Potential Estimates and Hodge Systems with $L^1$ data on compact manifolds

Daniel Spector; Felipe Hern\'andez; Jesse Goodman

arxiv: 2606.12891 · v1 · pith:MGL6LL36new · submitted 2026-06-11 · 🧮 math.AP · math.FA

Potential Estimates and Hodge Systems with L¹ data on compact manifolds

Jesse Goodman , Felipe Hern\'andez , Daniel Spector This is my paper

Pith reviewed 2026-06-27 06:26 UTC · model grok-4.3

classification 🧮 math.AP math.FA

keywords Riesz potentialsLorentz estimatesHodge systemscompact manifoldsL1 datadifferential formsPoisson equation

0 comments

The pith

Riesz potentials of L1 closed or co-closed k-forms on compact manifolds are bounded in the Lorentz space L^{n/(n-α),1}.

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

The paper proves that on any smooth compact Riemannian manifold of dimension n, the Riesz potential I_{α,k} applied to an L1 k-form F that is closed or co-closed and orthogonal to harmonics satisfies an inequality bounding the L^{n/(n-α),1} norm of the output by a constant times the L1 norm of F. This is the optimal Lorentz-space endpoint for potential estimates in this setting. The same bound then transfers directly to solutions of the associated Poisson equation and to the Hodge system when the data has finite mass. The result includes the three-dimensional torus case of the div-curl system as a special instance.

Core claim

For α ∈ (0,n) and k = 1,…,n−1 there exists C > 0 such that ∥I_{α,k} F∥_{L^{n/(n-α),1}(Λ^k)} ≤ C ∥F∥_{L^1(Λ^k)} whenever F ∈ L^1(Λ^k) is orthogonal to the harmonic k-forms and satisfies either dF = 0 or d^*F = 0. The same operator bound yields Lorentz-space estimates for the k-form Poisson equation and for the Hodge system with L1 data.

What carries the argument

The Riesz potential operator I_{α,k} acting on k-forms that satisfy the closed or co-closed condition and are orthogonal to harmonics.

Load-bearing premise

The manifold admits a Hodge decomposition that lets one project orthogonally away from the harmonics, and the Riesz potential is well-defined and continuous on the indicated spaces of forms.

What would settle it

A concrete L1 closed k-form on a specific compact manifold (for example the 3-torus) for which the L^{n/(n-α),1} norm of its Riesz potential grows faster than any multiple of its L1 norm.

read the original abstract

In this paper we establish optimal Lorentz estimates for the Riesz potentials acting on closed or co-closed $k$-forms of finite mass on a smooth, compact Riemannian manifold of dimension $n$: For $\alpha \in (0,n)$ and $k=1,\ldots,n-1$, there exists a constant $C>0$ such that \begin{align*} \| \mathcal{I}_{\alpha,k} F \|_{L^{n/(n-\alpha),1}(\Lambda^k)} \leq C \| F\|_{L^1(\Lambda^k)} \end{align*} for all $k$-forms $F \in L^1(\Lambda^k)$ orthogonal to the space of harmonic $k$-forms and satisfying $\mathrm{d} F=0$ or $\mathrm{d}^* F=0$. We show how this inequality implies analogous Lorentz bounds for solutions of the $k$-form Poisson equation and for the Hodge system with data having finite mass. These results include as a special case the div--curl system on the $3$-dimensional torus, where we answer an open question originally posed by J. Bourgain and H. Brezis.

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 proves the optimal Lorentz estimates for Riesz potentials on closed or co-closed L1 forms on compact manifolds and settles the Bourgain-Brezis div-curl question as a special case.

read the letter

The paper proves the optimal Lorentz estimates for Riesz potentials on closed or co-closed L1 forms on compact manifolds and settles the Bourgain-Brezis div-curl question as a special case.

It extends the known scalar and vector Riesz potential bounds to the Hodge setting for k-forms. The closed or co-closed condition supplies the cancellation that upgrades the estimate to the Lorentz space L^{n/(n-alpha),1} rather than the weaker weak-L^{n/(n-alpha),infty} bound that holds without it. Orthogonality to harmonics lets them define the potential via the spectral theorem on the Hodge Laplacian. They also record the immediate consequences for the k-form Poisson equation and the full Hodge system with L1 data.

The argument rests on standard Hodge theory plus the usual potential estimates, adapted to the manifold. The projection off harmonics is bounded on L1 because the harmonic forms are smooth, so that step is fine. The stress-test note finds no internal inconsistency or unsecured assumption in the stated claim.

A minor soft spot is that the abstract gives no explicit dependence of the constant on the manifold or on alpha and k, though existence is the main point. The result is stated for smooth compact manifolds, so no extra regularity issues appear.

This is for people working on endpoint estimates in geometric analysis or on the specific Bourgain-Brezis question. A reader who needs the div-curl case or the form version of these bounds will find a direct answer.

I would send it to referees.

Referee Report

0 major / 3 minor

Summary. The manuscript establishes optimal Lorentz estimates for the Riesz potentials I_{\alpha,k} acting on closed or co-closed k-forms F \in L^1(\Lambda^k) on a smooth compact Riemannian manifold of dimension n: for \alpha \in (0,n) and k=1 to n-1, \|I_{\alpha,k} F\|_{L^{n/(n-\alpha),1}} \leq C \|F\|_{L^1} whenever F is orthogonal to harmonic k-forms and satisfies dF=0 or d^*F=0. The authors derive analogous bounds for solutions of the k-form Poisson equation and the Hodge system with L^1 data; as a special case they resolve an open question of Bourgain-Brezis for the div-curl system on the 3-torus.

