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Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy

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Pith's one-line read Sharp exponential integrability for functions with finite non-local, non-convex energy

desk verdict The main interpolation result is false as stated—a linear counterexample breaks it—but the small-energy bound and the optimality examples are real and should be preserved. read the letter →

arxiv 1908.06179 v1 pith:6MIIPY3D submitted 2019-08-16 math.FA math.AP

Arka Mallick , Hoai-Minh Nguyen This is my paper · ORCID
classification math.FAmath.AP MSC 26D1026A54
keywords Moser-Trudingerinequalityexponentialintegrabilitynon-localenergySobolevPoincaréJohn-Nirenbergnon-convexBMO
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 asks how strongly a function must be integrable when its non-local 'jump energy' is finite—the energy that counts pairs of points whose values differ by more than δ, weighted by the singular kernel |x−y|^(−(d+p)). It proves that a small amount of this energy forces an exponential moment: for p>d, if the normalized energy is at most a small constant M0, then the average of exp(α(p/d)δ^(−1)|u−u_B|) over a ball B is bounded. For larger energy M, the same conclusion holds with a reduced exponent (p/d)^β, where β depends only on M and α. The rate p/d is optimal: for any larger γ, there are admissible functions with divergent exponential integrals. This closes the gap between the standard John-Nirenberg (BMO) exponential integrability and genuine boundedness for this class of non-convex, non-local energies.

What carries the argument

The central object is the non-local, non-convex jump energy I_{δ,p}(u,O). The load-bearing mechanism is a two-step transfer: first a radial reflection extends u from B_1 to B_{3/2} without increasing I_{δ,p} by more than a universal factor, using an extension lemma ([18, Lemma 17]) that also controls level sets, so that John-Nirenberg's inequality applies on the larger ball and gives a starting level-set bound; then two lemmas iterate that bound down to B_1—a geometric counting lemma (Lemma 2.1) that converts level-set volume into the p/d power of the energy, and a dyadic-decay lemma (Lemma 2.2) showing I_{2^k δ,p} ≤ $2^{{−k(p−1)}}$ I_{δ,p}. Iteration of these three ingredients yields the exponential tail estimate $e^{{−α(p/d)λ}}$ and hence the stated integrability.

What would settle it

Construct a sequence of functions u_n on B_1 with normalized energy |B|^((p−d)/d) δ^(−p) I_{δ,p}(u_n,B) → 0 but ∫_B exp(α(p/d)δ^(−1)|u_n − (u_n)_B|) → ∞; such a sequence would directly contradict Theorem 1.1(ii) and Theorem 1.2. Alternatively, compute I_{δ,p} of the radial reflection of oscillating step functions on B_1 and compare with (2.8): an unbounded ratio I_δ(ũ, B_{3/2})/I_δ(u, B_1) would break the extension lemma on which the proof rests.

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Extended reading notes

Core claim

Stated on the paper's own terms: for p>d, the sharp exponential integrability threshold for the non-local energy I_{δ,p}(u,B) = ∫∫_{|u(x)−u(y)|>δ} δ^p / |x−y|^{d+p} dx dy is the rate α(p/d)δ^(−1) in the exponent when the normalized energy δ^(−p)|B|^((d−p)/d) I_{δ,p}(u,B) is sufficiently small (Theorem 1.1(ii)), and the sharp threshold is αδ^(−1) when p=d (Theorem 1.2). For an arbitrary bound M on the normalized energy, the same conclusion holds with the exponent scaled by a factor (p/d)^β, β∈[0,1] depending only on M and α. Proposition 1.7 proves optimality: for p>d and any γ>p/d, there exists a function with I_{δ,p}(u,B)≤M yet ∫_B $e^{{αγδ^(−1)|u−u_B|}}$=+∞; for p=d, there exists a sequence with energy tending to 0 whose exponential integrals diverge at any rate γ>1.

Load-bearing premise

The proof assumes the radial-reflection extension of a function from a ball to a slightly larger ball carries the non-local energy I_{δ,p} with a universal constant and controls the level sets of the extension by those of the original function; if this extension lemma fails uniformly, the John-Nirenberg step on the larger ball cannot be transferred back to the original ball.

