appears to retain a weak spatial regularity hypothesis that is removed in the statement of the main theorem; this step is load-bearing for the extension claimed in Theorem 1.1.

Authors: We thank the referee for identifying this point. Upon re-examination of the argument in §3, the L^r → L^s bounds for T_σ are derived using only the measurability of the symbol in x together with the standard size and smoothness estimates in the frequency variable; no additional spatial regularity is invoked in the key estimates or in the application of the Beltran-Cladek technique. Any intermediate phrasing that might suggest otherwise is incidental and not used. We will revise §3 to state explicitly that the bounds hold under mere measurability, thereby removing any ambiguity and confirming the load-bearing step for Theorem 1.1. revision: yes
Referee: [§4.2] §4.2, the alternative proof that avoids geometrically decaying sparse bounds: the argument still routes through the same L^r→L^s estimates whose validity for merely measurable symbols is not independently established, so the claimed simplification does not yet circumvent the central technical gap.

Authors: We appreciate this observation. The alternative route in §4.2 deliberately bypasses geometrically decaying sparse bounds by appealing directly to the L^r → L^s estimates established in §3. As clarified in our response to the first comment, those estimates require only measurability in x. The simplification therefore rests on the same foundation, which we maintain is correctly established for the measurable-symbol case. We will insert a cross-reference in §4.2 to the explicit statement of assumptions now added in §3. revision: yes

Circularity Check

0 steps flagged

No significant circularity: extension builds on external prior bounds without reducing claims to self-definition or fitted inputs

full rationale

The paper extends the Beltran-Cladek technique from cited external work to obtain pointwise sparse bounds for rough pseudodifferential operators with merely measurable spatial symbols. It also supplies an alternative proof route that avoids geometrically decaying sparse bounds and reproves known results for S^0_{1,δ} symbols. No derivation step is shown to reduce by construction to its own inputs, fitted parameters, or a self-citation chain; the L^r-L^s bounds are invoked from the prior literature rather than derived tautologically within the present manuscript. The central claims therefore retain independent mathematical content relative to the cited foundation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are stated. The work rests on standard background facts about Hörmander symbol classes and operator boundedness that are assumed from prior literature.

pith-pipeline@v0.9.0 · 5646 in / 983 out tokens · 45630 ms · 2026-05-19T05:11:18.688020+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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

We further develop their technique, obtaining pointwise sparse bounds for rough pseudodifferential operators that are merely measurable in their spatial variables
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

Proposition 1.2 … |T(f)(x)| ≤ C ∑_{Q∈S} ⟨f⟩_{r,Q} χ_Q(x)

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

21 extracted references · 21 canonical work pages

[1]

Estimates for the kernel and continuity properties of pseudo-differential operators

Josefina ´Alvarez and Jorge Hounie. “Estimates for the kernel and continuity properties of pseudo-differential operators”. In: Ark. Mat. 28.1 (1990), pp. 1–22. issn: 0004-2080,1871-

work page 1990
[2]

url: https://doi.org/10.1007/BF02387364

doi: 10.1007/BF02387364. url: https://doi.org/10.1007/BF02387364

work page doi:10.1007/bf02387364
[3]

Sparse bounds for pseudodifferential operators

David Beltran and Laura Cladek. “Sparse bounds for pseudodifferential operators”. In: J. Anal. Math. 140.1 (2020), pp. 89–116. issn: 0021-7670,1565-8538. doi: 10.1007/s11854- 020-0083-x. url: https://doi.org/10.1007/s11854-020-0083-x

work page doi:10.1007/s11854- 2020
[4]

Sharp weighted norm estimates beyond Calder´ on-Zygmund theory

Fr´ ed´ eric Bernicot, Dorothee Frey, and Stefanie Petermichl. “Sharp weighted norm estimates beyond Calder´ on-Zygmund theory”. In:Anal. PDE 9.5 (2016), pp. 1079–1113. issn: 2157- 5045,1948-206X. doi: 10.2140/apde.2016.9.1079 . url: https://doi.org/10.2140/ apde.2016.9.1079

work page doi:10.2140/apde.2016.9.1079 2016
[5]

Sharp function and weighted Lp estimates for a class of pseudodifferential operators

Sagun Chanillo and Alberto Torchinsky. “Sharp function and weighted Lp estimates for a class of pseudodifferential operators”. In: Ark. Mat. 24.1 (1986), pp. 1–25. issn: 0004- 2080,1871-2487. doi: 10.1007/BF02384387. url: https://doi.org/10.1007/BF02384387

work page doi:10.1007/bf02384387 1986
[6]

A pointwise estimate for pseudo-differential operators

Wenyi Chen and Guangqing Wang. “A pointwise estimate for pseudo-differential operators”. In: Bull. Math. Sci. 13.2 (2023), Paper No. 2250001, 13. issn: 1664-3607,1664-3615. doi: 10.1142/S1664360722500011. url: https://doi.org/10.1142/S1664360722500011

work page doi:10.1142/s1664360722500011 2023
[7]