Significance. If the central estimates hold, the work supplies sharp Lorentz-space potential bounds under the closed/co-closed condition on forms, improving the generic weak-L^{n/(n-\alpha),\infty} bound that holds without the differential constraint. The resolution of the Bourgain-Brezis question on the torus is a concrete advance, and the extension to general compact manifolds via the Hodge Laplacian is of interest for geometric PDE.

minor comments (3)

[Abstract] The statement of the main inequality in the abstract (and presumably in Theorem 1.1) should explicitly record that the constant C depends on the manifold, \alpha, k and n but is independent of F.
[§2] The spectral definition of I_{\alpha,k} via the Hodge Laplacian (presumably in §2) should include a short reminder that orthogonality to harmonics guarantees the operator is well-defined on the L^1 complement.
Figure 1 (if present) comparing the Lorentz and weak-L^p norms would benefit from an explicit caption stating the precise exponents used.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript, including the recognition of the resolution of the Bourgain-Brezis question as a concrete advance. The recommendation for minor revision is noted. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents a new optimal Lorentz estimate for the Riesz potential I_{\alpha,k} acting on L^1 closed or co-closed k-forms orthogonal to harmonics on a compact Riemannian manifold. This bound is derived from the spectral definition of the potential via the Hodge Laplacian together with the cancellation supplied by the closed/co-closed condition, which upgrades the generic weak-L^{n/(n-\alpha),\infty} bound to the stronger L^{n/(n-\alpha),1} space. The Hodge projection is bounded on L^1 because harmonics are smooth, and the orthogonality condition is a standard, externally verifiable hypothesis that makes the operator well-defined; it does not reduce the target inequality to a tautology or to a fitted input. No self-citation chain, ansatz smuggling, or renaming of known results is required for the central claim, which remains independent of the paper's own fitted quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a Hodge decomposition on the compact manifold (standard in differential geometry) and on the well-definedness of the Riesz potential operator on the indicated spaces of forms. No free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption Compact smooth Riemannian manifold admits Hodge decomposition allowing orthogonal projection onto the complement of harmonic forms.
Invoked when restricting to forms orthogonal to harmonics.
domain assumption Riesz potential I_{alpha,k} is well-defined and maps the indicated L1 forms into the target Lorentz space under the closed or co-closed condition.
This is the operator whose boundedness is asserted.

pith-pipeline@v0.9.1-grok · 5739 in / 1432 out tokens · 18258 ms · 2026-06-27T06:26:38.629457+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

73 extracted references · 44 canonical work pages

[1]

D. R. Adams,Morrey Spaces, Lecture Notes in Applied and Numerical Harmonic Analysis, Birkh¨ auser/Springer, Cham, 2015. MR3467116↑

2015
[2]

D. R. Adams and L. I. Hedberg,Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR1411441↑

1996
[3]

Alvino,Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll

A. Alvino,Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital. A (5)14 (1977), no. 1, 148–156.↑

1977
[4]

Acta Math

L. Ambrosio and B. Kirchheim,Currents in metric spaces, Acta Math.185(2000), no. 1, 1–80, DOI 10.1007/BF02392711. MR1794185↑

work page doi:10.1007/bf02392711 2000
[5]

Ayoush, D

R. Ayoush, D. Stolyarov, and M. Wojciechowski,Sobolev martingales, Rev. Mat. Iberoam. 37(2021), no. 4, 1225–1246, DOI 10.4171/rmi/1224. MR4269395↑

work page doi:10.4171/rmi/1224 2021
[6]

Bergh and J

J. Bergh and J. L¨ ofstr¨ om,Interpolation spaces. An introduction, Grundlehren der Mathema- tischen Wissenschaften, vol. No. 223, Springer-Verlag, Berlin-New York, 1976. MR0482275 ↑

1976
[7]

Berline, E

N. Berline, E. Getzler, and M. Vergne,Heat Kernels and Dirac Operators, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1992 (eng).↑

1992
[8]

Bourgain and H

J. Bourgain and H. Brezis,New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris338(2004), no. 7, 539–543.↑

2004
[9]

,New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS)9(2007), no. 2, 277–315.↑ 50 JESSE GOODMAN, FELIPE HERN ´ANDEZ, AND DANIEL SPECTOR

2007
[10]

Bousquet, E

P. Bousquet, E. Russ, Y. Wang, and P.-L. Yung,Approximation in higher-order Sobolev spaces and Hodge systems, J. Funct. Anal.276(2019), no. 5, 1430–1478, DOI 10.1016/j.jfa.2018.08.003. MR3912781↑

work page doi:10.1016/j.jfa.2018.08.003 2019
[11]

Bousquet and J

P. Bousquet and J. Van Schaftingen,Hardy-Sobolev inequalities for vector fields and cancel- ing linear differential operators, Indiana Univ. Math. J.63(2014), no. 5, 1419–1445, DOI 10.1512/iumj.2014.63.5395. MR3283556↑

work page doi:10.1512/iumj.2014.63.5395 2014
[12]

Breit and A

D. Breit and A. Cianchi,Symmetric gradient Sobolev spaces endowed with rearrangement-invariant norms, Adv. Math.391(2021), Paper No. 107954, 101, DOI 10.1016/j.aim.2021.107954. MR4303731↑

work page doi:10.1016/j.aim.2021.107954 2021
[13]

Breit, A

D. Breit, A. Cianchi, and D. Spector,Sobolev inequalities for canceling operators, J. Math. Pures Appl. (9)207(2026), Paper No. 103844, 34, DOI 10.1016/j.matpur.2025.103844 (English, with English and French summaries). MR5013126↑

work page doi:10.1016/j.matpur.2025.103844 2026
[14]

,Riesz potential estimates under co-canceling constraints, preprint, arXiv:2512.06352. ↑

work page arXiv
[15]

Brezis,Laser beams and limiting cases of Sobolev inequality, Nonlinear Partial Differential Equations and Their ApplicationsII(1982), 86–97.↑

H. Brezis,Laser beams and limiting cases of Sobolev inequality, Nonlinear Partial Differential Equations and Their ApplicationsII(1982), 86–97.↑

1982
[16]

Brezis and J

H. Brezis and J. Van Schaftingen,Boundary estimates for elliptic systems withL 1-data, Calc. Var. Partial Differential Equations30(2007), no. 3, 369–388, DOI 10.1007/s00526-007-0094-

work page doi:10.1007/s00526-007-0094- 2007
[17]