Editorial extensions

If this is right

  • For p>d, a small normalized non-local energy implies a strong exponential moment: the average of e^{α(p/d)δ^(−1)|u−u_B|} is bounded by a constant independent of u.
  • For p=d, the same holds with rate δ^(−1), a Moser-Trudinger-type gain that John-Nirenberg alone cannot deliver because the BMO norm need not vanish as the energy vanishes.
  • For arbitrary energy level M, a reduced exponential rate (p/d)^β still holds, with β depending only on M and α; the proof makes no use of gradient or boundedness assumptions.
  • The rate p/d (and rate 1 when p=d) cannot be improved: any excess γ>p/d destroys the exponential integrability even among functions with arbitrarily small energy (p=d) or bounded energy (p>d).
  • On a smooth bounded domain the same exponential estimates hold for the function itself, up to a factor e^{α(p/d)^β δ^(−1)‖u‖_{L^1}}.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The dyadic-decay lemma (Lemma 2.2) is a generic mechanism: any energy functional with a comparable split-estimate under midpoint interpolation should inherit exponential integrability, so the result likely extends to kernels other than |x−y|^(−(d+p)) and to weighted or anisotropic variants.
  • The smallness threshold M0 is not made explicit; a natural next step would quantify M0 in terms of d, p, α, and identify whether the optimal constant in the exponential estimate matches the Moser-Trudinger constant in the limit p→d from above.
  • Proposition 1.7(i) constructs a function with finite energy but divergent exponential moment at supercritical rate; one might test numerically whether this construction also exhibits a 'truncation' phenomenon in the BMO sense, since the BMO norm stays bounded while the exponential integrability exponent exceeds the threshold.
  • The p=d sequence with energy tending to 0 suggests a possible non-local analogue of the Moser-Trudinger supremum: normalize the energy to a fixed level and look for the sharp constant multiplying δ^(−1); the paper's result guarantees finiteness but leaves the sharp constant open.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies exponential integrability properties of measurable functions with finite non-local, non-convex energy I_{δ,p}(u,O), defined in (1.3) as the double integral of δ^p |x-y|^{-(d+p)} over pairs where |u(x)-u(y)|>δ. The main results are stated as Theorem 1.1 for p>d: for every M>0 and α>0 there is β=β(α,M)∈[0,1] such that the normalized exponential integral with exponent α(p/d)^β over a ball B is bounded by a constant C depending only on d,p,α whenever |B|^{(p-d)/d}δ^{-p}I_{δ,p}(u,B)≤M, and for a small M0 the better exponent α(p/d) is admissible. Theorem 1.2 states an analogous small-energy exponential integrability result in the critical case p=d. Proposition 1.7 gives examples showing that the growth rate p/d cannot be exceeded. The proofs proceed by a radial extension to B_{3/2}, John-Nirenberg estimates, a dyadic level-set iteration, and an external extension lemma from [18].

Significance. If the results were correct, they would fill a natural gap between the John-Nirenberg exponential integrability available for BMO functions and the boundedness or Moser-Trudinger behavior available for Sobolev functions, using only the non-local, non-convex energy I_{δ,p}. The optimality examples in Proposition 1.7 are a useful contribution and appear to be correctly constructed. However, the central theorem, Theorem 1.1(i), is false as stated, and the proof of the other main theorem, Theorem 1.2, is omitted. The paper's advertised main claim is therefore not established, and the current version cannot be accepted.

