A dimension-free weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$

Cody B. Stockdale; Daniel Spector; Yuyuan Ouyang

REVIEW 1 major objections 3 minor 1 cited by

A dimension-free weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$

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Pith's one-line read The vector Riesz transform satisfies a weak-type (1,1) bound with constant 2 in all dimensions.

desk verdict A serious, largely convincing proof of Stein's dimension-free weak-type bound with constant 2; one displayed equality in the final estimate is wrong as written, but the triangle inequality repairs it, so the result should survive referee scrutiny. read the letter →

arxiv 2608.18068 v1 pith:P75KNR52 submitted 2026-08-18 math.CA math.AP

Yuyuan Ouyang , Daniel Spector , Cody B. Stockdale This is my paper · ORCID

classification math.CAmath.AP MSC 42B2035R1149J40

keywords vectorRiesztransformweaktype(11)dimension-freeboundobstacleproblemfractionalLaplacianLewy-Stampacchiaestimatepartialbalayagevariationalinequality

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 claims that the vector Riesz transform on $\mathbb{R}^n$ obeys the weak-type $(1,1)$ bound $\|Rf\|_{L^{1,\infty}(\mathbb{R}^n)} \le 2\|f\|_{L^1(\mathbb{R}^n)}$ with a constant that is independent of the dimension $n$. The proof decomposes a nonnegative function as $f = \mu + (-\Delta)^{1/2}u$, where $\mu$ is bounded by the level $\lambda$, carries the full $L^1$ mass of $f$, and equals $\lambda$ on the set where $u$ is positive; a Lewy–Stampacchia estimate for a fractional obstacle problem supplies this decomposition. Applying the decomposition separately to $f_+$ and $f_-$ turns the Riesz transform into $R\mu + \nabla u$, so the large level set of $Rf$ is controlled by the support of $u$ and by $R\mu$. If correct, the theorem improves the previous logarithmic-in-$n$ weak-type bound to an absolute constant and answers the 1986 problem from the International Congress of Mathematicians in the affirmative.

What carries the argument

The load-bearing object is a variational obstacle problem for the fractional Laplacian: minimize $J_\lambda(v)=\tfrac12 E_\alpha(v)-\int_{\mathbb{R}^n}(f-\lambda)v\,dx$ over nonnegative $v$ in the energy space $X_\alpha$, for $0<\alpha<2$. The unique minimizer $u$ determines a distribution $\eta=(-\Delta)^{\alpha/2}u-(f-\lambda)$ that is a nonnegative measure, and the Lewy–Stampacchia estimate $0\le\eta\le(\lambda-f)_+$ bounds it pointwise. Setting $\mu=\lambda-\eta$ then gives the decomposition $f=\mu+(-\Delta)^{\alpha/2}u$ with the three key properties: $\mu$ has the same $L^1$ mass as $f$, $\mu$ never exceeds $\lambda$, and $\mu$ coincides with $\lambda$ on the positivity set of $u$. For $\alpha\ge1$ the minimizer is in $H^1$, so $\nabla u$ vanishes outside that set; this gradient property is what converts the Riesz transform identity into a level-set estimate.

What would settle it

Compute the obstacle decomposition at some level $\lambda$ for a one-dimensional pair of adjacent blobs, such as $f_+=\mathbf{1}_{[-2,-1]}$ and $f_-=\mathbf{1}_{[1,2]}$; if the sets where $\mu_+$ and $\mu_-$ are positive overlap, the proof's displayed equality is false. Then evaluate $\lambda|\{|Rf|>\lambda\}|/\|f\|_1$ directly for that $f$; any value exceeding $2$ would disprove Theorem 1.1, while a consistent value below $2$ would leave the theorem intact and only require a repaired line in the proof.

