Bellman functions and L^p estimates for paraproducts

• Autorzy: Kovač V. , Škreb K. A.

Identyfikatory

DOI

10.19195/0208-4147.38.2.11

• Języki publikacji: EN

Abstrakty

EN

We give an explicit formula for one possible Bellman function associated with the L^p boundedness of dyadic paraproducts regarded as bilinear operators or trilinear forms. Then we apply the same Bellman function in various other settings, to give self-contained alternative proofs of the estimates for several classical operators. These include the martingale paraproducts of Bañuelos and Bennett and the paraproducts with respect to the heat flows.

Słowa kluczowe

EN

Bellman function; martingale; paraproduct

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2018
• Tom: Vol. 38, Fasc. 2
• Strony: 459--479
• Opis fizyczny: Bibliogr. 26 poz., rys.

Twórcy

autor

Kovač V.

• Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia

autor

Škreb K. A.

• Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića Miošića 26, 10000 Zagreb, Croatia

Bibliografia

• [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York 1992.
• [2] R. Bañuelos and A. G. Bennett, Paraproducts and commutators of martingale transforms, Proc. Amer. Math. Soc. 103 (4) (1988), pp. 1226-1234.
• [3] R. Bañuelos and A. Osękowski, On the Bellman function of Nazarov, Treil and Volberg, Math. Z. 278 (2014), pp. 385-399.
• [4] Á. Bényi, D. Maldonado, and V. Naibo, What is…a paraproduct?, Notices Amer. Math. Soc. 57 (7) (2010), pp. 858-860.
• [5] D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), pp. 1494-1504.
• [6] D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (3) (1984), pp. 647-702.
• [7] A. Carbonaro and O. Dragičević, Bellman function and linear dimension-free estimates in a theorem of Bakry, J. Funct. Anal. 265 (7) (2013), pp. 1085-1104.
• [8] A. Carbonaro and O. Dragičević, Functional calculus for generators of symmetric contraction semigroups, Duke Math. J. 166 (5) (2017), pp. 937-974.
• [9] D. L. Cohn, Measure Theory, second edition, Birkhäuser/Springer, New York 2013.
• [10] G. David and J.-L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (2) (1984), pp. 371-397.
• [11] B. Davis, On the Lp norms of stochastic integrals and other martingales, Duke Math. J. 43 (4) (1976), pp. 697-704.
• [12] P. Durcik, An L4 estimate for a singular entangled quadrilinear form, Math. Res. Lett. 22 (5) (2015), pp. 1317-1332.
• [13] P. Durcik, Lp estimates for a singular entangled quadrilinear form, Trans. Amer. Math. Soc. 369 (10) (2017), pp. 6935-6951.
• [14] S. Janson and J. Peetre, Paracommutators-boundedness and Schatten-von Neumann properties, Trans. Amer. Math. Soc. 305 (2) (1988), pp. 467-504.
• [15] V. Kovač, Boundedness of the twisted paraproduct, Rev. Mat. Iberoam. 28 (4) (2012), pp. 1143-1164.
• [16] V. Kovač, and K. A. Škreb, One modification of the martingale transform and its applications to paraproducts and stochastic integrals, J. Math. Anal. Appl. 426 (2) (2015), pp. 1143-1163.
• [17] F. L. Nazarov and S. R. Treil, The hunt for a Bellman function: Applications to estimates of singular integral operators and to other classical problems in harmonic analysis (in Russian), Algebra i Analiz 8 (1996), pp. 32-162; English transl. in St. Petersburg Math. J. 8 (1997), pp. 721-824.
• [18] F. L. Nazarov and A. Volberg, The Bellman function, the two-weight Hilbert transform, and embeddings of the model spaces KƟ, J. Anal. Math. 87 (2002), pp. 385-414.
• [19] F. L. Nazarov and A. Volberg, Heating of the Ahlfors-Beurling operator and estimates of its norm, St. Petersburg Math. J. 15 (2004), pp. 563-573.
• [20] A. Osękowski, Sharp Martingale and Semimartingale Inequalities, Birkhäuser/Springer, Basel 2012.
• [21] S. Petermichl and A. Volberg, Heating of the Ahlfors-Beurling operator: Weakly quasiregular maps on the plane are quasiregular, Duke Math. J. 112 (2) (2002), pp. 281-305.
• [22] P. E. Protter, Stochastic Integration and Differential Equations, second edition, Springer, Berlin 2005.
• [23] T. Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems 28 (2) (2008), pp. 657-688.
• [24] C. Thiele, Wave Packet Analysis, American Mathematical Society, Providence, RI, 2006.
• [25] V. Vasyunin and A. Volberg, Bellster and others (2008), preprint.
• [26] Wolfram Research, Inc., Mathematica, Version 9.0, Champaign, IL (2012).

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).