We obtain an exact estimate for a nonincreasing uniform rearrangement of a function of two variables from the Muckenhoupt class A 1.
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B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Trans. Amer. Math. Soc., 165, 207–226 (1972).
R. Hunt, B. Muckenhoupt, and R. L. Wheeden, “Weighted norm inequalities for the conjugate function and Hilbert transform,” Trans. Amer. Math. Soc., 176, 227–251 (1973).
A. Korenovskii, “Mean oscillations and equimeasurable rearrangements of functions,” Lect. Notes Unione Mat. Ital., No. 4 (2007).
E. Yu. Leonchik, “On the Muckenhoupt condition in the multidimensional case,” Visn. Odes. Nats. Univ., 12 , Issue 7, 80–84 (2007).
G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University, Cambridge (1934).
B. Bojarski, C. Sbordone, and I. Wik, “The Muckenhoupt class \( {A_1}\left( \mathbb{R} \right) \),” Stud. Math., 101 (2), 155–163 (1992).
I. Klemes, “A mean oscillation inequality,” Proc. Amer. Math. Soc., 93, No. 3, 497–500 (1985).
E. Yu. Leonchik and N. A. Malaksiano, “Exact indices of summability for functions from the classes A ∞,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 2, 17–26 (2007).
A. P. Calderón and A. Zygmund, “On the existence of certain singular integrals,” Acta Math., 88, 85–139 (1952).
A. A. Korenovskyy, A. K. Lerner, and A. M. Stokolos, “On a multidimensional form of F. Riesz’s “rising sun” lemma,” Proc. Amer. Math. Soc., 133, No. 5, 1437–1440 (2005).
J. B. Garnett, Bounded Analytic Functions [Russian translation], Mir, Moscow (1984).
E. M. Stein, Singular Integrals and Differential Properties of Functions, Princeton University, Princeton (1970).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 8, pp. 1145–1148, August, 2010.
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Leonchik, E.Y. On an estimate for the rearrangement of a function from the Muckenhoupt class A 1 . Ukr Math J 62, 1333–1338 (2011). https://doi.org/10.1007/s11253-011-0433-z
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DOI: https://doi.org/10.1007/s11253-011-0433-z