Watch

Extended reading notes

Core claim

The central discovery is that the full vector Riesz transform $R=(R_1,\dots,R_n)$ maps $L^1(\mathbb{R}^n)$ into weak $L^1$ with operator constant at most $2$ for every dimension $n$, settling a question posed in 1986. The route is structural rather than generic: instead of a Calderón–Zygmund decomposition, the paper uses the identity $R=\nabla(-\Delta)^{-1/2}$ and builds, for each nonnegative datum and each level $\lambda$, a decomposition $f=\mu+(-\Delta)^{1/2}u$ whose pieces satisfy $\|\mu\|_{L^\infty}\le\lambda$, $\|\mu\|_{L^1}=\|f\|_{L^1}$, $\mu=\lambda$ on $\{u>0\}$, and $\nabla u=0$ off that set. With $f=f_+-f_-$, the corresponding $\mu$ and $u$ give $Rf=R\mu+\nabla u$, the set $\{|Rf|>\lambda\}$ is contained in $\{u>0\}\cup\{|R\mu|>\lambda\}$, and the mass and support bounds make each part cost at most $\|f\|_{L^1}$. The stated theorem includes the component bounds as a direct consequence.

Load-bearing premise

The load-bearing premise is that the obstacle decomposition of Theorem 1.2 exists with its claimed mass, support, and gradient properties; in the proof of Theorem 1.1 this is applied to $f_+$ and $f_-$ separately, where a subsidiary equality $\int|\mu|=\int(\mu_++\mu_-)$ additionally assumes the supports of $\mu_+$ and $\mu_-$ are disjoint, which is not established, although the triangle inequality would repair that step.

Editorial extensions

If this is right

The component Riesz transforms $R_j$ inherit the bound with constant $2$, since $|R_jf|\le|Rf|$ pointwise.
The previous best dimensional dependence, logarithmic in $n$, is replaced by an absolute constant, resolving the 1986 weak-type question in the affirmative.
The constant $2$ coincides with the dimension-free strong-type $L^p$ constant, so the weak-type result sits on the same scale as the sharp martingale-based estimates.
Because the decomposition is proved for $0<\alpha<2$ while only $\alpha=1$ is used, the same obstacle machinery is available for other operators built from fractional Laplacians.
Interpolating the dimension-free weak-type estimate with the known $L^2$ bound yields dimension-free strong-type estimates for $R$ with constants independent of $n$.

Reading between the lines

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

The displayed equality $\int_{\mathbb{R}^n}|\mu|\,dx=\int_{\mathbb{R}^n}(\mu_++\mu_-)\,dx$ in the proof of Theorem 1.1 assumes $\mu_+$ and $\mu_-$ have disjoint supports; that disjointness is neither proved nor generally true, but replacing the equality by the triangle inequality preserves the estimate, so the gap is repairable.
The same partial-balayage construction may yield dimension-free weak-type bounds for other singular integrals with a 'gradient of $(-\Delta)^{-1/2}$' structure, provided the unbounded-domain fractional obstacle and Lewy–Stampacchia steps can be replicated.
A numerical check of the obstacle decomposition in low dimensions could test whether the constant $2$ is sharp or whether smaller constants hold for special classes of data.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper settles a real open problem: the vector Riesz transform has a weak-type (1,1) bound with a dimension-free constant, specifically 2. That's a genuine advance over Janakiraman's logarithmic bound for the components, which only gave a √n log n vector estimate. The proof is not a routine Calderón–Zygmund adaptation; the fractional obstacle decomposition in Theorem 1.2 is original and the authors prove the needed Lewy–Stampacchia estimate on unbounded domains rather than importing it. The technical work is detailed, self-contained, and mostly correct. The constant 2 matching the known strong-type constant is a nice bonus.

