Heating of the Ahlfors–Beurling operator, and estimates of its norm
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A. Volberg and F. Nazarov
Translated by: the authors - St. Petersburg Math. J. 15 (2004), 563-573
- DOI: https://doi.org/10.1090/S1061-0022-04-00822-2
- Published electronically: July 6, 2004
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Abstract:
A new estimate is established for the norm of the Ahlfors–Beurling transform $T\varphi (z):=\frac 1\pi \iint \frac {\varphi (\zeta ) dA(\zeta )}{(\zeta - z)^2}$ in $L^p(dA)$. Namely, it is proved that $\|T\|_{L^p\rightarrow L^p} \leq 2(p-1)$ for all $p\geq 2$. The method of Bellman function is used; however, the exact Bellman function of the problem has not been found. Instead, a certain approximation to the Bellman function is employed, which leads to the factor 2 on the right (in place of the conjectural $1$).References
- Kari Astala, Tadeusz Iwaniec, and Eero Saksman, Beltrami operators in the plane, Duke Math. J. 107 (2001), no. 1, 27–56. MR 1815249, DOI 10.1215/S0012-7094-01-10713-8
- Kari Astala, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), no. 1, 37–60. MR 1294669, DOI 10.1007/BF02392568
- Rodrigo Bañuelos and Gang Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), no. 3, 575–600. MR 1370109, DOI 10.1215/S0012-7094-95-08020-X
- R. Bañuelos and P. J. Méndez-Hernández, Space-time Brownian motion and the Beurling-Ahlfors transform, Indiana Univ. Math. J. 52 (2003), no. 4, 981–990. MR 2001941, DOI 10.1512/iumj.2003.52.2218
- B. V. Boyarskiĭ, Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 661–664 (Russian). MR 0071620
- B. V. Boyarskiĭ, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients, Mat. Sb. (N.S.) 43(85) (1957), 451–503 (Russian). MR 0106324
- Bogdan Bojarski, Quasiconformal mappings and general structural properties of systems of non linear equations elliptic in the sense of Lavrent′ev, Symposia Mathematica, Vol. XVIII, Academic Press, London, 1976, pp. 485–499. (Convegno sulle Transformazioni Quasiconformi e Questioni Connesse, INDAM, Rome, 1974),. MR 0507823
- Bogdan 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), 473–478; ibid. 22 (1974), 479–484 (English, with Russian summary). MR 364856
- Rodrigo Bañuelos and Pedro J. Méndez-Hernández, Sharp inequalities for heat kernels of Schrödinger operators and applications to spectral gaps, J. Funct. Anal. 176 (2000), no. 2, 368–399. MR 1784420, DOI 10.1006/jfan.2000.3611
- Donald L. Burkholder, Explorations in martingale theory and its applications, École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 1–66. MR 1108183, DOI 10.1007/BFb0085167
- D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702. MR 744226
- Stephen M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253–272. MR 1124164, DOI 10.1090/S0002-9947-1993-1124164-0
- R. A. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2) 134 (1991), no. 1, 65–124. MR 1114608, DOI 10.2307/2944333
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- F. W. Gehring, Open problems, Proceedings of Roumanian–Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings, 1969, p. 306.
- F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
- F. W. Gehring, Topics in quasiconformal mappings, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 62–80. MR 934216
- F. W. Gehring and E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I 388 (1966), 15. MR 0201635
- T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen 1 (1982), no. 6, 1–16 (English, with German and Russian summaries). MR 719167, DOI 10.4171/ZAA/37
- T. Iwaniec, The best constant in a BMO-inequality for the Beurling-Ahlfors transform, Michigan Math. J. 33 (1986), no. 3, 387–394. MR 856530, DOI 10.1307/mmj/1029003418
- Tadeusz Iwaniec, Hilbert transform in the complex plane and area inequalities for certain quadratic differentials, Michigan Math. J. 34 (1987), no. 3, 407–434. MR 911814, DOI 10.1307/mmj/1029003621
- Tadeusz Iwaniec, $L^p$-theory of quasiregular mappings, Quasiconformal space mappings, Lecture Notes in Math., vol. 1508, Springer, Berlin, 1992, pp. 39–64. MR 1187088, DOI 10.1007/BFb0094237
- Tadeusz Iwaniec and Gaven Martin, Quasiregular mappings in even dimensions, Acta Math. 170 (1993), no. 1, 29–81. MR 1208562, DOI 10.1007/BF02392454
- Tadeusz Iwaniec and Gaven Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25–57. MR 1390681
- St. Petermichl and J. Wittwer, A sharp weighted estimate on the norm of Hilbert transform via invariant $A_2$ characteristic of the weight, Preprint, Michigan State Univ., 2000.
- J. Wittwer, Thesis, Univ. Chicago, 2000.
- Olli Lehto, Quasiconformal mappings and singular integrals, Symposia Mathematica, Vol. XVIII, Academic Press, London, 1976, pp. 429–453. (Convegno sulle Transformazioni Quasiconformi e Questioni Connesse, INDAM, Rome, 1974),. MR 0492241
- O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 126, Springer-Verlag, New York-Heidelberg, 1973. Translated from the German by K. W. Lucas. MR 0344463
- F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909–928. MR 1685781, DOI 10.1090/S0894-0347-99-00310-0
- Stefanie Petermichl and Alexander Volberg, Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), no. 2, 281–305. MR 1894362, DOI 10.1215/S0012-9074-02-11223-X
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Daniel W. Stroock, Probability theory, an analytic view, Cambridge University Press, Cambridge, 1993. MR 1267569
- A. Volberg, Bellman approach to some problems in harmonic analysis, Séminaire sur les Équations aux Dérivées Partielles, École Polytech., 2002, Exp. No. XX.
Bibliographic Information
- A. Volberg
- Affiliation: Michigan State University, East Lansing, Michigan, USA, and Equipe d’Analyse Université Paris VI, 4 Place Jussieu, 75 252 Paris cédex 05, France
- Email: volberg@math.msu.edu
- F. Nazarov
- Affiliation: Michigan State University, East Lansing, Michigan, USA
- MR Author ID: 233855
- Email: fedja@math.msu.edu
- Received by editor(s): December 20, 2002
- Published electronically: July 6, 2004
- Additional Notes: Partially supported by the NSF grant DMS 0200713
- © Copyright 2004 American Mathematical Society
- Journal: St. Petersburg Math. J. 15 (2004), 563-573
- MSC (2000): Primary 42B20, 42C15, 42A50, 47B35
- DOI: https://doi.org/10.1090/S1061-0022-04-00822-2
- MathSciNet review: 2068982