major comments (4)
  1. [Section 1, Theorem 1.1(i), Eq. (1.11)] The statement of Theorem 1.1(i) is false as written. Fix B=B_1 and δ=1, and for M>0 take u_M(x)=A_M x_1 with A_M chosen so that I_{1,p}(u_M,B_1)=M. By (1.4)-(1.5), A_M is comparable to (M/K_{d,p})^{1/p} and hence A_M→∞ as M→∞. For every β∈[0,1] one has (p/d)^β≥1, so the exponent in (1.11) is at least α. A direct estimate gives ∫_{B_1} e^{α(p/d)^β A_M|x_1|} dx ≥ c e^{α A_M}/A_M →∞ as M→∞. Since the constant C in (1.11) is asserted to depend only on d, p, and α and not on M, the supremum in (1.11) is infinite. Thus the theorem cannot hold as stated; the constant would have to depend on M, which would be a substantially weaker statement.
  2. [Section 2.2, Eqs. (2.12)-(2.13)] The proof of part i) proves a different inequality from the one stated. The level-set estimate (2.13) reads |{|u|>λℓ0}| ≤ e^{-α(p/d)λ+2}|{|u|>ℓ0}|, which by the layer-cake formula yields exponential integrability of e^{b|u|} with b<α(p/d)/ℓ0, that is, with a multiplier ℓ0^{-1}. The sentence after (2.13), claiming that this implies ∫_{B_1} e^{α(p/d)|u|} dx ≤ C, is not justified unless ℓ0≤1, whereas ℓ0 defined in (2.12) is generally larger than 1 because it is at least c_1 M. Moreover, the final identification β(α,M)=ℓ0^{-1} does not match the exponent (p/d)^β in (1.11): the inequality actually obtained has exponent α(p/d)ℓ0^{-1}, not α(p/d)^{ℓ0^{-1}}. These mismatches confirm that Theorem 1.1(i) is not supported by the proof.
  3. [Section 2.4] The proof of Theorem 1.2, which is the main theorem for the critical case p=d, is omitted; the text says 'The proof is similar to the one of part ii) of Theorem 1.1 and is omitted.' The same is true for Propositions 1.5 and 1.6, whose derivations are only sketched via local charts and [18, Lemma 17]. A main theorem cannot rest on an entirely omitted proof, especially when the p=d case requires a separate treatment and the analogous p>d proof already contains the mismatches described above.
  4. [Section 2.2, Eqs. (2.7)-(2.8)] The passage from B_1 to B_{3/2} is load-bearing. The radial reflection defines ũ, and the two estimates (2.7) and (2.8) are quoted from [18, Lemma 17] without statement or proof. The subsequent John-Nirenberg argument is performed on B_{3/2}, and the conclusion is transferred back to B_1 only through these two estimates. Since the entire level-set iteration depends on this extension step, the manuscript should either state and prove the needed lemma in the present notation or provide a precise reference with all hypotheses. As written, the dependence on an unstated external result makes the proof incomplete.
minor comments (4)
  1. [Section 1, Theorem 1.1] There is a typo in the statement: 'let B be a an open ball' should read 'let B be an open ball.'
  2. [Section 1, Proposition 1.7] The two assertions in Proposition 1.7 are both labeled 'i)'; the second should be labeled 'ii)'.
  3. [Section 2.1, Lemma 2.2] The statement of Lemma 2.2 begins 'Let g ∈ L1_loc(O)' but the integrand is written in terms of u; the lemma should use u consistently.
  4. [Introduction and references] There are numerous typographical errors: 'Propposition' in the introduction, 'founded' for 'found', 'inequaliies' in the abstract, 'Soboleff' in reference [40], and misspelled names in reference [9]. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the exponential-integrability bound is derived, not assumed; self-citations are independent prior results.

full rationale

The main results Theorems 1.1 and 1.2 are proved from external ingredients: John-Nirenberg's inequality, the elementary geometric Lemma 2.1, the scaling Lemma 2.2, and an extension estimate (2.8) quoted as [18, Lemma 17]. The target inequality (1.11)-(1.13) does not appear among these inputs; the proof derives an exponential tail decay (2.13) and then integrates it. The cited [18] lemma is prior work by one of the current authors, but it is a parameter-free extension bound whose statement does not contain exponential integrability, so this self-citation is real independent support rather than a circular premise. Propositions 1.5 and 1.6 are explicitly consequences of Theorems 1.1 and 1.2, and Proposition 1.7 is a separate construction rather than a renamed version of the main estimate. Section 2.5 omits the chart/extension details for Propositions 1.5 and 1.6; this is an expositional omission, not evidence that the propositions were assumed. The only notable concern near the claimed result is not circularity: the proof ends by setting beta=ell_0^{-1}, while the printed statement of (1.11) claims a constant C independent of M, and an affine function A_M x_1 with energy M suggests the integral grows with M for every beta in [0,1]. That is a correctness or formulation issue, not a reduction of the prediction to its inputs. No fitted parameter is renamed as a prediction, and no uniqueness or extension theorem is invoked in a way that contains the target inequality by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The results are proved inside a well-established framework: the non-local energy I_{δ,p} is defined in prior work, and the proofs call on John-Nirenberg, BBM, Nguyen's characterization, and an extension lemma from the authors' earlier paper. No new entities or fitted physical parameters are introduced; the only hand-chosen parameters are the exponent β and the smallness threshold M0, both of whose explicit values are left unspecified.

