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A solution to Morrey's problem in $\mathbb{R}^{2\times m}$

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Pith's one-line read For all large m, there exist p-homogeneous rank-one convex integrands on 2×m matrices that are nowhere quasiconvex.

desk verdict A genuinely new approach to Morrey's problem, but the key lower bound (17) has a load-bearing gap in the passage from R^{2n} to the torus/ball; worth refereeing, not yet citable. read the letter →

arxiv 2608.03488 v1 pith:NTTNJGWE submitted 2026-08-04 math.AP

Gabriele Cassese This is my paper · ORCID
classification math.AP MSC 49J4546B0935A15
keywords rank-oneconvexityquasiconvexityMorrey'sproblemhomogeneousintegrandscalculusofvariationsUMDspacesmartingaletransformslaminates
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

Quasiconvexity is the convexity condition behind weak lower semicontinuity and existence of minimisers in the calculus of variations; rank-one convexity is a necessary and much weaker condition. This paper proves that, in the space of 2×m matrices with m large, rank-one convexity does not imply quasiconvexity even among positively p-homogeneous integrands: for every p in (1,∞) and all large m there is a p-homogeneous rank-one convex F that is not quasiconvex at any matrix. The construction separates two thresholds, one for rank-one convexity and one for quasiconvexity, and shows the second exceeds the first by a growing power of the dimension. Since the envelope of the constructed integrand remains rank-one convex and p-homogeneous, the failure is inherited everywhere, giving a negative answer to the homogeneous Morrey problem in 2×m for large m, and hence to the original problem there. The same scheme also settles the square case for all large d and the 4×2 case for every p different from 2.

What carries the argument

The carrying object is the blockwise holomorphic/anti-holomorphic pair (P_n,Q_n): each 2×2 block of a 2×2n matrix is split into a component that respects complex multiplication and a component that conjugates, with values in ℓ_q^n. The argument hinges on the exact rank-one identity Q_n(R)=λ(R)P_n(R) for a unimodular scalar λ(R), which converts the rank-one constraint along martingale increments into a complex unimodular martingale transform. UMD-space estimates—bounds on taking predictable ±1 or unimodular transforms of martingales—then control the rank-one convexity threshold by a constant independent of n. On the quasiconvexity side, the explicit test map u_n(z)=(∏_{j=1}^n ar z_j)e^{-|z|^2

What would settle it

Take a torus-periodic or ball-truncated version of u_n and compute the ratio ‖Q_n(∇u)‖_{L^p(ℓ_q^n)} / ‖P_n(∇u)‖_{L^p(ℓ_q^n)} for increasing n with q>p+2. If the ratio does not grow like n^{1/(p+2)−1/q}, the gap underpinning Theorem 1.2 collapses; if it does grow, the proof's limiting step is concretely justified.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: take p∈(1,∞); there is m0(p) such that for all m≥m0(p) there is an integrand f:R^{2×m}→R that is p-homogeneous, rank-one convex, and nowhere quasiconvex. The witness is built from f_{C,p}(A)=C^p‖P_n(A)‖_{ℓ_q^n}^p − ‖Q_n(A)‖_{ℓ_q^n}^p, where P_n and Q_n extract, blockwise on 2×2n matrices, the complex-linear and conjugate-linear parts. A rank-one identity makes Q_n(M) a unimodular predictable transform of P_n(M) along any rank-one martingale, so the rank-one convexity threshold is bounded above by a UMD constant independent of n; an explicit smooth rapidly decaying test map gives the opposite threshold growing like n^{1/(p+2)−1/q} when q>p+2. Choosing q>p+2 a

Load-bearing premise

Everything rests on the unshown step that the large-ratio calculation made on the whole plane, with the smooth test map u_n, survives passage to the torus or unit ball in the quasiconvexity formula; the paper invokes 'standard limiting arguments' without proving that the heavy-tail growth is preserved by periodization or truncation.

Editorial extensions

If this is right

  • In R^{2×m} with m large, the homogeneous Morrey problem has a negative answer for every p>1, and hence the original Morrey problem is also settled negatively there.
  • The same threshold-separation scheme gives, for all large d, a conjugation- and transposition-invariant p-homogeneous rank-one convex integrand on R^{d×d} that is nowhere quasiconvex.
  • In dimension 4×2, for every p∈(1,∞) with p≠2, the problem fails as well, so non-Hilbertian geometry alone can produce counterexamples in a fixed low dimension.
  • In 2×m for large m there exist smooth, coercive integrands satisfying a uniform Legendre–Hadamard inequality that are nevertheless not quasiconvex at 0; consequently quasiconvexity is a nonlocal condition in this setting.
  • In the square case with p=2 and q=4, a fourth-degree homogeneous polynomial gives an explicit rank-one convex but not quasiconvex integrand, and degree three is the lowest degree at which such a phenomenon can occur.