The soft spot is in the proof of Theorem 1.1, exactly where the stress-test note points. In bounding ∥μ∥²_{L²}, the paper writes ∫|μ|dx = ∫(μ_+ + μ_-)dx. That equality expects μ_+ and μ_- to have disjoint supports, which is not established and is generally false: the obstacle solutions can have overlapping positive parts even when f_+ and f_- are disjointly supported. The stress-test's example with f_+ = f_- = f is not the actual setup (the positive and negative parts of f are disjoint), but the concern itself stands. The fix is trivial: replace the equality with the triangle inequality ∫|μ|dx ≤ ∫(μ_+ + μ_-)dx, and the same bound follows. Everything else in the Theorem 1.1 argument—the set inclusion, the measure bound on Ω, and the Chebyshev step—is unaffected. So this is a minor, repairable slip in an otherwise sound proof.

The AI statement is unusual but not a problem. The authors disclose LLM involvement and state they have independently verified and rewritten the mathematical content. That is transparent and doesn't change the mathematics on the page.

My take: the central claim is very likely correct, and the proof is substantial enough to warrant serious refereeing. A referee should ask the authors to fix the equality and perhaps add a sentence clarifying that the L¹ mass preservation holds for each μ_±, which makes the repaired inequality immediate. I would also want the density argument checked—it's standard, but the constant should be seen to pass to L¹ uniformly.

This paper deserves peer review, not desk rejection. If you work in harmonic analysis or singular integrals, it's worth your time; the decomposition technique may propagate. For a reading group, it would make a good session on modern obstacle-problem methods in harmonic analysis.

Referee Report

1 major / 3 minor

Summary. This paper proves Theorem 1.1: for every f in L^1(R^n), the vector Riesz transform R=(R_1,...,R_n) satisfies the weak-type (1,1) bound ||Rf||_{L^{1,∞}} ≤ 2||f||_{L^1}, with the constant 2 independent of the dimension n. This would settle Stein's 1986 problem on dimension-free weak-type bounds. The proof introduces a decomposition theorem (Theorem 1.2) for nonnegative f in L^1∩L^2: for each λ>0 and 0<α<2 there exist nonnegative μ and u such that f = μ + (-Δ)^{α/2}u, with 0≤μ≤λ, ||μ||_1 = ||f||_1, μ=λ on the positivity set of u, and the measure of that set controlled by ||f||_1/λ. The decomposition is established by solving an obstacle problem for the fractional Laplacian on R^n, proving existence and uniqueness of minimizers, deriving variational inequalities, and proving a Lewy-Stampacchia estimate on unbounded domains. Theorem 1.1 follows by applying the decomposition with α=1 to f_+ and f_- and controlling the superlevel set of |Rf| by the union of the two positivity sets and the superlevel set of |Rμ|.

Significance. If correct, the paper settles a well-known open problem and improves the previously known dimension dependence for the component Riesz transforms from O(log n) to a dimension-free constant 2 for the vector Riesz transform. The paper's main strength is its self-contained development: the Lewy-Stampacchia estimate is proved on unbounded domains rather than imported as a black box, and the constant 2 is derived from the argument rather than fitted. The decomposition theorem is stated for general 0<α<2 and is likely to be of independent interest. The proof is detailed and largely checkable; the main written gap in the proof of Theorem 1.1 is an unjustified equality in one displayed chain, which is readily repaired without changing the conclusion. The authors' disclosure of AI assistance is transparent and does not itself affect the mathematical assessment.

major comments (1)

[Section 1, proof of Theorem 1.1] The chain displaying ||μ||_{L^2}^2 ≤ λ∫|μ| dx = λ∫(μ_+ + μ_-) dx = λ(||f_+||_1 + ||f_-||_1) = λ||f||_1 contains an equality that is not justified and is generally false. The functions μ_+ and μ_- are obtained by applying Theorem 1.2 separately to f_+ and f_-; they are nonnegative, but their supports need not be disjoint. For example, if f_+ = f_- = g with g a nonzero nonnegative function, then uniqueness of the minimizer in Theorem 4.1 gives μ_+ = μ_- and hence μ = 0, while μ_+ + μ_- is positive on the common support. The intended argument is repaired by replacing this equality with the pointwise inequality |μ| ≤ μ_+ + μ_-, which yields ||μ||_{L^2}^2 ≤ λ∫|μ| dx ≤ λ(||f_+||_1 + ||f_-||_1) = λ||f||_1. The remainder of the proof of Theorem 1.1 is unaffected, but as written this step is a genuine gap in the proof of the main theorem and must be corrected.