free parameters (2)
  • β(α, M) in Theorem 1.1(i) = unspecified; proof proposes β = ℓ0^{-1}
    An exponent 0 ≤ β ≤ 1 inserted into the statement to weaken the integrability exponent for large energy levels; its value is neither computable from the proof nor consistent with the displayed exponent α(p/d)^β.
  • M0 in Theorem 1.1(ii) and Theorem 1.2 = unspecified (small)
    A smallness threshold whose existence is asserted but not quantified; the proof claims it exists if M0 is small enough, but no explicit bound is given.
assumptions (5)
  • standard math John-Nirenberg inequality
    Used to establish the initial level-set estimate (2.9) on B_{3/2}.
  • domain assumption [18, Lemma 17] extension preserving I_{δ,p}
    External lemma on extending functions while preserving the non-local energy; no proof is given in this paper.
  • domain assumption Nguyen's characterization of Sobolev spaces (Proposition 1.2, from [28])
    Prior theorem in the same research program, used to justify the BMO bound (1.8) and the Poincaré inequality (1.7).
  • domain assumption BBM formula (Proposition 1.1, from [10])
    Background characterization of Sobolev spaces that motivates the non-local energy I_{δ,p}.
  • standard math Moser-Trudinger and Morrey inequalities
    Cited as background in the introduction for comparison with the new results.

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Pith. "Pith review of Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy." pith.science (2026). https://pith.science/paper/6MIIPY3D

@misc{pith2026190806179,
  author       = {Pith},
  title        = {Pith review of: Exponential integrability in the spirit of Moser-Trudinger's inequalities of functions with finite non-local, non-convex energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6MIIPY3D}},
  note         = {Machine review of arXiv:1908.06179}
}
abstract

Let $d \ge 1$, $p \ge d$, and let $\Omega$ be a smooth bounded open subset of $\mathbb{R}^d$. We prove some exponential integrability in the spirit of Moser-Trudinger's inequalities for measurable functions $u$ defined in $\Omega$ such that $$ \mathop{\int_{\Omega} \int_{\Omega}}_{|u(x) - u(y)| > \delta} \frac{1}{|x-y|^{d+p}} \, dx \, dy < + \infty, $$ for some $\delta > 0$. This double integral appeared in characterizations of Sobolev spaces and involved in improvements of the Sobolev inequaliies, Poincar\'e inequalities, and Hardy inequalities.

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Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

  1. [18]

    Brezis and H.-M

    H. Brezis and H.-M. Nguyen, Non-local functionals related to the total variation and co nnections with image processing, Ann. PDE 1 4 (2018)

    work page 2018

  2. [1]

    Adimurthi and A Mallick, A Hardy type inequality on fractional order Sobolev spaces o n the Heisenberg group , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 18 (2018), 917–949

    work page 2018

  3. [2]

    Ambrosio, G

    L. Ambrosio, G. De Philippis, and L. Martinazzi, Gamma-convergence of nonlocal perimeter functionals , Manuscripta Math. 134 (2011), 377–403

    work page 2011

  4. [3]

    Antonucci, M

    C. Antonucci, M. Gobbino, M. Migliorini, and N. Picenni, On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation . C. R. Math. Acad. Sci. Paris 356 (2018), 859–864

    work page 2018

  5. [4]

    Optimal constants for a non-local approximation of Sobolev norms and total variation

    C. Antonucci, M. Gobbino, M. Migliorini, N. Picenni, Opt imal constants for a non-local approximation of Sobolev norms and total variation, arXiv: 1708.01231

    work page Pith review arXiv

  6. [5]

    On the gap between Gamma-limit and pointwise limit for a non-local approximation of the total variation

    C. Antonucci, M. Gobbino, N. Picenni, On the gap between g amma-limit and pointwise limit for a non-local approximation of the total variation, arXiv:1712.04413

    work page Pith review arXiv

  7. [6]

    Aubert and P

    G. Aubert and P. Kornprobst, Can the nonlocal characterization of Sobolev spaces by Bour gain et al. be useful for solving variational problems? , SIAM J. Numer. Anal. 47 (2009), 844–860

    work page 2009

  8. [7]

    Bourgain and H.-M

    J. Bourgain and H.-M. Nguyen, A new characterization of Sobolev spaces , C. R. Math. Acad. Sci. Paris 343 1 (2006)

    work page 2006

Show all 48 references
  1. [8]

    Brezis, How to recognize constant functions

    H. Brezis, How to recognize constant functions. Connections with Sobo lev spaces , Volume in honor of M. Vishik, Uspekhi Mat. Nauk 57 (2002), 59–74; English translation in Russian Math. Survey s 57 (2002), 693–708

