A modification of Hardy-Littlewood maximal-function on Lie groups

Document Type : Original Article

Author

Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran

Abstract

For a real-valued function $f$ on a metric measure space $(X,d,\mu)$ the Hardy-Littlewood centered-ball maximal-function of $f$ is given by the `supremum-norm':
$$Mf(x):=\sup_{r>0}\frac{1}{\mu(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|d\mu.$$
In this note, we replace the supremum-norm on parameters $r$ by $\mathcal{L}_p$-norm with weight $w$ on parameters $r$ and define Hardy-Littlewood integral-function $I_{p,w}f$. It is shown that $I_{p,w}f$ converges pointwise to $Mf$ as $p\to\infty$. Boundedness of the sublinear operator $I_{p,w}$ and continuity of the function $I_{p,w}f$ in case that $X$ is a Lie group, $d$ is a left-invariant metric, and $\mu$ is a left Haar-measure (resp. right Haar-measure) are studied.

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References

  1. Chousionis, S. Li, and S. Zimmerman, Singular integrals on C1,α regular curves in Carnot groups, J. Anal. Math., 146 (2022), pp. 299–326.
  2. B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.
  3. Forni, Twisted cohomological equations for translation flows, Ergodic Theory Dynam. Systems, 42 (2022), pp. 881–916.
  4. Hormozi and K. Yabuta, Comments on: “Sharp weighted estimates for square functions associated to operators on spaces of homogeneous type”, J. Geom. Anal., 32 (2022), pp. Paper No. 1, 8.
  5. D. Ionescu, A. Magyar, M. Mirek, and T. Z. Szarek, Polynomial averages and pointwise ergodic theorems on nilpotent groups, Invent. Math., 231 (2023), pp. 1023–1140.
  6. Papageorgiou, Riesz means on locally symmetric spaces, Complex Anal. Oper. Theory, 16 (2022), pp. Paper No. 43, 14.
  7. M. Sadr and M. Barzegar Ganji, Abstract Hardy-Littlewood maximal inequality, Iran. J. Sci. Technol. Trans. A Sci., 45 (2021), pp. 1717–1724.
  8. Santagati, Hardy spaces on homogeneous trees with flow measures, J. Math. Anal. Appl., 510 (2022), pp. Paper No. 126015, 23.
  9. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.
  10. Yang and P. Li, Boundedness and compactness of commutators related with Schr¨odinger operators on Heisenberg groups, J. Pseudo-Differ. Oper. Appl., 14 (2023), pp. Paper No. 8, 59.
AUT Journal of Mathematics and Computing
Volume 5, Issue 2
2024
Pages 143-149

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