minor comments (3)

[Section 5, Lemma 5.1, proof] After applying (5.2) with φ = χ_ρ, the displayed identity reads ∫ μ χ_ρ dx = ∫ f χ_ρ dx + E_α(u,χ_ρ), but (5.2) gives the same expression with a minus sign before E_α(u,χ_ρ). The conclusion is unaffected because E_α(u,χ_ρ) → 0, but the sign should be corrected. In addition, the phrase 'and the fact that to see' is an incomplete sentence and should be rewritten.
[Section 1, proof of Theorem 1.1] The density reduction 'By density, we may assume f∈L^1∩L^2' is standard but is used before the Riesz transform is known to be bounded on L^1; a one-sentence justification, for example by truncation f_k = f 1_{|f|≤k} and convergence in measure together with lower semicontinuity of the weak-L^1 quasinorm, would make the proof fully self-contained.
[Section 3, Lemma 3.4] There is a typo 'it’s negative part' where 'its negative part' is intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the proof is self-contained; the only flagged issue, an unjustified equality in the final estimate of Theorem 1.1, is a correctable gap rather than a circular step.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. Theorem 1.1 is deduced from Theorem 1.2, and Theorem 1.2 is proved in Sections 2-6 directly: existence and uniqueness of the minimizer (Theorem 4.1), the variational inequality and energy identity (Theorem 4.2), the Lewy-Stampacchia estimate (Theorem 5.1), the mass identity (Lemma 5.1), and the properties of mu and u (Lemmas 5.2 and 5.3). None of these steps assumes the weak-type (1,1) bound being proved. The constant 2 is not fitted; it emerges from subadditivity, lambda |Omega| <= ||f||_1, and the L2 isometry of the Riesz transform. Self-citations, notably [18], appear only as historical remarks and are not used as black boxes; no author-generated uniqueness theorem is imported, since uniqueness is proved in Theorem 4.1. The Lewy-Stampacchia estimate is derived in full, not cited as an external result, and the paper explicitly distinguishes its unbounded-domain proof from the bounded-domain analogue in [17]. There is no ansatz smuggled in via citation and no renaming of a known result as a new derivation. The only substantive concern is a proof gap, not circularity: in the proof of Theorem 1.1 the chain ||mu||^2_L2 <= lambda integral |mu| dx = lambda integral (mu_+ + mu_-) dx assumes that mu_+ and mu_- have disjoint supports, which is not established and is generally false; the triangle inequality replaces the equality with an inequality and gives the same bound. That is a correctness issue to repair, not a circular step. Accordingly the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central derivation is essentially self-contained. It relies on standard fractional Sobolev space theory and variational inequality techniques; no empirical constants, fitted parameters, or new postulates are introduced. The only implicit assumption is the well-posedness of the fractional obstacle problem on the whole space, which the paper proves rather than assumes.

assumptions (4)

standard math Gagliardo representation of the fractional Laplacian and associated energy (Lemma 3.1)
Used throughout to express E_α as a double integral and to justify pointwise formulas for (-Δ)^{α/2}.
standard math Plancherel identity and Fourier transform conventions for (-Δ)^{α/2}
Used to relate the fractional Laplacian to Fourier multipliers and to prove that u ∈ H^α and that the Fourier multiplier of the Riesz transform has norm one.
standard math Weak lower semicontinuity of the fractional Sobolev norm and compactness in L^2 for energy-bounded sequences
Used in the direct method to establish existence of the minimizer of the obstacle energy.
domain assumption Completeness of the quasi-normed space L^{1,∞} and standard density of L^1∩L^2 in L^1 for singular integral extension
The proof extends the weak-type inequality from L^1∩L^2 to L^1 by density, which is standard but not explicitly justified.

