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$ L^p$-bounds for the Beurling-Ahlfors transform


Authors: Rodrigo Bañuelos and Prabhu Janakiraman
Journal: Trans. Amer. Math. Soc. 360 (2008), 3603-3612
MSC (2000): Primary 42B20, 60H05
DOI: https://doi.org/10.1090/S0002-9947-08-04537-6
Published electronically: February 13, 2008
MathSciNet review: 2386238
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Abstract: Let $ B$ denote the Beurling-Ahlfors transform defined on $ L^p(\mathbb{C})$, $ 1<p<\infty$. The celebrated conjecture of T. Iwaniec states that its $ L^p$ norm $ \Vert B\Vert _p=p^*-1$ where $ p^*= \max\{p,\frac{p}{p-1}\}$. In this paper the new upper estimate

$\displaystyle \Vert B\Vert _p\leq 1.575\,(p^*-1), \hspace{3mm} 1<p<\infty,$

is found.


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  • 1. K. Astala; Area distortion of quasiconformal mappings. Acta Math. 173 (1994), no. 1, 37-60. MR 1294669 (95m:30028b)
  • 2. R. Bañuelos, P. Méndez-Hernández; Space-Time Brownian Motion and the Beurling-Ahlfors Transform. Indiana University Math J. 52 (2003), 981-990. MR 2001941 (2004h:60067)
  • 3. R. Bañuelos, G. Wang; Sharp Inequalities for Martingales with Applications to the Beurling-Ahlfors and Riesz Transforms. Duke Math. J. 80 (1995), 575-600. MR 1370109 (96k:60108)
  • 4. D. L. Burkholder; Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12 (1984), 647-702. MR 744226 (86b:60080)
  • 5. D. L. Burkholder; An elementary proof of an inequality of R. E. A. C. Paley. Bull. London Math. Soc. 17 (1985), no. 5, 474-478. MR 806015 (87a:42041)
  • 6. D. L. Burkholder; Sharp inequalities for martingales and stochastic integrals. In: Colloque Paul Lévy (Palaiseau, 1987), Astérisque, 157-158 (1988) 75-94. MR 976214 (90b:60051)
  • 7. D. L. Burkholder; A proof of Pełczynśki's conjecture for the Haar system. Studia Math. 91 (1988), no. 1, 79-83. MR 957287 (89j:46026)
  • 8. D. L. Burkholder; Explorations in martingale theory and its applications. École d'Été de Probabilités de Saint-Flour XIX--1989, Lecture Notes in Math., 1464 (Springer, Berlin, 1991) 1-66. MR 1108183 (92m:60037)
  • 9. D. L. Burkholder; Strong differential subordination and stochastic integration. Ann. Probab. 22 (1994), no. 2, 995-1025. MR 1288140 (95h:60085)
  • 10. D. L. Burkholder; The best constant in the Davis inequality for the expectation of the martingale square function. Trans. Amer. Math. 354 (2002), 91-105. MR 1859027 (2002g:60063)
  • 11. S. Donaldson, D. Sullivan; Quasiconformal 4-manifolds. Acta Math. 163 (1989), 181-252. MR 1032074 (91d:57012)
  • 12. O. Dragičević, A. Volberg; Bellman functions, Littlewood-Paley estimates and asymptotics for the Ahlfors-Beurling operator in $ L^p(\mathbb{C})$. Indiana Univ. Math J. 54 (2005), no. 4, 971-995. MR 2164413 (2006i:30025)
  • 13. T. Iwaniec; Extremal inequalities in Sobolev spaces and quasiconformal mappings. Z. Anal. Anwendungen 1 (1982), 1-16. MR 719167 (85g:30027)
  • 14. T. Iwaniec; $ L^p$-theory of quasiregular mappings in Quasiconformal Space Mappings. Ed. Matti Vuorinen, Lecture Notes in Math. 1508, Springer, Berlin, 1992. MR 1187085 (94c:30030)
  • 15. T. Iwaniec, G. Martin; Quasiregular mappings in even dimensions. Acta Math. 170 (1993), 29-81. MR 1208562 (94m:30046)
  • 16. O. Lehto; Remarks on the integrability of the derivatives of quasiconformal mappings, Ann. Acad. Sci. Fenn. Series AI Math. 371 (1965), 8 pp. MR 0181748 (31:5975)
  • 17. F. Nazarov, A. Volberg; Heat extension of the Beurling operator and estimates for its norm. (Russian) Algebra i Analiz 15 (2003), no. 4, 142-158. MR 2068982 (2005f:30042)
  • 18. E. M. Stein; Singular integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. MR 0290095 (44:7280)
  • 19. D. W. Stroock; Probability Theory, an Analytic View, Cambridge University Press, 1993. MR 1267569 (95f:60003)

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Additional Information

Rodrigo Bañuelos
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-1395
Email: banuelos@math.purdue.edu

Prabhu Janakiraman
Affiliation: Department of Mathematics, University of Illinois, Urbana-Champaign, Illinois 61801
Email: pjanakir@math.uiuc.edu

DOI: https://doi.org/10.1090/S0002-9947-08-04537-6
Keywords: Singular integrals, stochastic integrals
Received by editor(s): November 15, 2005
Received by editor(s) in revised form: April 26, 2006
Published electronically: February 13, 2008
Additional Notes: The first author was supported in part by NSF grant #0603701-DMS
The second author was supported in part by an NSF VIGRE postdoctoral fellowship
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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