On the Bellman function of Nazarov, Treil and Volberg

Mathematische Zeitschrift - Tập 278 - Trang 385-399 - 2014
Rodrigo Bañuelos1, Adam Osȩkowski2
1Department of Mathematics Purdue University West Lafayette USA
2Department of Mathematics, Informatics and Mechanics, University of Warsaw, Warsaw, Poland

Tóm tắt

We give an explicit formula for the Bellman function associated with the dual bound related to the unconditional constant of the Haar system.

Tài liệu tham khảo

Bañuelos, R., Janakiraman, P.: \(L^p\)-bounds for the Beurling–Ahlfors transform. Trans. Am. Math. Soc. 360, 3604–3612 (2008) Bañuelos,R., Osȩkowski, A.: Sharp Martingale Inequalities and Applications to Riesz Transforms on Manifolds, Lie Groups and Gauss Space (submitted) Burkholder, D.L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12, 647–702 (1984) Burkholder, D.L.: A proof of Pełczyński’s conjecture for the Haar system. Studia Math. 91, 268–273 (1988) Burkholder, D.L. : Explorations in Martingale Theory and Its Applications, École d’Ete de Probabilités de Saint-Flour XIX–1989, pp. 1–66, Lecture Notes in Mathematics, vol. 1464. Springer, Berlin (1991) Dragičević, O., Volberg, A.: Bellman function and dimensionless estimates of classical and Ornstein–Uhlenbeck Riesz transforms. J. Oper. Theory 56, 167–198 (2006) Marcinkiewicz, J.: Quelques théorèmes sur les séries orthogonales. Ann. Soc. Polon. Math. 16, 84–96 (1937) Maurey, B.: Système de Haar, Séminaire Maurey-Schwartz, 1974–1975. Ecole Polytechnique, Paris (1975) Nazarov, F.L., Treil, S.R.: The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. St. Petersburg Math. J. 8, 721–824 (1997) Nazarov, F.L., Treil, S.R., Volberg, A.: The Bellman functions and two-weight inequalities for Haar multipliers. J. Am. Math. Soc. 12, 909–928 (1999) Nazarov, F.L., Volberg, A.: Heating of the Ahlfors–Beurling operator and estimates of its norm. St. Petersburg Math. J. 15, 563–573 (2004) Osȩkowski, A.: Sharp Martingale and Semimartingale Inequalities, Monografie Matematyczne, vol. 72. Birkhäuser Basel (2012) Paley, R.E.A.C.: A remarkable series of orthogonal functions. Proc. Lond. Math. Soc. 34, 241–264 (1932) Schauder, J.: Eine Eigenschaft des Haarschen Orthogonalsystems. Math. Z. 28, 317–320 (1928)