Chanillo, J

S. Chanillo, J. Van Schaftingen, and P.-L. Yung,Applications of Bourgain-Br´ ezis inequalities to fluid mechanics and magnetism, C. R. Math. Acad. Sci. Paris354(2016), no. 1, 51–55, DOI 10.1016/j.crma.2015.10.005 (English, with English and French summaries). MR3439724 ↑

work page doi:10.1016/j.crma.2015.10.005 2016
[18]

,Bourgain-Brezis inequalities on symmetric spaces of non-compact type, J. Funct. Anal.273(2017), no. 4, 1504–1547, DOI 10.1016/j.jfa.2017.05.005. MR3661407↑

work page doi:10.1016/j.jfa.2017.05.005 2017
[19]

Y.-W. B. Chen, J. Goodman, F. Hernandez, and D. Spector,An atomic decomposition of one-dimensional metric currents without boundary(Feb. 14, 2025), available athttps:// arxiv.org/abs/2502.09871.↑

work page arXiv 2025
[20]

Phase retrieval from Gabor measurements

T. Coulhon, G. Kerkyacharian, and P. Petrushev,Heat kernel generated frames in the setting of Dirichlet spaces, J. Fourier Anal. Appl.18(2012), no. 5, 995–1066, DOI 10.1007/s00041- 012-9232-7. MR2970038↑

work page doi:10.1007/s00041- 2012
[21]

Danshen,Axisymmetric incompressible flows with bounded vorticity, Uspekhi Mat

R. Danshen,Axisymmetric incompressible flows with bounded vorticity, Uspekhi Mat. Nauk 62(2007), no. 3(375), 73–94, DOI 10.1070/RM2007v062n03ABEH004412 (Russian, with Rus- sian summary); English transl., Russian Math. Surveys62(2007), no. 3, 475–496. MR2355419 ↑

work page doi:10.1070/rm2007v062n03abeh004412 2007
[22]

Federer,Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969

H. Federer,Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR0257325↑

1969
[23]

G. B. Folland,Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13(1975), no. 2, 161–207, DOI 10.1007/BF02386204. MR0494315↑

work page doi:10.1007/bf02386204 1975
[24]

Garofalo and G

N. Garofalo and G. Tralli,Feeling the heat in a group of Heisenberg type, Adv. Math.381 (2021), Paper No. 107635, 42, DOI 10.1016/j.aim.2021.107635. MR4214400↑

work page doi:10.1016/j.aim.2021.107635 2021
[25]

Gmeineder, B

F. Gmeineder, B. Rait ¸˘ a, and J. Van Schaftingen,Boundary ellipticity and limit- ingL 1-estimates on halfspaces, Adv. Math.439(2024), Paper No. 109490, 25, DOI 10.1016/j.aim.2024.109490. MR4690504↑

work page doi:10.1016/j.aim.2024.109490 2024
[26]

Goodman, F

J. Goodman, F. Hernandez, and D. Spector,Two approximation results for divergence free measures, Port. Math.81(2024), no. 3-4, 247–264, DOI 10.4171/pm/2126. MR4781637↑

work page doi:10.4171/pm/2126 2024
[27]

Grafakos,Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol

L. Grafakos,Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014.↑

2014
[28]

Hebey,Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol

E. Hebey,Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR1481970↑

1996
[29]

H´ elein,Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol

F. H´ elein,Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original; with a foreword by James Eells. MR1913803↑

2002
[30]

Hernandez, B

F. Hernandez, B. Rait ¸˘ a, and D. Spector,EndpointL 1 estimates for Hodge systems, Math. Ann.385(2023), no. 3-4, 1923–1946, DOI 10.1007/s00208-022-02383-y. MR4566709↑

work page doi:10.1007/s00208-022-02383-y 2023
[31]

Hernandez and D

F. Hernandez and D. Spector,Fractional integration and optimal estimates for elliptic sys- tems, Calc. Var. Partial Differential Equations63(2024), no. 5, Paper No. 117, 29, DOI 10.1007/s00526-024-02722-8. MR4739434↑

work page doi:10.1007/s00526-024-02722-8 2024
[32]

Hounie and T

J. Hounie and T. Picon,L 1 Sobolev estimates for (pseudo)-differential operators and ap- plications, Math. Nachr.289(2016), no. 14-15, 1838–1854, DOI 10.1002/mana.201500017. MR3563907↑ POTENTIAL ESTIMATES AND HODGE SYSTEMS ON COMPACT MANIFOLDS 51

work page doi:10.1002/mana.201500017 2016
[33]

Hounie and T

J. Hounie and T. Picon,Local Hardy-Littlewood-Sobolev inequalities for canceling elliptic differential operators, J. Math. Anal. Appl.494(2021), no. 1, Paper No. 124598, 24, DOI 10.1016/j.jmaa.2020.124598. MR4153251↑

work page doi:10.1016/j.jmaa.2020.124598 2021
[34]

and Scott, C

T. Iwaniec, C. Scott, and B. Stroffolini,Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pura Appl. (4)177(1999), 37–115, DOI 10.1007/BF02505905. MR1747627↑

work page doi:10.1007/bf02505905 1999
[35]

Keel and T

M. Keel and T. Tao,Endpoint Strichartz estimates, Amer. J. Math.120(1998), no. 5, 955–

1998
[36]

S. G. Krantz, M. M. Peloso, and D. Spector,Some remarks onL 1 embeddings in the subelliptic setting, Nonlinear Anal.202(2021), 112149, DOI 10.1016/j.na.2020.112149. MR4156975↑

work page doi:10.1016/j.na.2020.112149 2021
[37]

Lanzani and E

L. Lanzani and E. M. Stein,A note on div curl inequalities, Math. Res. Lett.12(2005), no. 1, 57–61, DOI 10.4310/MRL.2005.v12.n1.a6. MR2122730↑

work page doi:10.4310/mrl.2005.v12.n1.a6 2005
[38]

