Annals of Functional Analysis

Sharp weak estimates for Hardy-type operators

Guilian Gao, Xiaomin Hu, and Chunjie Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, we prove the weak bound for an n-dimensional Hardy operator on a central Morrey space. Meanwhile, we obtain the precise operator norm, and we give the weak bounds for the conjugate Hardy operator on Lebesgue space with power weights. The corresponding operator norms are also computed. As an application, we obtain an estimate for the gamma function.

Article information

Source
Ann. Funct. Anal., Volume 7, Number 3 (2016), 421-433.

Dates
Received: 4 August 2015
Accepted: 28 December 2015
First available in Project Euclid: 17 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1466164867

Digital Object Identifier
doi:10.1215/20088752-3605447

Mathematical Reviews number (MathSciNet)
MR3513126

Zentralblatt MATH identifier
1346.42029

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 26D10: Inequalities involving derivatives and differential and integral operators 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26D15: Inequalities for sums, series and integrals

Keywords
Hardy operator the conjugate Hardy operator central Morrey space weak central Morrey space duality

Citation

Gao, Guilian; Hu, Xiaomin; Zhang, Chunjie. Sharp weak estimates for Hardy-type operators. Ann. Funct. Anal. 7 (2016), no. 3, 421--433. doi:10.1215/20088752-3605447. https://projecteuclid.org/euclid.afa/1466164867


Export citation

References

  • [1] J. Alvarez, J. Lakey, and M. Guzmán-Partida, Spaces of bounded $\lambda$-central mean oscillation, Morrey spaces, and $\lambda$-central Carleson measures, Collect. Math. 51 (2000), no. 1, 1–47.
  • [2] K. F. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), no. 1, 9–26.
  • [3] G. A. Bliss, An integral inequality, J. Lond. Math. Soc. S1–5 (1930), no. 1, 40–46.
  • [4] M. Christ and L. Grafakos, Best constants for two nonconvolution inequalities, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1687–1693.
  • [5] Z. W. Fu, L. Grafakos, S. Z. Lu, and F.Y. Zhao, Sharp bounds for m-linear Hardy and Hilbert operators, Houston J. Math. 38 (2012), no. 1, 225–244.
  • [6] Z. W. Fu, Z. G. Liu, S. Z. Lu, and H. B. Wang, Characterization for commutators of $n$-dimensional fractional Hardy operators, Sci. China Ser. A 50 (2007), no. 10, 1418–1426.
  • [7] G. L. Gao and F. Y. Zhao, Sharp weak bounds for Hausdorff operators, Anal. Math. 41 (2015), no. 3, 163–173.
  • [8] A. Kufner, L. Maligranda, and L.-E. Persson, The Hardy Inequality: About its History and Some Related Results, Vydavatelský Servis, Plzeň, 2007.
  • [9] A. Kufner and L.-E. Persson, Weighted Inequalities of Hardy Type, World Scientific, Singapore, 2003.
  • [10] S. Z. Lu, D. Yan, and F. Zhao, Sharp bounds for Hardy type operators on higher-dimensional product spaces, J. Inequal. Appl. 2013 (2013), 11 pp.
  • [11] F. J. Martín-Reyes and S. J. Ombrosi, Mixed weak type inequalities for one-sided operators, Q. J. Math. 60 (2009), no. 1, 63–73.
  • [12] F. J. Martín-Reyes and P. Ortega Salvador, On weighted weak type inequalities for modified Hardy operators, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1739–1746.
  • [13] F. J. Martín-Reyes, P. Ortega Salvador, and M. D. Sarrión Gavilán, Boundedness of operators of Hardy type in $\Lambda^{p,q}$ spaces and weighted mixed inequalities for singular integral operators, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 1, 157–170.
  • [14] B. Opic and A. Kufner, Hardy-type Inequalities, Pitman Res. Notes Math. Ser. 219, Longman Scientific & Technical, Harlow, 1990.
  • [15] P. Ortega Salvador, Weighted generalized weak type inequalities for modified Hardy operators, Collect. Math. 51 (2000), no. 2, 149–155.
  • [16] L.-E. Persson and S. G. Samko, A note on the best constants in some Hardy inequalities, J. Math. Inequal. 9 (2015), no. 2, 437–447.
  • [17] F. Y. Zhao, Z. W. Fu, and S. Z. Lu, Endpoint estimates for $n$-dimensional Hardy operators and their commutators, Sci. China Math. 55 (2012), no. 10, 1977–1990.
  • [18] F. Y. Zhao and S. Z. Lu, The best bound for $n$-dimensional fractional Hardy operators, Math. Inequal. Appl. 18 (2015), no. 1, 233–240.