Reading between the lines

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

  • Editorial extension: since the separation mechanism needs q>p+2, the proof cannot be pushed to the Hilbertian 2×2 case; this suggests that any proof of a positive result there must exploit structure absent from the ℓ_q spaces used here.
  • Editorial extension: the threshold-separation template may be tested on other pairs of linear maps (P,Q); the only needed ingredients are a rank-one phase identity and a family of test maps whose ratio grows with n, so one can search for counterexamples below m0(p) by checking those two ingredients.
  • Editorial extension: the paper leaves m0(p) qualitative; a numerical experiment at moderate n, comparing the explicit heavy-tail lower bound with the best available UMD constants for ℓ_q^n, could locate the first separating dimension and support or refute the paper's suggestion that m0=4 may be reachable.
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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

2 major / 4 minor

Summary. This paper addresses Morrey's problem in the class of p-homogeneous integrands on R^{2×m}. The main theorem (Theorem 1.2) asserts that for every p∈(1,∞) and all m≥m_0(p) there is a continuous p-homogeneous rank-one convex integrand F:R^{2×m}→R that is nowhere quasiconvex. The proof uses a threshold-separation scheme: for projections P_n,Q_n onto ℓ_q^n, define F_{n,p,q,C}=C^p∥P_n A∥^p − ∥Q_n A∥^p. The rank-one convexity threshold is bounded above by a UMD martingale-transform constant (independent of n), while quasiconvexity at 0 is bounded below by the ratio for an explicit Schwartz test map u_n, which grows like n^{1/(p+2)−1/q} for q>p+2. Choosing C between the two thresholds and taking the rank-one convex envelope yields the counterexample. Variants give the square case and dimension 4×2.

Significance. If correct, this is a major advance: it settles the homogeneous Morrey problem negatively in all sufficiently large dimensions (and the ordinary Morrey problem in 2×m for large m), a question open since Morrey's original work. The martingale/UMD method is a new and potentially powerful tool in the calculus of variations, and the paper also contains several by-products (non-invariance of quasiconvexity under transposition, smooth counterexamples, non-locality, an explicit p=3 example, and a claimed Lean formalization). The estimates are parameter-free in the sense that no fitted constants are used: C_rc is controlled by UMD constants and C_qc by an explicit test map. However, the proof as written has a false algebraic identity in Lemma 4.1 and an unproved limiting step from (15) to (17); both are repairable but require non-negligible corrections.

major comments (2)
  1. [Section 4.2, Lemma 4.1] The identity Q_nR = lambda(R)P_nR is false as stated. With R=a⊗ξ, the proof's own formulas give alpha_j = a\bar v_j/2 and beta_j = a v_j/2, so beta_j = (a/\bar a) \overline{alpha_j}, not (a/\bar a)alpha_j. For a=(1,0), x=(1,0,0,1), P=(1/2,-i/2), Q=(1/2,i/2), no single lambda works. This invalidates the proofs of Proposition 4.1 and Proposition 4.2 as written. The repair is straightforward: replace Q_n by its coordinatewise conjugate (an isometry of ell_q^n) or apply the UMD transform bound to \overline{P_n(M)}. Please correct the lemma and all dependent statements.
  2. [Section 4.3, (15) to (17)] The passage from the unbounded-domain identity (15) for the Schwartz function u_n to the lower bound (17) for C_qc is not shown. C_qc is defined via Dacorogna's formula on the unit ball/torus, so one must truncate u_n to a bounded domain, make it zero on the boundary or periodic, rescale to the unit cell, and prove that the quotient converges to the unbounded ratio. This is load-bearing because the numerator is concentrated on the thin event {min_j R_j <= c n^{-1/(p+2)}}; the cutoff must be identically 1 on that event and the error must be small compared to n^{delta}. Please provide the full limiting argument; without it (17) is unsupported.
minor comments (4)
  1. [Section 4.3, Remark 4.2] The stated bounds for q=p+2 and q=∞ are given without proof. If they are not used in the main theorems, label them as conjectures or provide sketches; otherwise a reader cannot verify them.
  2. [Section 1.3 / Proposition 2.3] The statement 'quasiconvexity at any point implies quasiconvexity at 0' for homogeneous integrands is used to conclude 'nowhere quasiconvex' in Theorem 1.2. A short proof or reference would improve readability; the argument via the finite-valued homogeneous quasiconvex envelope is not spelled out.
  3. [Section 4.5, Proposition 4.2] After the conjugation fix in Lemma 4.1, the phrase 'thanks to Lemma 4.1 it suffices to prove that U is zig-zag concave' should be updated: the rank-one variation becomes (x+th, y+tεh) with ε = \bar λ, not ε=λ. This is not merely notational.
  4. [Throughout] There are a few typographical issues (e.g., the citation of Lemma 2.2 contains a stray 'and'; the acknowledgements contain a corrupted name). These do not affect the mathematics.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: thresholds are bounded by independent UMD and explicit test-map estimates; the flagged Section 4.3 transfer is an unproved gap, not a circular reduction.