J. M. Lee,Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR2954043↑

2013
[39]

V. G. Maz’ya,Classes of domains and imbedding theorems for function spaces, Dokl. Akad. Nauk SSSR133(1960), 527–530 (Russian); English transl., Soviet Math. Dokl.1(1960), 882–885.↑

1960
[40]

Maz’ya,Bourgain–Brezis type inequality with explicit constants, Contemp

V. Maz’ya,Bourgain–Brezis type inequality with explicit constants, Contemp. Math.445 (2007), 247–252.↑

2007
[41]

,Estimates for differential operators of vector analysis involvingL 1-norm, J. Eur. Math. Soc.12(2010), no. 1, 221–240.↑

2010
[42]

´A. D. Mart´ ınez and D. Spector,An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces, Adv. Nonlinear Anal.10(2021), no. 1, 877–894, DOI 10.1515/anona-2020-0157. MR4191703↑

work page doi:10.1515/anona-2020-0157 2021
[43]

C. B. Morrey Jr.,Multiple integrals in the calculus of variations, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1966 edition [MR0202511]. MR2492985↑

2008
[44]

O’Neil,Convolution operators andL(p, q)spaces, Duke Math

R. O’Neil,Convolution operators andL(p, q)spaces, Duke Math. J.30(1963), 129–142. MR146673↑

1963
[45]

Paolini and E

E. Paolini and E. Stepanov,Decomposition of acyclic normal currents in a metric space, J. Funct. Anal.263(2012), no. 11, 3358–3390, DOI 10.1016/j.jfa.2012.08.009. MR2984069↑

work page doi:10.1016/j.jfa.2012.08.009 2012
[46]

,Structure of metric cycles and normal one-dimensional currents, J. Funct. Anal. 264(2013), no. 6, 1269–1295, DOI 10.1016/j.jfa.2012.12.007. MR3017264↑

work page doi:10.1016/j.jfa.2012.12.007 2013
[47]

Peetre,New thoughts on Besov spaces, Duke University Mathematics Series, vol

J. Peetre,New thoughts on Besov spaces, Duke University Mathematics Series, vol. No. 1, Duke University, Mathematics Department, Durham, NC, 1976. MR0461123↑

1976
[48]

Rait ¸˘ a, D

B. Rait ¸˘ a, D. Spector, and D. Stolyarov,A trace inequality for solenoidal charges, Potential Anal.59(2023), no. 4, 2093–2104, DOI 10.1007/s11118-022-10008-x. MR4684387↑

work page doi:10.1007/s11118-022-10008-x 2023
[49]

Raita and D

B. Raita and D. Spector,A note on estimates for elliptic systems withL 1 data, C. R. Math. Acad. Sci. Paris357(2019), no. 11-12, 851–857, DOI 10.1016/j.crma.2019.11.007. MR4038260↑

work page doi:10.1016/j.crma.2019.11.007 2019
[50]

Schikorra, D

A. Schikorra, D. Spector, and J. Van Schaftingen,AnL 1-type estimate for Riesz potentials, Rev. Mat. Iberoam.33(2017), no. 1, 291–303, DOI 10.4171/RMI/937. MR3615452↑

work page doi:10.4171/rmi/937 2017
[51]

Seeger and W

A. Seeger and W. Trebels,Embeddings for spaces of Lorentz-Sobolev type, Math. Ann.373 (2019), no. 3-4, 1017–1056, DOI 10.1007/s00208-018-1730-8. MR3953119↑

work page doi:10.1007/s00208-018-1730-8 2019
[52]

S. K. Smirnov,Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz5(1993), no. 4, 206–238 (Russian, with Russian summary); English transl., St. Petersburg Math. J.5(1994), no. 4, 841–867. MR1246427↑

1993
[53]

Spector,New directions in harmonic analysis onL 1, Nonlinear Anal.192(2020), 111685, 20, DOI 10.1016/j.na.2019.111685

D. Spector,New directions in harmonic analysis onL 1, Nonlinear Anal.192(2020), 111685, 20, DOI 10.1016/j.na.2019.111685. MR4034690↑

work page doi:10.1016/j.na.2019.111685 2020
[54]

,An optimal Sobolev embedding forL 1, J. Funct. Anal.279(2020), no. 3, 108559, DOI 10.1016/j.jfa.2020.108559. MR4093790↑

work page doi:10.1016/j.jfa.2020.108559 2020
[55]

Spector and J

D. Spector and J. Van Schaftingen,Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30(2019), no. 3, 413–436, DOI 10.4171/RLM/854. MR4002205↑

work page doi:10.4171/rlm/854 2019
[56]

Spivak,A comprehensive introduction to differential geometry

M. Spivak,A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, DE, 1979. MR0532830↑

1979
[57]

E. M. Stein,Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies, vol. No. 63, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1970. MR0252961↑

1970
[58]

,Editor’s note: the differentiability of functions inR n, Ann. of Math. (2)113(1981), no. 2, 383–385. MR0607898↑

1981
[59]

E. M. Stein and G. Weiss,Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR0304972 ↑ 52 JESSE GOODMAN, FELIPE HERN ´ANDEZ, AND DANIEL SPECTOR

1971
[60]

D. M. Stolyarov,Weakly canceling operators and singular integrals, Tr. Mat. Inst. Steklova 312(2021), 259–271 (Russian, with Russian summary). English version published in Proc. Steklov Inst. Math.312(2021), no. 1, 249–260.↑

2021
[61]

Sb.213(2022), no

,Hardy-Littlewood-Sobolev inequality forp= 1, Mat. Sb.213(2022), no. 6, 125– 174, DOI 10.4213/sm9645 (Russian, with Russian summary); English transl., Sb. Math.213 (2022), no. 6, 844–889. MR4461456↑

work page doi:10.4213/sm9645 2022
[62]

