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.
Similar content being viewed by others
References
Andersen, K.F., Muckenhoupt, B.: Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions. Studia Math. 72, 9–26 (1982)
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)
Bañuelos, R., Janakiraman, P.: \(L^p\)-bounds for the Beurling–Ahlfors transform. Trans. Am. Math. Soc. 360, 3603–3612 (2008)
Bañuelos, R., Janakiraman, P.: On the weak-type constant of the Beurling–Ahlfors transform. Mich. Math. J. 58, 459–477 (2009)
Bañuelos, R., Osȩkowski, A.: Sharp inequalities for the Beurling–Ahlfors transform on radial functions. Duke Math. J. 162, 417–434 (2013)
Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)
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)
Boza, S., Soria, J.: Averaging operators on decreasing or positive functions: equivalence and optimal bounds. J. Approx. Theory 237, 135–152 (2019)
Brown, A., Halmos, P.R., Shields, A.L.: Cesàro operators. Acta Sci. Math. (Szeged) 26, 125–137 (1965)
Carro, M.J., Pick, L., Soria, J., Stepanov, V.: On embeddings between classical Lorentz spaces. Math. Inequal. Appl. 4, 397–428 (2001)
Carro, M.J., Soria, J.: Weighted Lorentz spaces and the Hardy operator. J. Funct. Anal. 112, 480–494 (1993)
Carro, M.J., Soria, J.: Boundedness of some integral operators. Can. J. Math. 45, 1155–1166 (1993)
Gao, G., Zhao, F.: Sharp weak bounds for Hausdorff operators. Anal. Math. 41, 163–173 (2015)
He, Q., Yan, D.: Sharp weak bounds and limiting weak-type behaviour for Hardy type operators, Preprint
Iwaniec, T.: Extremal inequalities in Sobolev spaces and quasiconformal mappings. Z. Anal. Anwend. 1, 1–16 (1982)
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)
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)
Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific Publishing Co., Inc., River Edge (2003)
Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972)
Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. Studia Math. 96, 145–158 (1990)
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)
Strzelecki, M.: The \(L^p\)-norms of the Beurling–Ahlfors transform on radial functions. Ann. Acad. Sci. Fenn. Math. 42, 73–93 (2017)
Strzelecki, M.: Hardy’s operator minus identity and power weights, Preprint
Acknowledgements
We would like to thank the referees for their careful revision, which has greatly improved the final version of this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Both authors have been partially supported by the Spanish Government Grant MTM2016-75196-P (MINECO/FEDER, UE) and the Catalan Autonomous Government Grant 2017SGR358.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-020-00973-8