full rationale

The central claim Theorem 1.2 rests on a threshold separation: C_rc is bounded above by the dimension-free UMD constant UMD_{C,p}(ℓ_q) via Prop 4.1, Prop 2.5 and Prop 2.7(5), while C_qc is bounded below by explicit Schwartz test maps producing identity (15) and the order-statistic bound (16). No fitted quantity is used to infer the result: C is chosen strictly between two independently estimated thresholds, and neither threshold is defined in terms of the other or of the conclusion. The martingale–laminate correspondence (Props 2.1–2.5) is either proved in the text or cited to standard external references ([63], [25], [68]); the self-citations [19] and [17] support auxiliary formulas and are not used to assume Theorem 1.2. The genuinely weak point is Section 4.3's one-sentence 'By standard limiting arguments, this implies (17)': the transfer of the unbounded-domain identity (15) to Dacorogna's torus/ball test maps is not shown, so if the heavy-tail estimate is lost under periodization the lower bound for C_qc, and hence the theorem, collapses. That is an omitted proof and a correctness risk, not a circular reduction: (17) is not equivalent to C_qc's definition by construction, and the test map is not chosen using C_qc. The formal Lean proof [18] is machine-checked independent support. Self-citations occur but are not load-bearing in a circular sense, so the score is 2 rather than 0.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central construction rests on three kinds of input: (i) standard envelope and quasiconvexity facts of the calculus of variations (Dacorogna formula, homogeneity propagation); (ii) imported quantitative martingale-transform bounds for non-Hilbertian spaces (UMD constants of ell_q from [67],[73], square-function and Khintchine inequalities for Schatten spaces from [66],[44], and the sharp norm of the squared Beurling-Ahlfors transform from [28]); (iii) the dyadic prelaminate-martingale correspondence, which is attributed to the author's own prior work [19] and only sketched here. The genuinely new content, whose correctness I could partially verify, is the exact rank-one identity for (P_n, Q_n), the explicit Schwartz witness, and the order-statistic separation of the two thresholds. There are no data-fitted parameters; the constants q, C, and m_0 are regime or existential choices, and no new entities are postulated.

free parameters (4)
  • C (threshold constant) = exists in the interval (C_rc, C_qc); not numerically specified
    The integrand family F_{C,p}(A) = C^p norm(P(A))^p - norm(Q(A))^p requires selecting C strictly between the two thresholds; existence of such C is proven via (10) and (17), so it is a region choice, not a fit to external data.
  • q (Schatten/ell_q exponent) = any q > p+2
    The separation exponent delta_{p,q} = 1/(p+2) - 1/q requires q > p+2; larger q gives a wider gap. A designer choice, not fitted.
  • m_0(p) (critical dimension) = exists; non-quantitative
    The theorem asserts existence of m_0(p) via the order-statistic estimate; Remark 4.3 states the estimates are qualitative and no quantitative m_0 is computed.
  • alpha = 6(5/6)^5 (Burkholder scaling, p=3 example) = 55/64, approximately 2.41 (explicit)
    Hand-chosen to make the zig-zag concavity computation work in Section 4.5; an explicit proof constant, not data-fitted.
assumptions (5)
  • domain assumption UMD_{C,p}(ell_q) < infinity with constant independent of n (uniform bound on unimodular martingale transforms on ell_q^n)
    Imported from Randrianantoanina [67] and Yaroslavtsev [73] via Prop 2.7(2),(5). Caps C_rc uniformly in n in Prop 4.1; load-bearing for the threshold separation.
  • domain assumption Dyadic prelaminate and rank-one Rademacher martingale correspondence (identity between inf over dyadic prelaminates and rank-one convex envelope)
    Props 2.4-2.5, attributed to the author's [19]; proof sketched in the paper. Converts the martingale estimate into a rank-one convexity statement. If the dyadic restriction were lossy, C_rc would be miscalculated.
  • standard math Standard quasiconvexity and envelope facts: Dacorogna formula, envelope duality for continuous integrands, homogeneity propagation (Prop 2.3)
    Used throughout to pass from threshold estimates to existence of rank-one convex nowhere-quasiconvex integrands (proof of Theorem 1.2 and Prop 4.3).
  • domain assumption norm(B^2)_{L^p(R^2)} > p* - 1 for p != 2 (sharp L^p norm of the square of the Beurling-Ahlfors transform)
    From [28, Theorem 1.1] (Dragicevic), cited in Section 6; the entire 4x2 example (Theorem 1.4) rests on this strict inequality.
  • domain assumption Noncommutative square-function and Khintchine inequalities for Schatten spaces (Props 5.3, 5.4)
    Used in the square case (Section 5.2) to convert rank-one increment estimates into L^p norm comparisons; cited from [66] and [44].