    2002

  2. [9]

    Bethuel, H

    F. Bethuel, H. Brezis, F. Heilein, Gnzburg-Landau vorti ces. Reprint of the 1994 edition [MR1269538]. Modern Birkhuser Classics. Birkhuser/Springer, Cham, 2017. xxix +158 pp

    1994

  3. [10]

    Bourgain, H

    J. Bourgain, H. Brezis and P. Mironescu, Another look at Sobolev spaces , Optimal control and partial differential equations, IOS, Amsterdam (2001), 439–455

    2001

  4. [11]

    Bourgain, H

    J. Bourgain, H. Brezis and P. Mironescu, Limiting embedding theorems for W s,p when s ↑ 1 and applications , J. Anal. Math. 87 (2002), 77–101

    2002

  5. [12]

    Bourgain, H

    J. Bourgain, H. Brezis and P. Mironescu, Lifting, degree, and distributional Jacobian revisited , Comm. Pure Appl. Math. 58 (2005), 529–551

    2005

  6. [13]

    Bourgain, H

    J. Bourgain, H. Brezis, H.-M. Nguyen, A new estimate for the topological degree. C. R. Math. Acad. Sci. Paris 340 (2005), 787–791

    2005

  7. [14]

    Brezis and H.-M

    H. Brezis and H.-M. Nguyen, On a new class of functions related to VMO , C. R. Acad. Sci. Paris 349 (2011), 157–160

    2011

  8. [15]

    Brezis, Functional analysis, Sobolev spaces and par tial differential equations

    H. Brezis, Functional analysis, Sobolev spaces and par tial differential equations. Universitext. Springer, New York, 2011. xiv+599 pp

    2011

  9. [16]

    Brezis and H.-M

    H. Brezis and H.-M. Nguyen, Two subtle convex nonlocal approximation of the BV -norm, Nonlinear Anal., 137 (2016), 222–245

    2016

  10. [17]

    Brezis and H.-M

    H. Brezis and H.-M. Nguyen, The BBM formula revisited , Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 27 (2016), 515–533

    2016

  11. [19]

    Buades, B

    A. Buades, B. Coll and J. M. Morel, Neighborhood filters and PDE’s , Numer. Math. 105 (2006), 1–34

    2006

  12. [20]

    Caffarelli and E

    L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal mini mal surfaces , Calc. Var. PDE 41 (2011), 203–240. EXPONENTIAL INTEGRABILITY IN THE SPIRIT OF MOSER-TRUDINGE R’S INEQUALITIES 11

    2011

  13. [21]

    Davila, On an open question about functions of bounded variation , Calc

    J. Davila, On an open question about functions of bounded variation , Calc. Var. Partial Differential Equations 15 (2002), 519–527

    2002

  14. [22]

    Di Nezza, G

    E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces , Bull. Sci. Math. 136 5 (2012), 521–573

    2012

  15. [23]

    Fefferman and E

    C. Fefferman and E. Stein, H p spaces of several variables , Acta. Math. 129 (1972) 137–193

    1972

  16. [24]

    John and L

    F. John and L. Nirenberg, On functions of bounded mean oscillation , Comm. Pure Appl. Math. 14 (1961) 415–426

    1961

  17. [25]

    Characterization of Sobolev and BV Spaces

    G. Leoni and D. Spector, Characterization of Sobolev and BV Spaces , J. Funct. Anal. 261 (2011), 2926–2958, Corrigendum to “Characterization of Sobolev and BV Spaces” , J. Funct. Anal. 266 (2014), 1106–1114

    2011

  18. [26]

    V. G. Mazya, Sobolev Spaces. Springer, Berlin (1985)

    1985

  19. [27]

    Moser, A sharp form of an inequality by N

    J. Moser, A sharp form of an inequality by N. Trudinger , Indiana Univ. Math. J. 20 11 (1971), 1077–1092

    1971

  20. [28]

    Nguyen, Some new characterizations of Sobolev spaces , J

    H.-M. Nguyen, Some new characterizations of Sobolev spaces , J. Funct. Anal. 237 (2006), 689–720

    2006

  21. [29]

    Nguyen, Optimal constant in a new estimate for the degree , J

    H.-M. Nguyen, Optimal constant in a new estimate for the degree , J. Anal. Math. 101 (2007), 367–395

