Bellman function, littlewood-paley estimates and asymptotics for the Ahlfors-Beurling operator in Lp(ℂ)

Indiana University Mathematics Journal

Volumn 54, Issue 4, 2005, Pages 971-995

  Dragičević, Oliver ^a,b   Volberg, Alexander ^c  

^a UNIVERSITY OF LJUBLJANA   (Slovenia)

^b SCUOLA NORMALE SUPERIORE   (Italy)

^c MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Bellman PDE; Quasiconformal mappings; Singular integrals; Stochastic optimal control

Indexed keywords

EID: 26444522275     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2005.54.2554     Document Type: Article

References (18)

1. KARI ASTALA, Area distortion of quasiconformal mappings, Acta Math. 173 (1994), 37-60. MR1294669 (95m:30028b)
2. ALBERT BAERNSTEIN II and STEPHEN J. MONTGOMERY-SMITH, Some conjectures about integral means of ∂f and ∂̄f, Complex Analysis and Differential Equations, Proc. of Wallenburg Symposium in Honor of Matts Essén (Uppsala, 1997) (C. . Kiselman, ed.), Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., vol. 64, Uppsala Univ., Uppsala, 1999, pp. 92-109. MR1758918 (20011:30002)
3. RODRIGO BAÑUELOS and PEDRO J. MÉNDEZ-HERNÁNDEZ, Space-time Brownian motion and the Beurling-Ahlfors transform, Indiana Univ. Math. J. 52 (2003), 981-990, http://dx.doi.org/10.1512/iumj.2003.52.2218. MR2001941 (2004h:60067)
4. RODRIGO BAÑUELOS and GANG WANG, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), 575-600. MR1370109 (96k:60108)
5. DONALD L. BURKHOLDER, Explorations in martingale theory and its applications, Ecole d'Eté de Probabilités de Saint-Flour XIX - 1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 1-66. MR1108183 (92m:60037)
6. _, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647-702. MR744226 (86b:60080)
7. OLIVER DRAGICEVIC and ALEXANDER VOLBERG, Bellman function and dimensionless estimates of Littlewood-Paley type, J. Operator Theory (to appear).
8. LAWRENCE C. EVANS, Partial Differential Equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998, ISBN 0-8218-0772-2. MR1625845 (99e:35001)
9. ROBERT A. FEFFERMAN, CARLOS E. KENIG, and JILL PIPHER, The theory of weights and the Dirichletproblem for elliptic equations, Ann. of Math. (2) 134 (1991), 65-124. MR1114608 (93h:31010)
10. TADEUSZ IWANIEC and GAVEN MARTIN, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996), 25-57. MR1390681 (97k:42033)
11. NIGEL J. KALTON and I. EVGENI VERBITSKY, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc. 351 (1999), 3441-3497, http://dx.doi.org/10.1090/S0002-9947-99-02215-1. MR1475688 (99m:35073)
12. P. EKKEHARD KOPP, Martingales and Stochastic Integrals, Cambridge University Press, Cambridge, 1984, ISBN 0-521-24758-6. MR774050 (86i:60004)
13. O. LEHTO and K.I. VIRTANEN, Quasiconformal Mappings in the Plane, Springer-Verlag, New York, 1973.
14. FEDOR NAZAROV and ALEXANDER VOLBERG, Heat extension of the Beurling operator and estimates for its norm, Algebra i Analiz 15 (2003), 142-158. MR2068982 (2005B0042)
15. (Russian, with Russian summary. English transl, in St. Petersburg Math. J. 15 (2004))
16. STEFANIE PETERMICHL and ALEXANDER VOLBERG, Heating of the Ahlfors-Beurling operator, weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2002), 281-305. MR1894362 (2003d:42025)
17. DANIEL W. STROOCK, Probability Theory, an Analytic View, Cambridge University Press, Cambridge, 1993, ISBN 0-521-43123-9. MR1267569 (95f:60003)
18. ALEXANDER VOLBERG, Bellman approach to some problems in harmonic analysis, Séminaire Sur équations Aux dérivées Partielles, n° XX, Ecole Polytéchnique, Avril 2002.