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Pith. "Pith review of A solution to Morrey's problem in $\mathbb{R}^{2\times m}$." pith.science (2026). https://pith.science/paper/NTTNJGWE

@misc{pith2026260803488,
  author       = {Pith},
  title        = {Pith review of: A solution to Morrey's problem in $\mathbbR^2\times m$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NTTNJGWE}},
  note         = {Machine review of arXiv:2608.03488}
}
abstract

We construct homogeneous rank-one convex integrands $F\colon \mathbb{R}^{2\times m} \to \mathbb R$ that are nowhere quasiconvex when $m$ is large.

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Works this paper leans on

73 extracted references · 73 canonical work pages

  1. [1]

    A regularity theorem for minimizers of quasiconvex integrals,

    E. Acerbi and N. Fusco, “A regularity theorem for minimizers of quasiconvex integrals,”Arch. Rat. Mech. and Anal., vol. 99, no. 3, pp. 261–281, 1987

    work page 1987

  2. [2]

    The local Burkholder functional, quasiconvexity and Geometric Function Theory

    K. Astala, D. Faraco, A. Guerra, A. Koski, and J. Kristensen, “The local Burkholder functional, quasiconvexity and Geometric Function Theory,” arXiv:2309.03495, 2023

    work page Pith review arXiv 2023

  3. [3]

    Lower semicontinuity, Stoilow factorization and principal maps,

    K. Astala, D. Faraco, A. Guerra, A. Koski, and J. Kristensen, “Lower semicontinuity, Stoilow factorization and principal maps,”Commun. Pure Appl. Anal., vol. 23, no. 10, pp. 1608–1645, 2024

    work page 2024

  4. [4]

    Burkholder integrals, Morrey’s problem and quasiconformal mappings,

    K. Astala, T. Iwaniec, I. Prause, and E. Saksman, “Burkholder integrals, Morrey’s problem and quasiconformal mappings,”J. Amer. Math. Soc., vol. 25, no. 2, pp. 507–531, 2012

    work page 2012

  5. [5]

    A hunt for sharpLp-estimates and rank-one convex variational integrals,

    K. Astala, T. Iwaniec, I. Prause, and E. Saksman, “A hunt for sharpLp-estimates and rank-one convex variational integrals,”Filomat, vol. 29, no. 2, pp. 245–261, 2015

    work page 2015

  6. [6]

    Sets of gradients with no rank-one connections,

    J. M. Ball, “Sets of gradients with no rank-one connections,”J. Math. Pures Appl., vol. 69, no. 3, pp. 241–259, 1990

    work page 1990

  7. [7]

    The foundational inequalities of D. L. Burkholder and some of their ramifications,

    R. Bañuelos, “The foundational inequalities of D. L. Burkholder and some of their ramifications,”Ill. J. Math., vol. 54, no. 3, pp. 789–868, 2010

    work page 2010

  8. [8]

    L p boundsfortheBeurling–Ahlforstransform,

    R.BañuelosandP.Janakiraman,“L p boundsfortheBeurling–Ahlforstransform,”Trans. Amer. Math. Soc.,vol.360, no. 7, pp. 3603–3612, 2008

    work page 2008

Show all 73 references
  1. [9]

    Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms,

    R. Bañuelos and G. Wang, “Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms,”Duke Math. J., vol. 80, no. 3, pp. 575–600, 1995

    1995

  2. [10]

    Bhatia,Matrix Analysis(Graduate Texts in Mathematics)

    R. Bhatia,Matrix Analysis(Graduate Texts in Mathematics). New York: Springer, 1997, vol. 169

    1997

  3. [11]

