Duke Mathematical Journal

Beltrami operators in the plane

Kari Astala, Tadeusz Iwaniec, and Eero Saksman

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We determine optimal Lp-properties for the solutions of the general nonlinear elliptic system in the plane of the form

f\overline{z}=H(z, fz), hLp(C),

where H is a measurable function satisfying |H(z,w1)−H(z,w2)|≤ k|w1w2| and k is a constant k<1.

We also establish the precise invertibility and spectral properties in Lp(C) for the operators

$I, IμT, and Tμ,

where T is the Beurling transform. These operators are basic in the theory of quasi-conformal mappings and in linear and nonlinear elliptic partial differential equations (PDEs) in two dimensions. In particular, we prove invertibility in Lp(C) whenever $1+||μ|| <p<1+1/||μ||.

We also prove related results with applications to the regularity of weakly quasiconformal mappings.

Article information

Duke Math. J. Volume 107, Number 1 (2001), 27-56.

First available in Project Euclid: 5 August 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 35J60: Nonlinear elliptic equations 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)


Astala, Kari; Iwaniec, Tadeusz; Saksman, Eero. Beltrami operators in the plane. Duke Math. J. 107 (2001), no. 1, 27--56. doi:10.1215/S0012-7094-01-10713-8. http://projecteuclid.org/euclid.dmj/1091736135.

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  • L. V. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math. (2) 72 (1960), 385--404.
  • L. V. Ahlfors and A. Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101--129.
  • K. Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37--60.
  • R. Bañuelos and G. Wang, Sharp inequalities for martingales and applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), 575--600.
  • B. V. Bojarski, Homeomorphic solutions of Beltrami systems (in Russian), Dokl. Akad. Nauk. SSSR (N.S.) 102 (1955), 661--664.
  • --. --. --. --., ``Quasiconformal mappings and general structure properties of systems of nonlinear equations elliptic in the sense of Lavrentiev'' in Convegno sulle Transformazioni Quasiconformi e Questioni Connesse (Rome, 1974), Sympos. Math. 18, Academic Press, London, 1976, 485--499.
  • B. Bojarski and T. Iwaniec, Quasiconformal mappings and non-linear elliptic equations in two variables I, II, Bull. Acad. Polon. Sci. Sèr. Sci. Math. Astronom. Phys. 22 (1974), 473--478., 479--484.
  • S. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), 253--272.
  • R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241--250.
  • F. W. Gehring and E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 388 (1966), 1--15.
  • J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Math. Monogr., Oxford Univ. Press,New York, (1993).
  • T. Iwaniec, ``Quasiconformal mapping problem for general nonlinear systems of partial differential equations'' in Convegno sulle Transformazioni Quasiconformi e Questioni Connesse (Rome, 1974), Sympos. Math. 18, Academic Press, London, 1976, 501--517.
  • --. --. --. --., Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen 1 (1982), 1--16.
  • --. --. --. --., The best constant in a BMO-inequality for the Beurling-Ahlfors transform, Michigan Math. J. 33 (1987), 387--394.
  • --. --. --. --., Hilbert transform in the complex plane and area inequalities for certain quadratic differentials, Michigan Math. J. 34 (1987), 407--434.
  • --------, ``$L^p$-theory of quasiregular mappings'' in Quasiconformal Space Mappings, Lecture Notes in Math. 1508, Springer, Berlin, 1992.
  • T. Iwaniec, P. Koskela, and G. Martin, Mappings of, BMO-bounded distortion and Beltrami-type operators, preprint, Univ. of Jyväskylä, Jyväskylä, Finland, 1998.
  • T. Iwaniec and A. Mamourian, ``On the first-order nonlinear differential systems with degeneration of ellipticity'' in Proceedings of the Second Finnish-Polish Summer School in Complex Analysis (Jyväskylä, 1983), Bericht 28, Univ. Jyväskylä, Jyväskylä, Finland, 1984, 41--52.
  • T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), 29--81.
  • --. --. --. --., Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25--57.
  • S. Janson, Characterizations of $H^1$ by singular integral transforms on martingales and $\real^n$, Math. Scand. 41 (1977), 140--152.
  • M. A. Lavrentiev, A general problem of the theory of quasi-conformal representation of plane regions (in Russian), Mat. Sb. 21 (1947), 285--320.
  • --------, The fundamental theorem of the theory of quasi-conformal mappings of two-dimensional domains, Izvestia Acad. Sc. USSR 12 (1948).
  • O. Lehto, ``Quasiconformal mappings and singular integrals'' in Convegno sulle Transformazioni Quasiconformi e Questioni Connesse (Rome, 1974), Sympos. Math. 18, Academic Press, London, 1976, 429--453.
  • O. Lehto and K. Virtanen, Quasiconformal Mappings in the Plane, 2d ed., Grundlehren Math. Wiss. 126, Springer, New York, 1973.
  • C. B. Morrey, On the solution of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126--166.
  • S. Müller, T. QI, and B. S. Yan, On a new class of elastic deformations not allowing for cavitation, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 217--243.
  • F. Nazarov and A. Volberg, Heating of the Beurling operator and the estimates of its norm, preprint, 2000.
  • St. Petermichl and A. Volberg, Heating of the Beurling operator and the critical exponents for Beltrami equation, preprint, 2000.
  • E. Reich, Some estimates for the two-dimensional Hilbert transform, J. Analyse Math. 18 (1967), 279--293.
  • H. M. Reimann and T. Rychener, Funktionen beschränkter mittlerer Oszillation, Lecture Notes in Math., 487, Springer, Berlin, 1975.
  • E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Princeton Univ. Press, Princeton, 1993.
  • A. Uchiyama, On the compactness of operators of Hankel type, Tohuku Math. J. (2) 30 (1978), 163--171.