Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the $ p-1$ problem

Oliver Dragičević; Alexander Volberg

doi:10.1016/j.crma.2005.03.021

Harmonic Analysis

Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the

p - 1

problem

Presented by: Gilles Pisier

Oliver Dragičević ¹; Alexander Volberg ²

¹ Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 340 (2005) no. 10, pp. 731-734.

Abstract
Résumé

We utilize the method of Bellman functions to derive new $L^{p}$ -estimates of Littlewood–Paley type involving $p - 1$ . Among the applications to singular integrals we improve the $2 (p - 1)$ bounds for the Ahlfors–Beurling operator on $L^{p} (C)$ when $p \to \infty$ . In addition, dimensionless estimates of Riesz transforms in the classical as well as in the Ornstein–Uhlenbeck setting are attained.

On utilise la technique de la fonction de Bellman pour obtenir les estimations nouvelles et assez générales du type de Littlewood–Paley. Comme la premier consequence de nos estimation du type de Littlewood–Paley on derive les resultats classiques concernants les bornes libre de dimension pour les transformations de Riesz. La deuxième consequence est une amilioration de la borne dans $L^{p} (C)$ de transformation de Ahlfors–Beurling quand $p \to \infty$ .

Received: 2005-03-03
Accepted: 2005-03-22
Published online: 2005-05-10

DOI: 10.1016/j.crma.2005.03.021

Author's affiliations:

Oliver Dragičević ¹; Alexander Volberg ²

¹ Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

@article{CRMATH_2005__340_10_731_0,
     author = {Oliver Dragi\v{c}evi\'c and Alexander Volberg},
     title = {Bellman function for the estimates of {Littlewood{\textendash}Paley} type and asymptotic estimates in the $ p-1$ problem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {731--734},
     publisher = {Elsevier},
     volume = {340},
     number = {10},
     year = {2005},
     doi = {10.1016/j.crma.2005.03.021},
     language = {en},
}

TY  - JOUR
AU  - Oliver Dragičević
AU  - Alexander Volberg
TI  - Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the $ p-1$ problem
JO  - Comptes Rendus. Mathématique
PY  - 2005
SP  - 731
EP  - 734
VL  - 340
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2005.03.021
LA  - en
ID  - CRMATH_2005__340_10_731_0
ER  -

%0 Journal Article
%A Oliver Dragičević
%A Alexander Volberg
%T Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the $ p-1$ problem
%J Comptes Rendus. Mathématique
%D 2005
%P 731-734
%V 340
%N 10
%I Elsevier
%R 10.1016/j.crma.2005.03.021
%G en
%F CRMATH_2005__340_10_731_0

Oliver Dragičević; Alexander Volberg. Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the $ p-1$ problem. Comptes Rendus. Mathématique, Volume 340 (2005) no. 10, pp. 731-734. doi : 10.1016/j.crma.2005.03.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.03.021/

References
Cited by

[1] R. Bañuelos; P. Méndez-Hernández Space–time Brownian motion and the Beurling–Ahlfors transform, Indiana Univ. Math. J., Volume 52 (2003) no. 4, pp. 981-990

[2] D.L. Burkholder Sharp inequalities for martingales and stochastic integrals, Astérisque, Volume 157–158 (1988), pp. 75-94

[3] O. Dragičević, A. Volberg, Bellman functions and dimensionless estimates of Littlewood–Paley type, J. Operator Theory, in press

[4] O. Dragičević, A. Volberg, Bellman function, Littlewood–Paley estimates and asymptotics for the Ahlfors–Beurling operator in $L^{p} (C)$ , preprint

[5] T. Iwaniec Extremal inequalities in Sobolev spaces and quasiconformal mappings, Z. Anal. Anwendungen, Volume 1 (1982), pp. 1-16

[6] N. Krylov Optimal Control of Diffusion Processes, Springer-Verlag, New York, 1980 (308 p)

[7] F. Nazarov; S. Treil 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., Volume 8 (1997) no. 5, pp. 721-824

[8] F. Nazarov; S. Treil; A. Volberg Bellman function and two-weight inequality for martingale transform, J. Amer. Math. Soc., Volume 12 (1999) no. 4, pp. 909-928

[9] F. Nazarov; S. Treil; A. Volberg Bellman function in stochastic control and harmonic analysis, Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 393-423

[10] F. Nazarov; A. Volberg Heat extension of the Beurling operator and estimates for its norm, Algebra i Analiz, Volume 15 (2003) no. 4, pp. 142-158 (in Russian)

[11] S. Petermichl; A. Volberg Heating of the Ahlfors–Beurling operator: weakly quasiregular maps on the plane are quasiregular, Duke Math. J., Volume 112 (2002) no. 2, pp. 281-305

Cited by Sources:

Comments - Policy