    2007

  22. [30]

    Nguyen, Γ -convergence and Sobolev norms , C

    H.-M. Nguyen, Γ -convergence and Sobolev norms , C. R. Acad. Sci. Paris 345 (2007), 679–684

    2007

  23. [31]

    Nguyen, Further characterizations of Sobolev spaces , J

    H.-M. Nguyen, Further characterizations of Sobolev spaces , J. European Math. Soc. 10 (2008), 191–229

    2008

  24. [32]

    Nguyen, Γ -convergence, Sobolev norms, and BV functions , Duke Math

    H.-M. Nguyen, Γ -convergence, Sobolev norms, and BV functions , Duke Math. J. 157 (2011), 495–533

    2011

  25. [33]

    Nguyen, Some inequalities related to Sobolev norms , Calc

    H.-M. Nguyen, Some inequalities related to Sobolev norms , Calc. Var. Partial Differential Equations 41 (2011), 483–509

    2011

  26. [34]

    Nguyen, A refined estimate for the topological degree , C

    H.-M. Nguyen, A refined estimate for the topological degree , C. R. Math. Acad. Sci. Paris 355 (2017), 1046–1049

    2017

  27. [35]

    Nguyen, A

    H.-M. Nguyen, A. Pinamonti, M. Squassina, and E. Vecchi , New characterizations of magnetic Sobolev spaces , Adv. Nonlinear Anal. 7 (2018), 227–245

    2018

  28. [36]

    Nguyen and M

    H.-M. Nguyen and M. Squassina, Marco Logarithmic Sobolev inequality revisited , C. R. Math. Acad. Sci. Paris 355 (2017), 447–451

    2017

  29. [37]

    Nguyen; M

    H.-M. Nguyen; M. Squassina, On Hardy and Caffarelli-Kohn-Nirenberg inequalities , Journal d’Analyse Mathe- matique. 2017, to appear

    2017

  30. [38]

    Nguyen; M

    H.-M. Nguyen; M. Squassina, On anisotropic Sobolev spaces , Commun. Contemp. Math. 21 (2019), 1850017, 13 pp

    2019

  31. [39]

    Squassina, B

    M. Squassina, B. Volzone, Bourgain-Brezis-Mironescu formula for magnetic operator s, C. R. Math. Acad. Sci.Paris 354 (2016), 825–831

    2016

  32. [40]

    Peetre, Espaces d’interpolation et theoreme de Soboleff, Ann

    J. Peetre, Espaces d’interpolation et theoreme de Soboleff, Ann. Inst. Fourier (Grenoble) 16 (1966), 279–317

    1966

  33. [41]

    S. I. Pohozhaev, The Sobolev imbedding in the case pl = n. Proc. Tech. Sci. Conf. on Adv. Sci. Research 1964-1965, Mathematics Section, 158– 170, Moskov. Energet. Inst., Mos cow 1965

    1964

  34. [42]

    Ponce, A new approach to Sobolev spaces and connections to Γ -convergence, Calc

    A. Ponce, A new approach to Sobolev spaces and connections to Γ -convergence, Calc. Var. Partial Differential Equations 19 (2004), 229–255

    2004

  35. [43]

    Ponce, An estimate in the spirit of Poincar´ e’s inequality , J

    A. Ponce, An estimate in the spirit of Poincar´ e’s inequality , J. Eur. Math. Soc. 6 (2004), 1–15

    2004

  36. [44]

    Ponce and S

    A. Ponce and S. Daniel On formulae decoupling the total variation of BV functions. , Nonlinear Anal. 154 (2017), 241–257

    2017

  37. [45]

    J. V. Schaftingen, Estimates by gap potentials of free homotopy decomposition s of critical Sobolev maps , arXiv:1811.01706v2

    arXiv

  38. [46]

    N. S. Trudinger, On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 (1967), 473–483

    1967

  39. [47]

    L. P. Yaroslavsky, Digital Picture Processing. An introduction , Springer, 1985

    1985

  40. [48]

    L. P. Yaroslavsky and M.Eden, Fundamentals of Digital Optics , Springer, 1996. (A. Mallick) Department of Mathematics EPFL SB CAMA Station 8 CH-1015 Lausanne, Switzerland E-mail address : arka.mallick@epfl.ch (H.-M. Nguyen) Department of Mathematics EPFL SB CAMA Station 8 CH-1...

    1996

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