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Weak-type and end-point norm estimates for Hardy operators

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Abstract

We explicitly calculate the best constants for weak-type and other end-point estimates for the Hardy operator and its adjoint. In particular, we find the right value for decreasing power weights, fixing some previously unclear results.

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References

  1. Andersen, K.F., Muckenhoupt, B.: Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions. Studia Math. 72, 9–26 (1982)

    Article  MathSciNet  Google Scholar 

  2. Ariño, M.A., Muckenhoupt, B.: Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions. Trans. Am. Math. Soc. 320, 727–735 (1990)

    MathSciNet  MATH  Google Scholar 

  3. Bañuelos, R., Janakiraman, P.: \(L^p\)-bounds for the Beurling–Ahlfors transform. Trans. Am. Math. Soc. 360, 3603–3612 (2008)

    Article  Google Scholar 

  4. Bañuelos, R., Janakiraman, P.: On the weak-type constant of the Beurling–Ahlfors transform. Mich. Math. J. 58, 459–477 (2009)

    Article  MathSciNet  Google Scholar 

  5. Bañuelos, R., Osȩkowski, A.: Sharp inequalities for the Beurling–Ahlfors transform on radial functions. Duke Math. J. 162, 417–434 (2013)

    Article  MathSciNet  Google Scholar 

  6. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)

    MATH  Google Scholar 

  7. Boza, S., Soria, J.: Solution to a conjecture on the norm of the Hardy operator minus the identity. J. Funct. Anal. 260(4), 1020–1028 (2011)

    Article  MathSciNet  Google Scholar 

  8. Boza, S., Soria, J.: Averaging operators on decreasing or positive functions: equivalence and optimal bounds. J. Approx. Theory 237, 135–152 (2019)

    Article  MathSciNet  Google Scholar 

  9. Brown, A., Halmos, P.R., Shields, A.L.: Cesàro operators. Acta Sci. Math. (Szeged) 26, 125–137 (1965)

    MathSciNet  MATH  Google Scholar 

  10. Carro, M.J., Pick, L., Soria, J., Stepanov, V.: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 4, 397–428 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Carro, M.J., Soria, J.: Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112, 480–494 (1993)

    Article  MathSciNet  Google Scholar 

  12. Carro, M.J., Soria, J.: Boundedness of some integral operators. Can. J. Math. 45, 1155–1166 (1993)

    Article  MathSciNet  Google Scholar 

  13. Gao, G., Zhao, F.: Sharp weak bounds for Hausdorff operators. Anal. Math. 41, 163–173 (2015)

    Article  MathSciNet  Google Scholar 

  14. He, Q., Yan, D.: Sharp weak bounds and limiting weak-type behaviour for Hardy type operators, Preprint

  15. Iwaniec, T.: Extremal inequalities in Sobolev spaces and quasiconformal mappings. Z. Anal. Anwend. 1, 1–16 (1982)

    Article  MathSciNet  Google Scholar 

  16. Kolyada, V.I.: Optimal relationships between \(L^p\)-norms for the Hardy operator and its dual. Ann. Mat. Pura Appl. (4) 193, 423–430 (2014)

    Article  MathSciNet  Google Scholar 

  17. Kruglyak, N., Setterqvist, E.: Sharp estimates for the Identity minus Hardy operator on the cone of decreasing functions. Proc. Am. Math. Soc. 136(7), 2505–2513 (2008)

    Article  MathSciNet  Google Scholar 

  18. Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific Publishing Co., Inc., River Edge (2003)

    Book  Google Scholar 

  19. Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972)

    Article  MathSciNet  Google Scholar 

  20. Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. Studia Math. 96, 145–158 (1990)

    Article  MathSciNet  Google Scholar 

  21. Sinnamon, G., Stepanov, V.D.: The weighted Hardy inequality: new proofs and the case \(p=1\). J. Lond. Math. Soc. (2) 54(1), 89–101 (1996)

    Article  MathSciNet  Google Scholar 

  22. Strzelecki, M.: The \(L^p\)-norms of the Beurling–Ahlfors transform on radial functions. Ann. Acad. Sci. Fenn. Math. 42, 73–93 (2017)

    Article  MathSciNet  Google Scholar 

  23. Strzelecki, M.: Hardy’s operator minus identity and power weights, Preprint

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Acknowledgements

We would like to thank the referees for their careful revision, which has greatly improved the final version of this work.

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Correspondence to Santiago Boza.

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Both authors have been partially supported by the Spanish Government Grant MTM2016-75196-P (MINECO/FEDER, UE) and the Catalan Autonomous Government Grant 2017SGR358.

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Boza, S., Soria, J. Weak-type and end-point norm estimates for Hardy operators. Annali di Matematica 199, 2381–2393 (2020). https://doi.org/10.1007/s10231-020-00973-8

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  • DOI: https://doi.org/10.1007/s10231-020-00973-8

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