R. S. Strichartz,Analysis of the Laplacian on the complete Riemannian manifold, J. Func- tional Analysis52(1983), no. 1, 48–79, DOI 10.1016/0022-1236(83)90090-3. MR0705991 ↑

work page doi:10.1016/0022-1236(83)90090-3 1983
[63]

Tartar,Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll

L. Tartar,Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)1(1998), no. 3, 479–500 (English, with Italian summary). MR1662313↑

1998
[64]

Triebel,Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark

H. Triebel,Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark. Mat.24(1986), no. 2, 299–337, DOI 10.1007/BF02384402. MR0884191↑

work page doi:10.1007/bf02384402 1986
[65]

II, Monographs in Mathematics, vol

,Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkh¨ auser Ver- lag, Basel, 1992. MR1163193↑

1992
[66]

Van Schaftingen,A simple proof of an inequality of Bourgain, Brezis and Mironescu, C

J. Van Schaftingen,A simple proof of an inequality of Bourgain, Brezis and Mironescu, C. R. Math. Acad. Sci. Paris338(2004), no. 1, 23–26, DOI 10.1016/j.crma.2003.10.036 (English, with English and French summaries). MR2038078↑

work page doi:10.1016/j.crma.2003.10.036 2004
[67]

,Estimates forL 1-vector fields, C. R. Math. Acad. Sci. Paris339(2004), no. 3, 181–186.↑

2004
[68]

,Estimates forL 1 vector fields under higher-order differential conditions, J. Eur. Math. Soc. (JEMS)10(2008), no. 4, 867–882, DOI 10.4171/JEMS/133. MR2443922↑

work page doi:10.4171/jems/133 2008
[69]

,Limiting fractional and Lorentz space estimates of differential forms, Proc. Amer. Math. Soc.138(2010), no. 1, 235–240, DOI 10.1090/S0002-9939-09-10005-9. MR2550188↑

work page doi:10.1090/s0002-9939-09-10005-9 2010
[70]

,Limiting Sobolev inequalities for vector fields and canceling linear differential op- erators, J. Eur. Math. Soc. (JEMS)15(2013), no. 3, 877–921, DOI 10.4171/JEMS/380. MR3085095↑

work page doi:10.4171/jems/380 2013
[71]

Fixed Point Theory Appl.15(2014), no

,Limiting Bourgain-Brezis estimates for systems of linear differential equations: theme and variations, J. Fixed Point Theory Appl.15(2014), no. 2, 273–297, DOI 10.1007/s11784-014-0177-0. MR3298002↑

work page doi:10.1007/s11784-014-0177-0 2014
[72]

47–56, DOI 10.1007/978-3-031-48579-4 5

,Endpoint Sobolev inequalities for vector fields and cancelling operators, Extended ab- stracts 2021/2022—Methusalem lectures, Trends Math., Birkh¨ auser/Springer, Cham, [2024] ©2024, pp. 47–56, DOI 10.1007/978-3-031-48579-4 5. MR4738462↑

work page doi:10.1007/978-3-031-48579-4 2021
[73]

Yung,Sobolev inequalities for(0, q)forms on CR manifolds of finite type, Math

P.-L. Yung,Sobolev inequalities for(0, q)forms on CR manifolds of finite type, Math. Res. Lett.17(2010), no. 1, 177–196, DOI 10.4310/MRL.2010.v17.n1.a14. MR2592736↑ Department of Statistics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Email address, J. Goodman:jesse.goodman@auckland.ac.nz Department of Mathematics, Penn State Uni...

work page doi:10.4310/mrl.2010.v17.n1.a14 2010

[1] [1]

D. R. Adams,Morrey Spaces, Lecture Notes in Applied and Numerical Harmonic Analysis, Birkh¨ auser/Springer, Cham, 2015. MR3467116↑

2015

[2] [2]

D. R. Adams and L. I. Hedberg,Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR1411441↑

1996

[3] [3]

Alvino,Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll

A. Alvino,Sulla diseguaglianza di Sobolev in spazi di Lorentz, Boll. Un. Mat. Ital. A (5)14 (1977), no. 1, 148–156.↑

1977

[4] [4]

Acta Math

L. Ambrosio and B. Kirchheim,Currents in metric spaces, Acta Math.185(2000), no. 1, 1–80, DOI 10.1007/BF02392711. MR1794185↑

work page doi:10.1007/bf02392711 2000

[5] [5]

Ayoush, D

R. Ayoush, D. Stolyarov, and M. Wojciechowski,Sobolev martingales, Rev. Mat. Iberoam. 37(2021), no. 4, 1225–1246, DOI 10.4171/rmi/1224. MR4269395↑

work page doi:10.4171/rmi/1224 2021

[6] [6]

Bergh and J

J. Bergh and J. L¨ ofstr¨ om,Interpolation spaces. An introduction, Grundlehren der Mathema- tischen Wissenschaften, vol. No. 223, Springer-Verlag, Berlin-New York, 1976. MR0482275 ↑

1976

[7] [7]

Berline, E

N. Berline, E. Getzler, and M. Vergne,Heat Kernels and Dirac Operators, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1992 (eng).↑

1992

[8] [8]

Bourgain and H

J. Bourgain and H. Brezis,New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Math. Acad. Sci. Paris338(2004), no. 7, 539–543.↑

2004

[9] [9]

,New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc. (JEMS)9(2007), no. 2, 277–315.↑ 50 JESSE GOODMAN, FELIPE HERN ´ANDEZ, AND DANIEL SPECTOR

2007

[10] [10]

Bousquet, E

P. Bousquet, E. Russ, Y. Wang, and P.-L. Yung,Approximation in higher-order Sobolev spaces and Hodge systems, J. Funct. Anal.276(2019), no. 5, 1430–1478, DOI 10.1016/j.jfa.2018.08.003. MR3912781↑

work page doi:10.1016/j.jfa.2018.08.003 2019

[11] [11]