Conde-Alonso

Jos´ e M. Conde-Alonso. Lecture 2: Sparse domination in harmonic analysis . London Mathe- matical Society. 2021. url: https://www.youtube.com/watch?v=EuH_uq2qrEY

work page 2021
[8]

A sparse domination principle for rough singular integrals

Jos´ e M. Conde-Alonso et al. “A sparse domination principle for rough singular integrals”. In: Anal. PDE 10.5 (2017), pp. 1255–1284. issn: 2157-5045,1948-206X. doi: 10.2140/apde. 2017.10.1255. url: https://doi.org/10.2140/apde.2017.10.1255

work page doi:10.2140/apde 2017
[9]

In: Proc

Loukas Grafakos. Classical Fourier analysis. Third. Vol. 249. Graduate Texts in Mathematics. Springer, New York, 2014, pp. xviii+638. isbn: 978-1-4939-1193-6. doi: 10.1007/978- 1- 4939-1194-3. url: https://doi.org/10.1007/978-1-4939-1194-3

work page doi:10.1007/978- 2014
[10]

Equivalence of sparse and Carleson coefficients for general sets

Timo S. H¨ anninen. “Equivalence of sparse and Carleson coefficients for general sets”. In: Ark. Mat. 56.2 (2018), pp. 333–339. issn: 0004-2080,1871-2487. doi: 10.4310/ARKIV.2018. v56.n2.a8. url: https://doi.org/10.4310/ARKIV.2018.v56.n2.a8

work page doi:10.4310/arkiv.2018 2018
[11]

On the L2 continuity of pseudodifferential operators

Jorge Hounie. “On the L2 continuity of pseudodifferential operators”. In: Comm. Partial Differential Equations 11.7 (1986), pp. 765–778. issn: 0360-5302,1532-4133. doi: 10.1080/ 03605308608820444. url: https://doi.org/10.1080/03605308608820444

work page doi:10.1080/03605308608820444 1986
[12]

Quasi-isometric mappings

F. John. “Quasi-isometric mappings”. In: Seminari 1962/63 Anal. Alg. Geom. e Topol., Vol. 2, Ist. Naz. Alta Mat . Ed. Cremonese, Rome, 1965, pp. 462–473

work page 1962
[13]

The sparse T1 theorem

Michael T. Lacey and Dar´ ıo Mena Arias. “The sparse T1 theorem”. In: Houston J. Math. 43.1 (2017), pp. 111–127. issn: 0362-1588

work page 2017
[14]

Weighted norm inequalities for rough singular integral operators

Kangwei Li et al. “Weighted norm inequalities for rough singular integral operators”. In: J. Geom. Anal. 29.3 (2019), pp. 2526–2564. issn: 1050-6926,1559-002X. doi: 10.1007/s12220- 018-0085-4. url: https://doi.org/10.1007/s12220-018-0085-4

work page doi:10.1007/s12220- 2019
[15]

Bilinear sparse domination for oscillatory integral operators

Tobias Mattsson. “Bilinear sparse domination for oscillatory integral operators”. In: Anal. Math. Phys. 14.3 (2024), Paper No. 37, 33. issn: 1664-2368,1664-235X. doi: 10 . 1007 / s13324-024-00895-1 . url: https://doi.org/10.1007/s13324-024-00895-1

work page doi:10.1007/s13324-024-00895-1 2024
[16]

Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

Nicholas Michalowski, David Rule, and Wolfgang Staubach. “Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes”. In: J. Funct. Anal. 258.12 (2010), pp. 4183–4209. issn: 0022-1236. doi: 10.1016/j.jfa.2010.03.013 . url: https: //doi.org/10.1016/j.jfa.2010.03.013

work page doi:10.1016/j.jfa.2010.03.013 2010
[17]

Weighted Lp Boundedness of Pseudodifferential Operators and Applications

Nicholas Michalowski, David J. Rule, and Wolfgang Staubach. “Weighted Lp Boundedness of Pseudodifferential Operators and Applications”. In: Canadian Mathematical Bulletin 55.3 (2012), pp. 555–570. doi: 10.4153/CMB-2011-122-7

work page doi:10.4153/cmb-2011-122-7 2012
[18]

Weighted Sobolev spaces and pseudodifferential operators with smooth symbols

Nicholas Miller. “Weighted Sobolev spaces and pseudodifferential operators with smooth symbols”. In: Trans. Amer. Math. Soc. 269.1 (1982), pp. 91–109. issn: 0002-9947,1088-6850. doi: 10.2307/1998595. url: https://doi.org/10.2307/1998595. 22

work page doi:10.2307/1998595 1982
[19]