    Subordination by conformal martingales inLp and zeros of Laguerre polynomials,

    A. Borichev, P. Janakiraman, and A. Volberg, “Subordination by conformal martingales inLp and zeros of Laguerre polynomials,”Duke Math. J., vol. 162, no. 5, pp. 889–924, 2013

    2013

  4. [12]

    Laminates meet Burkholder functions,

    N. Boros, L. Székelyhidi Jr., and A. Volberg, “Laminates meet Burkholder functions,”J. Math. Pures Appl., vol. 100, no. 5, pp. 687–700, 2013. 24 REFERENCES

    2013

  5. [13]

    Some remarks on banach spaces in which martingale difference sequences are unconditional,

    J. Bourgain, “Some remarks on banach spaces in which martingale difference sequences are unconditional,”Ark. Mat., vol. 21, no. 2, pp. 163–168, Dec. 1983

    1983

  6. [14]

    Martingale transforms,

    D. L. Burkholder, “Martingale transforms,”Ann. Math. Stat., vol. 37, no. 6, pp. 1494–1504, Dec. 1966

    1966

  7. [15]

    Sharp inequalities for martingales and stochastic integrals,

    D. L. Burkholder, “Sharp inequalities for martingales and stochastic integrals,” inColloque Paul Lévy sur les pro- cessus stochastiques, ser. Astérisque 157-158, Société mathématique de France, 1988, pp. 75–94

    1988

  8. [16]

    SharpL p estimates of powers of the complex Riesz transform,

    A. Carbonaro, O. Dragičević, and V. Kovač, “SharpL p estimates of powers of the complex Riesz transform,” Mathematische Annalen, vol. 386, no. 1–2, pp. 1081–1125, 2023

    2023

  9. [17]

    Cassese,Korn’s inequality from the viewpoint of calculus of variations, Preprint, 2026

    G. Cassese,Korn’s inequality from the viewpoint of calculus of variations, Preprint, 2026. eprint:2603.22431

    2026 arXiv

  10. [18]

    Lean formalisation of morrey counterexamples in 2×m

    G. Cassese. “Lean formalisation of morrey counterexamples in 2×m.” GitHub repository, Accessed: Jul. 31, 2026. [Online]. Available:https://github.com/Gabriele-Cassese/MorreyTwoComponentsLean

    2026

  11. [19]

    Martingales, laminates and minimal Korn inequalities,

    G. Cassese, “Martingales, laminates and minimal Korn inequalities,”J. Math. Pures Appl., vol. 213, Sep. 2026, Art. no. 103941

    2026

  12. [20]

    Rank-One Convexity Implies Quasi-Convexity on Certain Hypersurfaces,

    N. Chaudhuri and S. Müller, “Rank-One Convexity Implies Quasi-Convexity on Certain Hypersurfaces,”Proc. Roy. Soc. Edinburgh Sect. A, vol. 133, no. 6, pp. 1263–1272, 2003

    2003

  13. [21]

    On coercive variational integrals,

    C. Y. Chen and J. Kristensen, “On coercive variational integrals,”Nonlinear Anal., vol. 153, pp. 213–229, 2017

    2017

  14. [22]

    Polyconvexity Equals Rank-One Convexity for Connected Isotropic Sets inM 2×2,

    S. Conti, C. De Lellis, S. Müller, and M. Romeo, “Polyconvexity Equals Rank-One Convexity for Connected Isotropic Sets inM 2×2,”C. R. Math. Acad. Sci. Paris, vol. 337, no. 4, pp. 233–238, 2003

    2003

  15. [23]

    Rank-one convex functions on2×2symmetric matrices and laminates on rank-three lines,

    S. Conti, D. Faraco, F. Maggi, and S. Müller, “Rank-one convex functions on2×2symmetric matrices and laminates on rank-three lines,”Calc. Var. Partial Differential Equations, vol. 24, no. 4, pp. 479–493, 2005

    2005

  16. [24]

    Some remarks on tangent martingale difference sequences inL1-spaces,

    S. Cox and M. Veraar, “Some remarks on tangent martingale difference sequences inL1-spaces,”Electronic Com- munications in Probability, vol. 12, no. 40, pp. 421–433, 2007

    2007

  17. [25]

    Dacorogna,Direct Methods in the Calculus of Variations(Applied Mathematical Sciences, 78), 2nd ed

    B. Dacorogna,Direct Methods in the Calculus of Variations(Applied Mathematical Sciences, 78), 2nd ed. Springer, 2008

    2008

  18. [26]

