Abstract
Let \(A_\infty ^+\) denote the class of one-sided Muckenhoupt weights, namely all the weights w for which \(\mathsf {M}^+:L^p(w)\rightarrow L^{p,\infty }(w)\) for some \(p>1\), where \(\mathsf {M}^+\) is the forward Hardy–Littlewood maximal operator. We show that \(w\in A_\infty ^+\) if and only if there exist numerical constants \(\gamma \in (0,1)\) and \(c>0\) such that
for all measurable sets \(E\subset \mathbb R\). Furthermore, letting
we show that for all \(w\in A_\infty ^+\) we have the asymptotic estimate \(\mathsf {C_w ^+}(\alpha )-1\lesssim (1-\alpha )^\frac{1}{c[w]_{A_\infty ^+}}\) for \(\alpha \) sufficiently close to 1 and \(c>0\) a numerical constant, and that this estimate is best possible. We also show that the reverse Hölder inequality for one-sided Muckenhoupt weights, previously proved by Martín-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of \(A_\infty ^+\). Our methods also allow us to show that a weight \(w\in A_\infty ^+\) satisfies \(w\in A_p ^+\) for all \(p>e^{c[w]_{A_\infty ^+}}\).
Similar content being viewed by others
References
Forzani, L., Martín-Reyes, F.J., Ombrosi, S.: Weighted inequalities for the two-dimensional one-sided Hardy–Littlewood maximal function. Trans. Am. Math. Soc. 363(4), 1699–1719 (2011). doi:10.1090/S0002-9947-2010-05343-7
Fujii, N.: Weighted bounded mean oscillation and singular integrals. Math. Jpn. 22(5), 529–534 (1977/78)
Hagelstein, P., Luque, T., Parissis, I.: Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases. Trans. Am. Math. Soc. 367(11), 7999–8032 (2015)
Hagelstein, P., Parissis, I.: Weighted Solyanik estimates for the Hardy–Littlewood maximal operator and embedding of \(A_\infty \) into \(A_p\). J. Geom. Anal. 26(2), 924–946 (2016). doi:10.1007/s12220-015-9578-6
Hagelstein, P., Parissis, I.: Solyanik estimates in harmonic analysis. In: Special Functions, Partial Differential Equations, and Harmonic Analysis, Springer Proceedings of Mathematical Statistics, vol. 108, Springer, Cham, pp. 87–103 (2014)
Hewitt, E., Stromberg, K.: Real and abstract analysis: a modern treatment of the theory of functions of a real variable. Third printing; Graduate Texts in Mathematics, No. 25. Springer, New York (1975)
Hytönen, T., Pérez, C.: Sharp weighted bounds involving \(A_\infty \). Anal. PDE 6(4), 777–818 (2013)
Kinnunen, J., Saari, O.: On weights satisfying parabolic Muckenhoupt conditions. Nonlinear Anal. 131, 289–299 (2016)
Lerner, A.K., Ombrosi, S.: A boundedness criterion for general maximal operators. Publ. Mat. 54(1), 53–71 (2010)
Martín-Reyes, F.J.: New proofs of weighted inequalities for the one-sided Hardy–Littlewood maximal functions. Proc. Am. Math. Soc. 117(3), 691–698 (1993)
Martín-Reyes, F.J., Pick, L., de la Torre, A.: \(A^+_\infty \) condition. Can. J. Math. 45(6), 1231–1244 (1993)
Martín-Reyes, F.J., Salvador, P.O., de la Torre, A.: Weighted inequalities for one-sided maximal functions. Trans. Am. Math. Soc. 319(2), 517–534 (1990)
Martín-Reyes, F.J., de la Torre, A.: Sharp weighted bounds for one-sided maximal operators. Collect. Math. 66(2), 161–174 (2015)
Ombrosi, S.: Weak weighted inequalities for a dyadic one-sided maximal function in \(\mathbb{R}^{n}\). Proc. Am. Math. Soc. 133(6), 1769–1775 (2005). doi:10.1090/S0002-9939-05-07830-5
Ortega Salvador, P.: Weighted inequalities for one-sided maximal functions in Orlicz spaces. Studia Math. 131(2), 101–114 (1998)
Saari, O.: Parabolic BMO and global integrability of supersolutions to doubly nonlinear parabolic equations. Rev. Mat. Iberoam. 32(3), 1001–1018 (2016). doi:10.4171/RMI/906
Sawyer, E.: Weighted inequalities for the one-sided Hardy–Littlewood maximal functions. Trans. Am. Math. Soc. 297(1), 53–61 (1986)
Solyanik, A.A.: On halo functions for differentiation bases. Mat. Zametki 54(6), 82–89 (1993), 160 (Russian, with Russian summary); English transl., Math. Notes 54 (1993), no. 5-6, 1241–1245 (1994)
Wilson, J.M.: Weighted inequalities for the dyadic square function without dyadic \(A_{\infty }\). Duke Math. J. 55(1), 19–50 (1987)
Wilson, J.M.: Weighted Littlewood–Paley theory and exponential-square integrability. Lecture Notes in Mathematics, vol. 1924. Springer, Berlin (2008)
Acknowledgements
We are indebted to Francisco Javier Martín-Reyes for enlightening discussions related to the subject of the paper. The authors thank the referee for an expert reading and suggestions that helped improve the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
P. Hagelstein: is partially supported by a grant from the Simons Foundation (#208831 to Paul Hagelstein).
I. Parissis: is supported by Grant MTM2014-53850 of the Ministerio de Economía y Competitividad (Spain), Grant IT-641-13 of the Basque Government, and IKERBASQUE.
O. Saari: is supported by the Academy of Finland and the Väisälä Foundation.
Rights and permissions
About this article
Cite this article
Hagelstein, P., Parissis, I. & Saari, O. Sharp inequalities for one-sided Muckenhoupt weights. Collect. Math. 69, 151–161 (2018). https://doi.org/10.1007/s13348-017-0201-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13348-017-0201-y