Elias M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory in- tegrals. Vol. 43. Princeton Mathematical Series. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993, pp. xiv+695. isbn: 0-691-03216-5

work page 1993
[20]

Bounded mean oscillation with Orlicz norms and duality of Hardy spaces

Jan-Olov Str¨ omberg. “Bounded mean oscillation with Orlicz norms and duality of Hardy spaces”. In: Indiana Univ. Math. J. 28.3 (1979), pp. 511–544. issn: 0022-2518,1943-5258. doi: 10.1512/iumj.1979.28.28037. url: https://doi.org/10.1512/iumj.1979.28.28037

work page doi:10.1512/iumj.1979.28.28037 1979
[21]

Sharp function and weighted Lp estimates for pseudo-differential opera- tors with symbols in general H¨ ormander classes

Guangqing Wang. Sharp function and weighted Lp estimates for pseudo-differential opera- tors with symbols in general H¨ ormander classes. 2022. arXiv: 2206.09825 [math.AP] . url: https://arxiv.org/abs/2206.09825. S. Mukeshimana, College of Science and Technology, University of R wanda, P.O. Box: 3900, Kigali, R wanda E-mail address : sosmukish@gmail.com D...

work page arXiv 2022

[1] [1]

Estimates for the kernel and continuity properties of pseudo-differential operators

Josefina ´Alvarez and Jorge Hounie. “Estimates for the kernel and continuity properties of pseudo-differential operators”. In: Ark. Mat. 28.1 (1990), pp. 1–22. issn: 0004-2080,1871-

work page 1990

[2] [2]

url: https://doi.org/10.1007/BF02387364

doi: 10.1007/BF02387364. url: https://doi.org/10.1007/BF02387364

work page doi:10.1007/bf02387364

[3] [3]

Sparse bounds for pseudodifferential operators

David Beltran and Laura Cladek. “Sparse bounds for pseudodifferential operators”. In: J. Anal. Math. 140.1 (2020), pp. 89–116. issn: 0021-7670,1565-8538. doi: 10.1007/s11854- 020-0083-x. url: https://doi.org/10.1007/s11854-020-0083-x

work page doi:10.1007/s11854- 2020

[4] [4]

Sharp weighted norm estimates beyond Calder´ on-Zygmund theory

Fr´ ed´ eric Bernicot, Dorothee Frey, and Stefanie Petermichl. “Sharp weighted norm estimates beyond Calder´ on-Zygmund theory”. In:Anal. PDE 9.5 (2016), pp. 1079–1113. issn: 2157- 5045,1948-206X. doi: 10.2140/apde.2016.9.1079 . url: https://doi.org/10.2140/ apde.2016.9.1079

work page doi:10.2140/apde.2016.9.1079 2016

[5] [5]

Sharp function and weighted Lp estimates for a class of pseudodifferential operators

Sagun Chanillo and Alberto Torchinsky. “Sharp function and weighted Lp estimates for a class of pseudodifferential operators”. In: Ark. Mat. 24.1 (1986), pp. 1–25. issn: 0004- 2080,1871-2487. doi: 10.1007/BF02384387. url: https://doi.org/10.1007/BF02384387

work page doi:10.1007/bf02384387 1986

[6] [6]

A pointwise estimate for pseudo-differential operators

Wenyi Chen and Guangqing Wang. “A pointwise estimate for pseudo-differential operators”. In: Bull. Math. Sci. 13.2 (2023), Paper No. 2250001, 13. issn: 1664-3607,1664-3615. doi: 10.1142/S1664360722500011. url: https://doi.org/10.1142/S1664360722500011

work page doi:10.1142/s1664360722500011 2023

[7] [7]

Conde-Alonso

Jos´ e M. Conde-Alonso. Lecture 2: Sparse domination in harmonic analysis . London Mathe- matical Society. 2021. url: https://www.youtube.com/watch?v=EuH_uq2qrEY

work page 2021

[8] [8]

A sparse domination principle for rough singular integrals

Jos´ e M. Conde-Alonso et al. “A sparse domination principle for rough singular integrals”. In: Anal. PDE 10.5 (2017), pp. 1255–1284. issn: 2157-5045,1948-206X. doi: 10.2140/apde. 2017.10.1255. url: https://doi.org/10.2140/apde.2017.10.1255

work page doi:10.2140/apde 2017

[9] [9]

In: Proc

Loukas Grafakos. Classical Fourier analysis. Third. Vol. 249. Graduate Texts in Mathematics. Springer, New York, 2014, pp. xviii+638. isbn: 978-1-4939-1193-6. doi: 10.1007/978- 1- 4939-1194-3. url: https://doi.org/10.1007/978-1-4939-1194-3

work page doi:10.1007/978- 2014

[10] [10]