Bousquet and J

P. Bousquet and J. Van Schaftingen,Hardy-Sobolev inequalities for vector fields and cancel- ing linear differential operators, Indiana Univ. Math. J.63(2014), no. 5, 1419–1445, DOI 10.1512/iumj.2014.63.5395. MR3283556↑

work page doi:10.1512/iumj.2014.63.5395 2014

[12] [12]

Breit and A

D. Breit and A. Cianchi,Symmetric gradient Sobolev spaces endowed with rearrangement-invariant norms, Adv. Math.391(2021), Paper No. 107954, 101, DOI 10.1016/j.aim.2021.107954. MR4303731↑

work page doi:10.1016/j.aim.2021.107954 2021

[13] [13]

Breit, A

D. Breit, A. Cianchi, and D. Spector,Sobolev inequalities for canceling operators, J. Math. Pures Appl. (9)207(2026), Paper No. 103844, 34, DOI 10.1016/j.matpur.2025.103844 (English, with English and French summaries). MR5013126↑

work page doi:10.1016/j.matpur.2025.103844 2026

[14] [14]

,Riesz potential estimates under co-canceling constraints, preprint, arXiv:2512.06352. ↑

work page arXiv

[15] [15]

Brezis,Laser beams and limiting cases of Sobolev inequality, Nonlinear Partial Differential Equations and Their ApplicationsII(1982), 86–97.↑

H. Brezis,Laser beams and limiting cases of Sobolev inequality, Nonlinear Partial Differential Equations and Their ApplicationsII(1982), 86–97.↑

1982

[16] [16]

Brezis and J

H. Brezis and J. Van Schaftingen,Boundary estimates for elliptic systems withL 1-data, Calc. Var. Partial Differential Equations30(2007), no. 3, 369–388, DOI 10.1007/s00526-007-0094-

work page doi:10.1007/s00526-007-0094- 2007

[17] [17]

Chanillo, J

S. Chanillo, J. Van Schaftingen, and P.-L. Yung,Applications of Bourgain-Br´ ezis inequalities to fluid mechanics and magnetism, C. R. Math. Acad. Sci. Paris354(2016), no. 1, 51–55, DOI 10.1016/j.crma.2015.10.005 (English, with English and French summaries). MR3439724 ↑

work page doi:10.1016/j.crma.2015.10.005 2016

[18] [18]

,Bourgain-Brezis inequalities on symmetric spaces of non-compact type, J. Funct. Anal.273(2017), no. 4, 1504–1547, DOI 10.1016/j.jfa.2017.05.005. MR3661407↑

work page doi:10.1016/j.jfa.2017.05.005 2017

[19] [19]

Y.-W. B. Chen, J. Goodman, F. Hernandez, and D. Spector,An atomic decomposition of one-dimensional metric currents without boundary(Feb. 14, 2025), available athttps:// arxiv.org/abs/2502.09871.↑

work page arXiv 2025

[20] [20]

Phase retrieval from Gabor measurements

T. Coulhon, G. Kerkyacharian, and P. Petrushev,Heat kernel generated frames in the setting of Dirichlet spaces, J. Fourier Anal. Appl.18(2012), no. 5, 995–1066, DOI 10.1007/s00041- 012-9232-7. MR2970038↑

work page doi:10.1007/s00041- 2012

[21] [21]

Danshen,Axisymmetric incompressible flows with bounded vorticity, Uspekhi Mat

R. Danshen,Axisymmetric incompressible flows with bounded vorticity, Uspekhi Mat. Nauk 62(2007), no. 3(375), 73–94, DOI 10.1070/RM2007v062n03ABEH004412 (Russian, with Rus- sian summary); English transl., Russian Math. Surveys62(2007), no. 3, 475–496. MR2355419 ↑

work page doi:10.1070/rm2007v062n03abeh004412 2007

[22] [22]

Federer,Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969

H. Federer,Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR0257325↑

1969

[23] [23]

G. B. Folland,Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13(1975), no. 2, 161–207, DOI 10.1007/BF02386204. MR0494315↑

work page doi:10.1007/bf02386204 1975

[24] [24]

Garofalo and G

N. Garofalo and G. Tralli,Feeling the heat in a group of Heisenberg type, Adv. Math.381 (2021), Paper No. 107635, 42, DOI 10.1016/j.aim.2021.107635. MR4214400↑

work page doi:10.1016/j.aim.2021.107635 2021

[25] [25]

Gmeineder, B

F. Gmeineder, B. Rait ¸˘ a, and J. Van Schaftingen,Boundary ellipticity and limit- ingL 1-estimates on halfspaces, Adv. Math.439(2024), Paper No. 109490, 25, DOI 10.1016/j.aim.2024.109490. MR4690504↑

work page doi:10.1016/j.aim.2024.109490 2024

[26] [26]

Goodman, F

J. Goodman, F. Hernandez, and D. Spector,Two approximation results for divergence free measures, Port. Math.81(2024), no. 3-4, 247–264, DOI 10.4171/pm/2126. MR4781637↑

work page doi:10.4171/pm/2126 2024

[27] [27]

Grafakos,Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol

L. Grafakos,Classical Fourier analysis, 3rd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2014.↑

2014

[28] [28]

Hebey,Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol

E. Hebey,Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR1481970↑

1996

[29] [29]

H´ elein,Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol

F. H´ elein,Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original; with a foreword by James Eells. MR1913803↑

2002

[30] [30]

Hernandez, B

F. Hernandez, B. Rait ¸˘ a, and D. Spector,EndpointL 1 estimates for Hodge systems, Math. Ann.385(2023), no. 3-4, 1923–1946, DOI 10.1007/s00208-022-02383-y. MR4566709↑

work page doi:10.1007/s00208-022-02383-y 2023

[31] [31]

Hernandez and D

F. Hernandez and D. Spector,Fractional integration and optimal estimates for elliptic sys- tems, Calc. Var. Partial Differential Equations63(2024), no. 5, Paper No. 117, 29, DOI 10.1007/s00526-024-02722-8. MR4739434↑

work page doi:10.1007/s00526-024-02722-8 2024

[32] [32]

