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On a multidimensional form of F. Riesz's ``rising sun" lemma

Authors: A. A. Korenovskyy, A. K. Lerner and A. M. Stokolos
Journal: Proc. Amer. Math. Soc. 133 (2005), 1437-1440
MSC (2000): Primary 42B25
Published electronically: November 22, 2004
MathSciNet review: 2111942
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Abstract: A multidimensional version of the Riesz rising sun lemma is proved by means of a generalized dyadic process.

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  • 1. A.S. Besicovitch, On differentiation of Lebesgue double integrals, Fundam. Math. 25 (1935), 209-216.
  • 2. A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 0052553
  • 3. Loukas Grafakos and Stephen Montgomery-Smith, Best constants for uncentred maximal functions, Bull. London Math. Soc. 29 (1997), no. 1, 60–64. MR 1416408, 10.1112/S0024609396002081
  • 4. Miguel de Guzmán, Differentiation of integrals in 𝑅ⁿ, Lecture Notes in Mathematics, Vol. 481, Springer-Verlag, Berlin-New York, 1975. With appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón. MR 0457661
  • 5. Ivo Klemes, A mean oscillation inequality, Proc. Amer. Math. Soc. 93 (1985), no. 3, 497–500. MR 774010, 10.1090/S0002-9939-1985-0774010-0
  • 6. A. A. Korenovskiĭ, The connection between mean oscillations and exact exponents of summability of functions, Mat. Sb. 181 (1990), no. 12, 1721–1727 (Russian); English transl., Math. USSR-Sb. 71 (1992), no. 2, 561–567. MR 1099524
  • 7. A. A. Korenovskiĭ, The inverse Hölder inequality, the Muckenhoupt condition, and equimeasurable permutations of functions, Dokl. Akad. Nauk 323 (1992), no. 2, 229–232 (Russian); English transl., Russian Acad. Sci. Dokl. Math. 45 (1992), no. 2, 301–304 (1993). MR 1191537
  • 8. A. A. Korenovskiĭ, Sharp extension of a reverse Hölder inequality and the Muckenhoupt condition, Mat. Zametki 52 (1992), no. 6, 32–44, 158 (Russian, with Russian summary); English transl., Math. Notes 52 (1992), no. 5-6, 1192–1201 (1993). MR 1208001, 10.1007/BF01209371
  • 9. A.A. Korenovskyy, On the Gurov-Reshetnyak class of functions, submitted.
  • 10. A.A. Korenovskyy, The Riesz rising sun lemma for several variables and John-Nirenberg inequality, Math. Notes, to appear.
  • 11. F. Riesz, Sur l'existence de la dérivée des fonctions monotones et sur quelques problèmes qui s'y rattachent, Acta Sci. Math. Szeged 5 (1932), 208-221.
  • 12. F. Riesz, Sur un théorème de maximum de MM. Hardy et Littlewood, J. London Math. Soc. 7(1932), 10-13.
  • 13. Peter Sjögren, A remark on the maximal function for measures in 𝑅ⁿ, Amer. J. Math. 105 (1983), no. 5, 1231–1233. MR 714775, 10.2307/2374340
  • 14. Elias M. Stein, Singular integrals: the roles of Calderón and Zygmund, Notices Amer. Math. Soc. 45 (1998), no. 9, 1130–1140. MR 1640159

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Additional Information

A. A. Korenovskyy
Affiliation: Department of Mathematical Analysis, IMEM, National University of Odessa, Dvoryanskaya, 2, 65026 Odessa, Ukraine

A. K. Lerner
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel

A. M. Stokolos
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois, 60614

Keywords: ``Rising sun" lemma, dyadic property, differential basis
Received by editor(s): August 13, 2003
Received by editor(s) in revised form: January 15, 2004
Published electronically: November 22, 2004
Additional Notes: The work of the first author was partially supported by the France-Ukraine program of scientific collaboration “DNIPRO"
Communicated by: Andreas Seeger
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.