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Pith. "Pith review of A dimension-free weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$." pith.science (2026). https://pith.science/paper/P75KNR52

@misc{pith2026260818068,
  author       = {Pith},
  title        = {Pith review of: A dimension-free weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbbR^n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P75KNR52}},
  note         = {Machine review of arXiv:2608.18068}
}

abstract

We show that the best constant in the weak-type $(1,1)$ bound for the vector Riesz transform on $\mathbb{R}^n$ is at most $2$, independent of the dimension $n$. The proof relies on a new decomposition of the input data involving an obstacle problem for the fractional Laplacian and an associated Lewy-Stampacchia type estimate on an unbounded domain. This settles a problem posed by E. M. Stein at the 1986 International Congress of Mathematicians.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A $p = 2$ dichotomy for uniform Riesz transform bounds on stratified Lie groups
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The full horizontal Riesz transform on any stratified Lie group satisfies a weak type (1,1) inequality with constant at most 2, uniformly in the dimension, step, and group structure.

Reference graph

Works this paper leans on

20 extracted references · 18 canonical work pages · cited by 1 Pith paper

[1]

J. M. Aldaz,The weak type(1,1)bounds for the maximal function associated to cubes grow to infinity with the dimension, Ann. of Math. (2)173(2011), no. 2, 1013–1023. MR2776368

work page 2011
[2]

Cotlar martingale transforms and related singular integrals

R. Ba˜ nuelos,Cotlar martingale transforms and related singular integrals, arXiv:2604.09365 (2026). 12 YUYUAN OUYANG, DANIEL SPECTOR, AND CODY B. STOCKDALE

work page Pith review arXiv 2026
[3]

Ba˜ nuelos and A

R. Ba˜ nuelos and A. Osekowski,Sharp martingale inequalities and applications to Riesz transforms on manifolds, Lie groups and Gauss space, J. Funct. Anal.269(2015), no. 6, 1652-1713. MR3373431

work page 2015
[4]

Ba˜ nuelos and G

R. Ba˜ nuelos and G. Wang,Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J.80(1995), no. 3, 575–600. MR1370111

work page 1995
[5]

A. P. Calderon and A. Zygmund,On the existence of certain singular integrals, Acta Math.88(1952), 85–139. MR52553

work page 1952
[6]

Di Nezza, G

E. Di Nezza, G. Palatucci, and E. Valdinoci,Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math.136(2012), no. 5, 521–573. MR2944369

work page 2012
[7]

Dragiˇ cevi´ c and A

O. Dragiˇ cevi´ c and A. Volberg,Bellman function and dimensionless estimates of classical and Ornstein- Uhlenbeck Riesz transforms, J. Operator Theory56(2006), no. 1, 167–198. MR2254388

work page 2006
[8]

Gustafsson and M

B. Gustafsson and M. Sakai,Properties of some balayage operators, with applications to quadrature domains and moving boundary problems, Nonlinear Anal.22(1994), no. 10, 1221–1245. MR1279981

work page 1994

Show all 20 references

[9]

Gustafsson and J

B. Gustafsson and J. Roos,Partial balayage on Riemannian manifolds, J. Math. Pures Appl. (9)118 (2018), 82–127. MR3852470

2018
[10]

Janakiraman,Weak-type estimates for singular integrals and the Riesz transform, Indiana Univ

P. Janakiraman,Weak-type estimates for singular integrals and the Riesz transform, Indiana Univ. Math. J.53(2004), no. 2, 533–555. MR2060044

2004
[11]

H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, S. Liu, L. Chen, Y. Wang, and Z. Wang,Automated conjecture resolution with formal verification, arXiv:2604.03789 (2026)

2026 arXiv
[12]

Kinderlehrer and G

D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, vol. 88, Academic Press, New York, 1980. Reprinted as Classics in Applied Mathematics31, SIAM, Philadelphia, 2000; Chapter II,§3 (Truncation) an...