Hounie and T

J. Hounie and T. Picon,L 1 Sobolev estimates for (pseudo)-differential operators and ap- plications, Math. Nachr.289(2016), no. 14-15, 1838–1854, DOI 10.1002/mana.201500017. MR3563907↑ POTENTIAL ESTIMATES AND HODGE SYSTEMS ON COMPACT MANIFOLDS 51

work page doi:10.1002/mana.201500017 2016

[33] [33]

Hounie and T

J. Hounie and T. Picon,Local Hardy-Littlewood-Sobolev inequalities for canceling elliptic differential operators, J. Math. Anal. Appl.494(2021), no. 1, Paper No. 124598, 24, DOI 10.1016/j.jmaa.2020.124598. MR4153251↑

work page doi:10.1016/j.jmaa.2020.124598 2021

[34] [34]

and Scott, C

T. Iwaniec, C. Scott, and B. Stroffolini,Nonlinear Hodge theory on manifolds with boundary, Ann. Mat. Pura Appl. (4)177(1999), 37–115, DOI 10.1007/BF02505905. MR1747627↑

work page doi:10.1007/bf02505905 1999

[35] [35]

Keel and T

M. Keel and T. Tao,Endpoint Strichartz estimates, Amer. J. Math.120(1998), no. 5, 955–

1998

[36] [36]

S. G. Krantz, M. M. Peloso, and D. Spector,Some remarks onL 1 embeddings in the subelliptic setting, Nonlinear Anal.202(2021), 112149, DOI 10.1016/j.na.2020.112149. MR4156975↑

work page doi:10.1016/j.na.2020.112149 2021

[37] [37]

Lanzani and E

L. Lanzani and E. M. Stein,A note on div curl inequalities, Math. Res. Lett.12(2005), no. 1, 57–61, DOI 10.4310/MRL.2005.v12.n1.a6. MR2122730↑

work page doi:10.4310/mrl.2005.v12.n1.a6 2005

[38] [38]

J. M. Lee,Introduction to Smooth Manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR2954043↑

2013

[39] [39]

V. G. Maz’ya,Classes of domains and imbedding theorems for function spaces, Dokl. Akad. Nauk SSSR133(1960), 527–530 (Russian); English transl., Soviet Math. Dokl.1(1960), 882–885.↑

1960

[40] [40]

Maz’ya,Bourgain–Brezis type inequality with explicit constants, Contemp

V. Maz’ya,Bourgain–Brezis type inequality with explicit constants, Contemp. Math.445 (2007), 247–252.↑

2007

[41] [41]

,Estimates for differential operators of vector analysis involvingL 1-norm, J. Eur. Math. Soc.12(2010), no. 1, 221–240.↑

2010

[42] [42]

´A. D. Mart´ ınez and D. Spector,An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces, Adv. Nonlinear Anal.10(2021), no. 1, 877–894, DOI 10.1515/anona-2020-0157. MR4191703↑

work page doi:10.1515/anona-2020-0157 2021

[43] [43]

C. B. Morrey Jr.,Multiple integrals in the calculus of variations, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1966 edition [MR0202511]. MR2492985↑

2008

[44] [44]

O’Neil,Convolution operators andL(p, q)spaces, Duke Math

R. O’Neil,Convolution operators andL(p, q)spaces, Duke Math. J.30(1963), 129–142. MR146673↑

1963

[45] [45]

Paolini and E

E. Paolini and E. Stepanov,Decomposition of acyclic normal currents in a metric space, J. Funct. Anal.263(2012), no. 11, 3358–3390, DOI 10.1016/j.jfa.2012.08.009. MR2984069↑

work page doi:10.1016/j.jfa.2012.08.009 2012

[46] [46]

,Structure of metric cycles and normal one-dimensional currents, J. Funct. Anal. 264(2013), no. 6, 1269–1295, DOI 10.1016/j.jfa.2012.12.007. MR3017264↑

work page doi:10.1016/j.jfa.2012.12.007 2013

[47] [47]

Peetre,New thoughts on Besov spaces, Duke University Mathematics Series, vol

J. Peetre,New thoughts on Besov spaces, Duke University Mathematics Series, vol. No. 1, Duke University, Mathematics Department, Durham, NC, 1976. MR0461123↑

1976

[48] [48]

Rait ¸˘ a, D

B. Rait ¸˘ a, D. Spector, and D. Stolyarov,A trace inequality for solenoidal charges, Potential Anal.59(2023), no. 4, 2093–2104, DOI 10.1007/s11118-022-10008-x. MR4684387↑

work page doi:10.1007/s11118-022-10008-x 2023

[49] [49]

Raita and D

B. Raita and D. Spector,A note on estimates for elliptic systems withL 1 data, C. R. Math. Acad. Sci. Paris357(2019), no. 11-12, 851–857, DOI 10.1016/j.crma.2019.11.007. MR4038260↑

work page doi:10.1016/j.crma.2019.11.007 2019

[50] [50]

Schikorra, D

A. Schikorra, D. Spector, and J. Van Schaftingen,AnL 1-type estimate for Riesz potentials, Rev. Mat. Iberoam.33(2017), no. 1, 291–303, DOI 10.4171/RMI/937. MR3615452↑

work page doi:10.4171/rmi/937 2017

[51] [51]

Seeger and W

A. Seeger and W. Trebels,Embeddings for spaces of Lorentz-Sobolev type, Math. Ann.373 (2019), no. 3-4, 1017–1056, DOI 10.1007/s00208-018-1730-8. MR3953119↑

work page doi:10.1007/s00208-018-1730-8 2019

[52] [52]

S. K. Smirnov,Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows, Algebra i Analiz5(1993), no. 4, 206–238 (Russian, with Russian summary); English transl., St. Petersburg Math. J.5(1994), no. 4, 841–867. MR1246427↑

1993

[53] [53]

Spector,New directions in harmonic analysis onL 1, Nonlinear Anal.192(2020), 111685, 20, DOI 10.1016/j.na.2019.111685