    Some examples of rank one convex functions in dimension two,

    B. Dacorogna, J. Douchet, W. Gangbo, and J. Rappaz, “Some examples of rank one convex functions in dimension two,”Proc. Roy. Soc. Edinburgh Sect. A, vol. 114, no. 1-2, pp. 135–150, 1990

    1990

  19. [27]

    A counterexample in the vectorial calculus of variations,

    B. Dacorogna and P. Marcellini, “A counterexample in the vectorial calculus of variations,” inMaterial Instabilities in Continuum Mechanics, ser. Oxford Science Publications, J. M. Ball, Ed., New York: Oxford University Press, 1988, pp. 77–83

    1988

  20. [28]

    Some remarks on theLp estimates for powers of the Ahlfors–Beurling operator,

    O. Dragičević, “Some remarks on theLp estimates for powers of the Ahlfors–Beurling operator,”Archiv der Mathe- matik, vol. 96, no. 5, pp. 463–471, 2011

    2011

  21. [29]

    Quasiconvexity and partial regularity in the calculus of variations,

    L. C. Evans, “Quasiconvexity and partial regularity in the calculus of variations,”Arch. Rat. Mech. Anal., vol. 95, no. 3, pp. 227–252, Sep. 1986

    1986

  22. [30]

    Tartar’s conjecture and localization of the quasiconvex hull inR2×2,

    D. Faraco and L. Székelyhidi Jr., “Tartar’s conjecture and localization of the quasiconvex hull inR2×2,”Acta Math., vol. 200, no. 2, pp. 279–305, 2008

    2008

  23. [31]

    Analysis of concentration and oscillation effects generated by gradients,

    I. Fonseca, S. Müller, and P. Pedregal, “Analysis of concentration and oscillation effects generated by gradients,” SIAM J. Math. Anal., vol. 29, no. 3, pp. 736–756, 1998

    1998

  24. [32]

    C1,α Partial regularity of functions minimising quasiconvex integrals,

    N. Fusco and J. E. Hutchinson, “C1,α Partial regularity of functions minimising quasiconvex integrals,”Manuscripta Math., vol. 54, no. 1–2, pp. 121–143, 1985

    1985

  25. [33]

    On singular integral and martingale transforms,

    S. Geiss, S. Montgomery-Smith, and E. Saksman, “On singular integral and martingale transforms,”Trans. Amer. Math. Soc., vol. 362, no. 2, pp. 553–575, 2010

    2010

  26. [34]

    Partial Regularity of Minimizers of Quasiconvex Integrals,

    M. Giaquinta and G. Modica, “Partial Regularity of Minimizers of Quasiconvex Integrals,”Ann. Inst. H. Poincaré Anal. Non Linéaire, vol. 3, no. 3, pp. 185–208, 1986

    1986

  27. [35]

    From Microstructure-Independent Formulas for Composite Materials to Rank-One Convex, Non- quasiconvex Functions,

    Y. Grabovsky, “From Microstructure-Independent Formulas for Composite Materials to Rank-One Convex, Non- quasiconvex Functions,”Arch. Ration. Mech. Anal., vol. 227, no. 2, pp. 607–636, 2018

    2018

  28. [36]

    Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands,

    A. Guerra and J. Kristensen, “Automatic Quasiconvexity of Homogeneous Isotropic Rank-One Convex Integrands,” Arch. Ration. Mech. Anal., vol. 245, no. 1, pp. 479–500, 2022

    2022

  29. [37]

    Guerra, J

    A. Guerra, J. Kristensen, and B. Rait,ă,Oscillation and concentration under constant rank constraints, In prepara- tion

  30. [38]

    Numerical evidence towards a positive answer to Morrey’s problem,

    A. Guerra and R. Teixeira da Costa, “Numerical evidence towards a positive answer to Morrey’s problem,”Rev. Mat. Iberoam., vol. 38, no. 2, pp. 601–614, 2022

    2022

  31. [39]

    Two-by-two upper triangular matrices and Morrey’s conjecture,

    T. L. J. Harris, B. Kirchheim, and C.-C. Lin, “Two-by-two upper triangular matrices and Morrey’s conjecture,”Calc. Var. Partial Differential Equations, vol. 57, no. 3, pp. 1–12, 3 Apr. 2018, Art. no. 73, 12 pages

    2018

  32. [40]

    Preservation of quasiconvexity and quasimonotonicity in polynomial approximation of variational prob- lems,

    S. Heinz, “Preservation of quasiconvexity and quasimonotonicity in polynomial approximation of variational prob- lems,” Ph.D. thesis, Humboldt-Universität zu Berlin, 2008

    2008

  33. [41]

