Abstract
We study dimension-free Lp inequalities for r-variations of the Hardy–Littlewood averaging operators defined over symmetric convex bodies in \({\mathbb{R}^d}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bourgain, J.: On high dimensional maximal functions associated to convex bodies. Amer. J. Math. 108, 1467–1476 (1986)
Bourgain, J.: On \(L^p\) bounds for maximal functions associated to convex bodies in \(\mathbb{R}^n\). Israel J. Math. 54, 257–265 (1986)
Bourgain, J.: On dimension free maximal inequalities for convex symmetric bodies in \({\mathbf{R}}^n\). Geometrical aspects of functional analysis (1985/86). Lecture Notes in Math. 1267, 168–176 (1987)
Bourgain, J.: On the Hardy-Littlewood maximal function for the cube. Israel J. Math. 203, 275–293 (2014)
J. Bourgain, M. Mirek, E. Stein, and B. Wróbel. Dimension-free estimates for maximal functions: discrete and continuous perspective. Preprint (2017).
Carbery, A.: An almost-orthogonality principle with applications to maximal functions associated to convex bodies. Bull. Amer. Math. Soc. 14(2), 269–274 (1986)
L. Delaval, O. Guédon, and B. Maurey. Dimension-free bounds for the Hardy–Littlewood maximal operator associated to convex sets. Preprint (2016) https://arxiv.org/abs/1602.02015.
J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116. Notas de Matemática 104. North-Holland Publishing Co., Amsterdam (1985).
Jones, R.L.; Reinhold, K.: Oscillation and variation inequalities for convolution powers. Ergodic Theory Dyn. Syst. 21(6), 1809–1829 (2001)
Jones, R.L.; Seeger, A.; Wright, J.: Strong Variational and Jump Inequalities in Harmonic Analysis. Trans. Amer. Math. Soc. 360(12), 6711–6742 (2008)
Mirek, M.; Stein, E.M.; Trojan, B.: \(\ell ^p(\mathbb{Z}^d)\)-estimates for discrete operators of Radon type: variational estimates. Invent. Math. 209(3), 665–748 (2017)
Müller, D.: A geometric bound for maximal functions associated to convex bodies. Pac. J. Math. 142(2), 297–312 (1990)
Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Princeton University Press, Princeton, Annals of Mathematics Studies (1970)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1971)
Stein, E.M.: The development of square functions in the work of A. Zygmund. Bull. Amer. Math. Soc. 7, 359–376 (1982)
Stein, E.M.: Some results in harmonic analysis in \(\mathbb{R}^n, n\rightarrow \infty \). Bull. Amer. Math. Soc. 9, 71–73 (1983)
Stein, E.M.; Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)
Author information
Authors and Affiliations
Corresponding author
Additional information
Jean Bourgain was partially supported by NSF Grant DMS-1301619. Mariusz Mirek was partially supported by the Schmidt Fellowship and the IAS Found for Math. and by the National Science Center, NCN Grant DEC-2015/19/B/ST1/01149. Elias M. Stein was partially supported by NSF Grant DMS-1265524. Błażej Wróbel was partially supported by the National Science Centre, NCN Grant 2014/ 15/D/ST1/00405.
Rights and permissions
About this article
Cite this article
Bourgain, J., Mirek, M., Stein, E.M. et al. On Dimension-Free Variational Inequalities for Averaging Operators in \({\mathbb{R}^d}\). Geom. Funct. Anal. 28, 58–99 (2018). https://doi.org/10.1007/s00039-018-0433-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-018-0433-3