D. Spector,New directions in harmonic analysis onL 1, Nonlinear Anal.192(2020), 111685, 20, DOI 10.1016/j.na.2019.111685. MR4034690↑

work page doi:10.1016/j.na.2019.111685 2020

[54] [54]

,An optimal Sobolev embedding forL 1, J. Funct. Anal.279(2020), no. 3, 108559, DOI 10.1016/j.jfa.2020.108559. MR4093790↑

work page doi:10.1016/j.jfa.2020.108559 2020

[55] [55]

Spector and J

D. Spector and J. Van Schaftingen,Optimal embeddings into Lorentz spaces for some vector differential operators via Gagliardo’s lemma, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 30(2019), no. 3, 413–436, DOI 10.4171/RLM/854. MR4002205↑

work page doi:10.4171/rlm/854 2019

[56] [56]

Spivak,A comprehensive introduction to differential geometry

M. Spivak,A comprehensive introduction to differential geometry. Vol. I, 2nd ed., Publish or Perish, Inc., Wilmington, DE, 1979. MR0532830↑

1979

[57] [57]

E. M. Stein,Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies, vol. No. 63, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1970. MR0252961↑

1970

[58] [58]

,Editor’s note: the differentiability of functions inR n, Ann. of Math. (2)113(1981), no. 2, 383–385. MR0607898↑

1981

[59] [59]

E. M. Stein and G. Weiss,Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR0304972 ↑ 52 JESSE GOODMAN, FELIPE HERN ´ANDEZ, AND DANIEL SPECTOR

1971

[60] [60]

D. M. Stolyarov,Weakly canceling operators and singular integrals, Tr. Mat. Inst. Steklova 312(2021), 259–271 (Russian, with Russian summary). English version published in Proc. Steklov Inst. Math.312(2021), no. 1, 249–260.↑

2021

[61] [61]

Sb.213(2022), no

,Hardy-Littlewood-Sobolev inequality forp= 1, Mat. Sb.213(2022), no. 6, 125– 174, DOI 10.4213/sm9645 (Russian, with Russian summary); English transl., Sb. Math.213 (2022), no. 6, 844–889. MR4461456↑

work page doi:10.4213/sm9645 2022

[62] [62]

R. S. Strichartz,Analysis of the Laplacian on the complete Riemannian manifold, J. Func- tional Analysis52(1983), no. 1, 48–79, DOI 10.1016/0022-1236(83)90090-3. MR0705991 ↑

work page doi:10.1016/0022-1236(83)90090-3 1983

[63] [63]

Tartar,Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll

L. Tartar,Imbedding theorems of Sobolev spaces into Lorentz spaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)1(1998), no. 3, 479–500 (English, with Italian summary). MR1662313↑

1998

[64] [64]

Triebel,Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark

H. Triebel,Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark. Mat.24(1986), no. 2, 299–337, DOI 10.1007/BF02384402. MR0884191↑

work page doi:10.1007/bf02384402 1986

[65] [65]

II, Monographs in Mathematics, vol

,Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkh¨ auser Ver- lag, Basel, 1992. MR1163193↑

1992

[66] [66]

Van Schaftingen,A simple proof of an inequality of Bourgain, Brezis and Mironescu, C

J. Van Schaftingen,A simple proof of an inequality of Bourgain, Brezis and Mironescu, C. R. Math. Acad. Sci. Paris338(2004), no. 1, 23–26, DOI 10.1016/j.crma.2003.10.036 (English, with English and French summaries). MR2038078↑

work page doi:10.1016/j.crma.2003.10.036 2004

[67] [67]

,Estimates forL 1-vector fields, C. R. Math. Acad. Sci. Paris339(2004), no. 3, 181–186.↑

2004

[68] [68]

,Estimates forL 1 vector fields under higher-order differential conditions, J. Eur. Math. Soc. (JEMS)10(2008), no. 4, 867–882, DOI 10.4171/JEMS/133. MR2443922↑

work page doi:10.4171/jems/133 2008

[69] [69]

,Limiting fractional and Lorentz space estimates of differential forms, Proc. Amer. Math. Soc.138(2010), no. 1, 235–240, DOI 10.1090/S0002-9939-09-10005-9. MR2550188↑

work page doi:10.1090/s0002-9939-09-10005-9 2010

[70] [70]

,Limiting Sobolev inequalities for vector fields and canceling linear differential op- erators, J. Eur. Math. Soc. (JEMS)15(2013), no. 3, 877–921, DOI 10.4171/JEMS/380. MR3085095↑

work page doi:10.4171/jems/380 2013

[71] [71]

Fixed Point Theory Appl.15(2014), no

,Limiting Bourgain-Brezis estimates for systems of linear differential equations: theme and variations, J. Fixed Point Theory Appl.15(2014), no. 2, 273–297, DOI 10.1007/s11784-014-0177-0. MR3298002↑

work page doi:10.1007/s11784-014-0177-0 2014

[72] [72]

47–56, DOI 10.1007/978-3-031-48579-4 5

,Endpoint Sobolev inequalities for vector fields and cancelling operators, Extended ab- stracts 2021/2022—Methusalem lectures, Trends Math., Birkh¨ auser/Springer, Cham, [2024] ©2024, pp. 47–56, DOI 10.1007/978-3-031-48579-4 5. MR4738462↑

work page doi:10.1007/978-3-031-48579-4 2021

[73] [73]

Yung,Sobolev inequalities for(0, q)forms on CR manifolds of finite type, Math

P.-L. Yung,Sobolev inequalities for(0, q)forms on CR manifolds of finite type, Math. Res. Lett.17(2010), no. 1, 177–196, DOI 10.4310/MRL.2010.v17.n1.a14. MR2592736↑ Department of Statistics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand Email address, J. Goodman:jesse.goodman@auckland.ac.nz Department of Mathematics, Penn State Uni...

work page doi:10.4310/mrl.2010.v17.n1.a14 2010