    Quasiconvexity Equals Lamination Convexity for Isotropic Sets of2×2Matrices,

    S. Heinz, “Quasiconvexity Equals Lamination Convexity for Isotropic Sets of2×2Matrices,”Adv. Calc. Var., vol. 8, no. 1, pp. 43–53, 2015

    2015

  34. [42]

    Hytönen, J

    T. Hytönen, J. van Neerven, M. Veraar, and L. Weis,Analysis in Banach Spaces. Volume I: Martingales and Littlewood-Paley Theory(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge). Springer, 2016, vol. 63

    2016

  35. [43]

    Hytönen, J

    T. Hytönen, J. van Neerven, M. Veraar, and L. Weis,Analysis in Banach Spaces. Volume III: Harmonic Analysis and Spectral Theory(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge). Springer, 2023, vol. 76

    2023

  36. [44]

    Hytönen, J

    T. Hytönen, J. van Neerven, M. Veraar, and L. Weis,Analysis in Banach Spaces. Volume II: Probabilistic Methods and Operator Theory(Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge). Cham: Springer, 2017, vol. 67

    2017

  37. [45]

    Extremal inequalities in Sobolev spaces and quasiconformal mappings,

    T. Iwaniec, “Extremal inequalities in Sobolev spaces and quasiconformal mappings,”Z. Anal. Anwend., vol. 1, no. 6, pp. 1–16, 1982

    1982

  38. [46]

    Nonlinear Cauchy-Riemann operators inRn,

    T. Iwaniec, “Nonlinear Cauchy-Riemann operators inRn,”Trans. Amer. Math. Soc., vol. 354, no. 5, pp. 1961–1995, 2002

    1961

  39. [47]

    Riesz transforms and related singular integrals,

    T. Iwaniec and G. Martin, “Riesz transforms and related singular integrals,”J. Reine Angew. Math., vol. 473, pp. 25– 58, 1996

    1996

  40. [48]

    Noncommutative good-λinequalities,

    Y. Jiao, A. Osękowski, and L. Wu, “Noncommutative good-λinequalities,”Journal of the European Mathematical Society, 2025, Published online first, 7 May 2025. REFERENCES 25

    2025

  41. [49]

    Characterizations of Young measures generated by gradients,

    D. Kinderlehrer and P. Pedregal, “Characterizations of Young measures generated by gradients,”Arch. Ration. Mech. Anal., vol. 115, no. 4, pp. 329–365, 1991

    1991

  42. [50]

    On rank-one convex functions that are homogeneous of degree one,

    B. Kirchheim and J. Kristensen, “On rank-one convex functions that are homogeneous of degree one,”Arch. Ration. Mech. Anal., vol. 221, no. 1, pp. 527–558, 2016

    2016

  43. [51]

    On the non-locality of quasiconvexity,

    J. Kristensen, “On the non-locality of quasiconvexity,”Ann. Inst. H. Poincaré C Anal. Non Linéaire, vol. 16, no. 1, pp. 1–13, 1999

    1999

  44. [52]

    An introduction to generalized young measures,

    J. Kristensen and B. Raiţă, “An introduction to generalized young measures,” Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany, Lecture Note 45/2020, Nov. 2020, Published 3 November 2020

    2020

  45. [53]

    Inégalités de Khintchine dansC p (1< p <∞),

    F. Lust-Piquard, “Inégalités de Khintchine dansC p (1< p <∞),”C. R. Acad. Sci. Paris Sér. I Math., vol. 303, no. 7, pp. 289–292, 1986

    1986

  46. [54]

    Non-commutative Khintchine and Paley inequalities,

    F. Lust-Piquard and G. Pisier, “Non-commutative Khintchine and Paley inequalities,”Arkiv för Matematik, vol. 29, no. 1–2, pp. 241–260, 1991

    1991

  47. [55]

    Quasi-convexity and the Lower Semicontinuity of Multiple Integrals,

    C. B. Morrey Jr., “Quasi-convexity and the Lower Semicontinuity of Multiple Integrals,”Pacific Journal of Mathe- matics, vol. 2, no. 1, pp. 25–53, 1952

    1952

  48. [56]

    Multiple Integral Problems in the Calculus of Variations and Related Topics,

    C. B. Morrey Jr., “Multiple Integral Problems in the Calculus of Variations and Related Topics,”Annali della Scuola Normale Superiore di Pisa – Classe di Scienze, 3, vol. 14, no. 1, pp. 1–61, 1960

    1960

  49. [57]

    C. B. Morrey Jr.,Multiple Integrals in the Calculus of Variations(Die Grundlehren der mathematischen Wis- senschaften). Berlin: Springer-Verlag, 1966, vol. 130, 506 pp

