Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular. (English) Zbl 1025.30018
In a recent paper by K. Astala, T. Iwaniec and E. Saksman [Duke Math. J. 107, 27-56 (2002; Zbl 1009.30015)] it is shown that any solution \(f\in W^{1,q}_{\text{loc}}\) of the Beltrami equation with \(|\mu|_\infty= k<1\) is continuous, thus quasiregular, if \(q>k+1\) and that \(q<k+1\) is not sufficient for this result. The authors show that \(q=k+1\) is sufficient. The proof is based on a sharp weighted estimate of the Ahlfors-Beurling operator.
Reviewer: O.Martio (Helsinki)
MSC:
| 30C62 | Quasiconformal mappings in the complex plane |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 35K05 | Heat equation |
| 42C15 | General harmonic expansions, frames |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
| 47B38 | Linear operators on function spaces (general) |
Citations:
Zbl 1009.30015References:
| [1] | K. Astala, Area distortion of quasiconformal mappings , Acta Math. 173 (1994), 37–60. · Zbl 0815.30015 · doi:10.1007/BF02392568 |
| [2] | K. Astala, T. Iwaniec, and E. Saksman, Beltrami operators , preprint, 1999. |
| [3] | A. Baernstein II and S. J. Montgomery-Smith, “Some conjectures about integral means of \(\partial f\) and \(\overline\partial f\)” in Complex Analysis and Differential Equations (Uppsala, Sweden, 1999) , ed. Ch. Kiselman, Acta. Univ. Upsaliensis Univ. C Organ. Hist. 64 , Uppsala Univ. Press, Uppsala, Sweden, 1999, 92–109. · Zbl 0966.30001 |
| [4] | R. Bañuelos and A. Lindeman, A martingale study of the Beurling-Ahlfors transform in \(\mathbbR^n\) , J. Funct. Anal. 145 (1997), 224–265. · Zbl 0876.60026 · doi:10.1006/jfan.1996.3022 |
| [5] | R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms , Duke Math. J. 80 (1995), 575–600. · Zbl 0853.60040 · doi:10.1215/S0012-7094-95-08020-X |
| [6] | B. V. Bojarski, Homeomorphic solutions of Beltrami systems (in Russian), Dokl. Akad. Nauk. SSSR (N.S.) 102 (1955), 661–664. |
| [7] | –. –. –. –., Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients , Mat. Sb. (N.S.) 85 , no. 43 (1957), 451–503. |
| [8] | –. –. –. –., “Quasiconformal mappings and general structure properties of systems of non linear equations elliptic in the sense of Lavrentiev” in Convegno sulle Transformazioni Quasiconformie Questioni Connesse (Rome, 1974) , Sympos. Math. 18 , Academic Press, London, 1976, 485–499. |
| [9] | B. V. Bojarski and T. Iwaniec, Quasiconformal mappings and non-linear elliptic equations in two variables, I, II , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 473–484. · Zbl 0317.35038 |
| [10] | S. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities , Trans. Amer. Math. Soc. 340 (1993), 253–272. · Zbl 0795.42011 · doi:10.2307/2154555 |
| [11] | R. Fefferman, C. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations , Ann. of Math. (2) 134 (1991), 65–124. JSTOR: · Zbl 0770.35014 · doi:10.2307/2944333 |
| [12] | J. Garcia-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities And Related Topics , North-Holland Math. Stud. 116 , North-Holland, Amsterdam, 1985. · Zbl 0578.46046 |
| [13] | F. W. Gehring, “Open problems” in Proceedings of the Romanian-Finnish Seminar on Teichmuller Spaces and Quasiconformal Mappings (Brasov, Romania) , Acad. Soc. Rep. Romania, Bucharest, 1969, 306. |
| [14] | –. –. –. –., The \(L^p\)-integrability of the partial derivatives of a quasiconformal mapping , Acta Math. 130 (1973), 265–277. · Zbl 0258.30021 · doi:10.1007/BF02392268 |
| [15] | –. –. –. –., “Topics in quasiconformal mappings” in Proceedings of the International Congress of Mathematicians (Berkeley, 1986), Vols. I, II , Amer. Math. Soc., Providence, 1987, 62–80. |
| [16] | F. W. Gehring and E. Reich, Area distortion under quasiconformal mappings , Ann. Acad. Sci. Fenn. Ser A I No. 388 (1966), 1–15. · Zbl 0141.27105 |
| [17] | T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings , Z. Anal. Anwendungen 1 (1982), 1–16. · Zbl 0577.46038 |
| [18] | –. –. –. –., The best constant in a BMO-inequality for the Beurling-Ahlfors transform , Michigan Math. J. 33 (1986), 387–394. · Zbl 0624.42006 · doi:10.1307/mmj/1029003418 |
| [19] | –. –. –. –., Hilbert transform in the complex plane and the area inequalities for certain quadratic differentials , Michigan Math. J. 34 (1987), 407–434. · Zbl 0641.30036 · doi:10.1307/mmj/1029003621 |
| [20] | ——–, “\(L^p\)-theory of quasiregular mappings” in Quasiconformal Space Mappings , ed. Matti Vuorinen, Lecture Notes in Math. 1508 , Springer, Berlin, 1992. \CMP1 187 088 |
| [21] | T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions , Acta Math. 170 (1993), 29–81. · Zbl 0785.30008 · doi:10.1007/BF02392454 |
| [22] | –. –. –. –., Riesz transforms and related singular integrals , J. Reine Angew. Math. 473 (1996), 25–57. · Zbl 0847.42015 |
| [23] | O. Lehto, Remarks on the integrability of the derivatives of quasiconformal mappings , Ann. Acad. Sci. Fenn. Ser. A I No. 371 (1965), 3–8. · Zbl 0137.05503 |
| [24] | –. –. –. –., “Quasiconformal mappings and singular integrals” in Convegno sulle Transformazioni Quasiconformi e Questioni Connesse (Rome, 1974) , Sympos. Math. 18 , Academic Press, London, 1976, 429–453. |
| [25] | O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane , 2d ed., Grundlehren Math. Wiss. 126 , Springer, New York, 1973. · Zbl 0267.30016 |
| [26] | F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers , J. Amer. Math. Soc. 12 (1999), 909–928. JSTOR: · Zbl 0951.42007 · doi:10.1090/S0894-0347-99-00310-0 |
| [27] | F. Nazarov and A. Volberg, Heating of the Ahlfors-Beurling operator and the estimates of its norms , preprint, 2000, · Zbl 1175.47030 |
| [28] | S. Petermichl and J. Wittwer, A sharp estimate for the weighted Hilbert transform via Bellman functions , preprint, 2000, · Zbl 1040.42008 · doi:10.1307/mmj/1022636751 |
| [29] | E. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, And Oscillatory Integrals , Princeton Math. Ser. 43 , Monogr. Harmon. Anal. 3 , Princeton Univ. Press, Princeton, 1993. · Zbl 0821.42001 |
| [30] | G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities , Ann. Probab. 23 (1995), 522–551. · Zbl 0832.60055 · doi:10.1214/aop/1176988278 |
| [31] | J. Wittwer, A sharp estimate on the norm of the martingale transform , Math. Res. Lett. 7 (2000), 1–12. · Zbl 0951.42008 · doi:10.4310/MRL.2000.v7.n1.a1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