Equivalence of sparse and Carleson coefficients for general sets

Timo S. H¨ anninen. “Equivalence of sparse and Carleson coefficients for general sets”. In: Ark. Mat. 56.2 (2018), pp. 333–339. issn: 0004-2080,1871-2487. doi: 10.4310/ARKIV.2018. v56.n2.a8. url: https://doi.org/10.4310/ARKIV.2018.v56.n2.a8

work page doi:10.4310/arkiv.2018 2018

[11] [11]

On the L2 continuity of pseudodifferential operators

Jorge Hounie. “On the L2 continuity of pseudodifferential operators”. In: Comm. Partial Differential Equations 11.7 (1986), pp. 765–778. issn: 0360-5302,1532-4133. doi: 10.1080/ 03605308608820444. url: https://doi.org/10.1080/03605308608820444

work page doi:10.1080/03605308608820444 1986

[12] [12]

Quasi-isometric mappings

F. John. “Quasi-isometric mappings”. In: Seminari 1962/63 Anal. Alg. Geom. e Topol., Vol. 2, Ist. Naz. Alta Mat . Ed. Cremonese, Rome, 1965, pp. 462–473

work page 1962

[13] [13]

The sparse T1 theorem

Michael T. Lacey and Dar´ ıo Mena Arias. “The sparse T1 theorem”. In: Houston J. Math. 43.1 (2017), pp. 111–127. issn: 0362-1588

work page 2017

[14] [14]

Weighted norm inequalities for rough singular integral operators

Kangwei Li et al. “Weighted norm inequalities for rough singular integral operators”. In: J. Geom. Anal. 29.3 (2019), pp. 2526–2564. issn: 1050-6926,1559-002X. doi: 10.1007/s12220- 018-0085-4. url: https://doi.org/10.1007/s12220-018-0085-4

work page doi:10.1007/s12220- 2019

[15] [15]

Bilinear sparse domination for oscillatory integral operators

Tobias Mattsson. “Bilinear sparse domination for oscillatory integral operators”. In: Anal. Math. Phys. 14.3 (2024), Paper No. 37, 33. issn: 1664-2368,1664-235X. doi: 10 . 1007 / s13324-024-00895-1 . url: https://doi.org/10.1007/s13324-024-00895-1

work page doi:10.1007/s13324-024-00895-1 2024

[16] [16]

Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes

Nicholas Michalowski, David Rule, and Wolfgang Staubach. “Weighted norm inequalities for pseudo-pseudodifferential operators defined by amplitudes”. In: J. Funct. Anal. 258.12 (2010), pp. 4183–4209. issn: 0022-1236. doi: 10.1016/j.jfa.2010.03.013 . url: https: //doi.org/10.1016/j.jfa.2010.03.013

work page doi:10.1016/j.jfa.2010.03.013 2010

[17] [17]

Weighted Lp Boundedness of Pseudodifferential Operators and Applications

Nicholas Michalowski, David J. Rule, and Wolfgang Staubach. “Weighted Lp Boundedness of Pseudodifferential Operators and Applications”. In: Canadian Mathematical Bulletin 55.3 (2012), pp. 555–570. doi: 10.4153/CMB-2011-122-7

work page doi:10.4153/cmb-2011-122-7 2012

[18] [18]

Weighted Sobolev spaces and pseudodifferential operators with smooth symbols

Nicholas Miller. “Weighted Sobolev spaces and pseudodifferential operators with smooth symbols”. In: Trans. Amer. Math. Soc. 269.1 (1982), pp. 91–109. issn: 0002-9947,1088-6850. doi: 10.2307/1998595. url: https://doi.org/10.2307/1998595. 22

work page doi:10.2307/1998595 1982

[19] [19]

Elias M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory in- tegrals. Vol. 43. Princeton Mathematical Series. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993, pp. xiv+695. isbn: 0-691-03216-5

work page 1993

[20] [20]

Bounded mean oscillation with Orlicz norms and duality of Hardy spaces

Jan-Olov Str¨ omberg. “Bounded mean oscillation with Orlicz norms and duality of Hardy spaces”. In: Indiana Univ. Math. J. 28.3 (1979), pp. 511–544. issn: 0022-2518,1943-5258. doi: 10.1512/iumj.1979.28.28037. url: https://doi.org/10.1512/iumj.1979.28.28037

work page doi:10.1512/iumj.1979.28.28037 1979

[21] [21]

Sharp function and weighted Lp estimates for pseudo-differential opera- tors with symbols in general H¨ ormander classes

Guangqing Wang. Sharp function and weighted Lp estimates for pseudo-differential opera- tors with symbols in general H¨ ormander classes. 2022. arXiv: 2206.09825 [math.AP] . url: https://arxiv.org/abs/2206.09825. S. Mukeshimana, College of Science and Technology, University of R wanda, P.O. Box: 3900, Kigali, R wanda E-mail address : sosmukish@gmail.com D...

work page arXiv 2022