    1966

  50. [58]

    Rank-one Convexity Implies Quasiconvexity on Diagonal Matrices,

    S. Müller, “Rank-one Convexity Implies Quasiconvexity on Diagonal Matrices,”Int. Math. Res. Not., vol. 1999, no. 20, pp. 1087–1095, 1999

    1999

  51. [59]

    Quasiconvexity is not invariant under transposition,

    S. Müller, “Quasiconvexity is not invariant under transposition,”Proc. R. Soc. Edinburgh Sec. A Math., vol. 130, no. 2, pp. 389–395, 2000

    2000

  52. [60]

    Sharp localized inequalities for Fourier multipliers,

    A. Osękowski, “Sharp localized inequalities for Fourier multipliers,”Canad. J. Math., vol. 66, no. 6, pp. 1358–1381, 2014

    2014

  53. [61]

    Osękowski,Sharp Martingale and Semimartingale inequalities(IMPAN Monogr

    A. Osękowski,Sharp Martingale and Semimartingale inequalities(IMPAN Monogr. Mat. (N. S.), 72). Springer, 2012, vol. 72

    2012

  54. [62]

    On the UMD constant of the spaceℓN 1 ,

    A. Osękowski, “On the UMD constant of the spaceℓN 1 ,”Colloq. Math., vol. 142, no. 1, pp. 135–147, 2016

    2016

  55. [63]

    Laminates and microstructure,

    P. Pedregal, “Laminates and microstructure,”European J. Appl. Math., vol. 4, no. 2, pp. 121–149, 1993

    1993

  56. [64]

    A Note on Quasiconvexity and Rank-One Convexity for2×2Matrices,

    P. Pedregal and V. Šverák, “A Note on Quasiconvexity and Rank-One Convexity for2×2Matrices,”J. Convex Anal., vol. 5, no. 1, pp. 107–117, 1998

    1998

  57. [65]

    Pisier,Martingales in Banach Spaces(Cambridge Studies in Advanced Mathematics 155)

    G. Pisier,Martingales in Banach Spaces(Cambridge Studies in Advanced Mathematics 155). Cambridge University Press, 2016, 590 pp

    2016

  58. [66]

    Non-commutativeLp-spaces,

    G. Pisier and Q. Xu, “Non-commutativeLp-spaces,” inHandbook of the Geometry of Banach Spaces, W. B. Johnson and J. Lindenstrauss, Eds., vol. 2, Amsterdam: North-Holland, 2003, pp. 1459–1517

    2003

  59. [67]

    Non-commutative martingale transforms,

    N. Randrianantoanina, “Non-commutative martingale transforms,”Journal of Functional Analysis, vol. 194, no. 1, pp. 181–212, 2002

    2002

  60. [68]

    Rindler,Calculus of Variations(Universitext)

    F. Rindler,Calculus of Variations(Universitext). Springer, 2018

    2018

  61. [69]

    Laminates supported on cubes,

    G. Sebestyén and L. Székelyhidi Jr., “Laminates supported on cubes,”Journal of Convex Analysis, vol. 24, no. 4, pp. 1217–1237, 2017

    2017

  62. [70]

    Rank-one convexity does not imply quasiconvexity,

    V. Šverák, “Rank-one convexity does not imply quasiconvexity,”Proc. R. Soc. Edinb., Sect. A, Math., vol. 120, no. 1–2, pp. 185–189, 1992

    1992

  63. [71]

    Tomczak-Jaegermann,Banach–Mazur Distances and Finite-Dimensional Operator Ideals(Pitman Monographs and Surveys in Pure and Applied Mathematics)

    N. Tomczak-Jaegermann,Banach–Mazur Distances and Finite-Dimensional Operator Ideals(Pitman Monographs and Surveys in Pure and Applied Mathematics). Harlow, Essex: Longman Scientific & Technical, 1989, vol. 38, Copublished in the United States with John Wiley & Sons, Inc., New York

    1989

  64. [72]

    Burkholder’s function via Monge–Ampère equation,

    V. Vasyunin and A. Volberg, “Burkholder’s function via Monge–Ampère equation,”Ill. J. Math., vol. 54, no. 4, pp. 1393–1428, 2010

    2010

  65. [73]

    Even fourier multipliers and martingale transforms in infinite dimensions,

    I. S. Yaroslavtsev, “Even fourier multipliers and martingale transforms in infinite dimensions,”Indag. Math., vol. 29, no. 5, pp. 1290–1309, 2018

    2018

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