1980
[13]

Levine and Y

L. Levine and Y. Peres,Strong spherical asymptotics for rotor-router aggregation and the divisible sandpile, Potential Anal.30(2009), no. 1, 1–27. MR2465710

2009
[14]

,Scaling limits for internal aggregation models with multiple sources, J. Anal. Math.111(2010), 151–219. MR2747064

2010
[15]

J. Liu, G. Gao, Z. Sun, B. Wu, S. Liu, J. Jiang, H. Ju, L. Chen, R. Cheng, and X. Zhang,Danus: Orchestrating Mathematical Reasoning Agents with Fact-Graph Memory, arXiv:2607.06447 (2026)

2026 arXiv
[16]

Polya,How to solve it: A new aspect of mathematical method, Princeton University Press, Princeton, NJ, 1945

G. Polya,How to solve it: A new aspect of mathematical method, Princeton University Press, Princeton, NJ, 1945. MR0011666

1945
[17]

Servadei and E

R. Servadei and E. Valdinoci,Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam.29(2013), no. 3, 1091–1126. MR3090147

2013
[18]

Spector and C

D. Spector and C. B. Stockdale,On the dimensional weak-type(1,1)bound for Riesz transforms, Com- mun. Contemp. Math.23(2021), no. 7, Paper No. 2050072, 19 pp. MR4329315

2021
[19]

E. M. Stein,Some results in harmonic analysis inR n, forn→∞, Bull. Amer. Math. Soc. (N.S.)9 (1983), no. 1, 71–73. MR699317

1983
[20]

E. M. Stein,Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 196–221. MR934224 Yuyuan Ouyang, School of Mat...

1986

Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

J. M. Aldaz,The weak type(1,1)bounds for the maximal function associated to cubes grow to infinity with the dimension, Ann. of Math. (2)173(2011), no. 2, 1013–1023. MR2776368

work page 2011

[2] [2]

Cotlar martingale transforms and related singular integrals

R. Ba˜ nuelos,Cotlar martingale transforms and related singular integrals, arXiv:2604.09365 (2026). 12 YUYUAN OUYANG, DANIEL SPECTOR, AND CODY B. STOCKDALE

work page Pith review arXiv 2026

[3] [3]

Ba˜ nuelos and A

R. Ba˜ nuelos and A. Osekowski,Sharp martingale inequalities and applications to Riesz transforms on manifolds, Lie groups and Gauss space, J. Funct. Anal.269(2015), no. 6, 1652-1713. MR3373431

work page 2015

[4] [4]

Ba˜ nuelos and G

R. Ba˜ nuelos and G. Wang,Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J.80(1995), no. 3, 575–600. MR1370111

work page 1995

[5] [5]

A. P. Calderon and A. Zygmund,On the existence of certain singular integrals, Acta Math.88(1952), 85–139. MR52553

work page 1952

[6] [6]

Di Nezza, G

E. Di Nezza, G. Palatucci, and E. Valdinoci,Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math.136(2012), no. 5, 521–573. MR2944369

work page 2012

[7] [7]

Dragiˇ cevi´ c and A

O. Dragiˇ cevi´ c and A. Volberg,Bellman function and dimensionless estimates of classical and Ornstein- Uhlenbeck Riesz transforms, J. Operator Theory56(2006), no. 1, 167–198. MR2254388

work page 2006

[8] [8]

Gustafsson and M

B. Gustafsson and M. Sakai,Properties of some balayage operators, with applications to quadrature domains and moving boundary problems, Nonlinear Anal.22(1994), no. 10, 1221–1245. MR1279981

work page 1994