The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function

Yıl: 2019 Cilt: 43 Sayı: 3 Sayfa Aralığı: 1755 - 1769 Metin Dili: İngilizce DOI: 10.3906/mat-1901-41 İndeks Tarihi: 15-05-2020

The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function

Öz:
The aim of this work is to extend the recent work of the author on the discrete frequency function to themore delicate continuous frequency function T , and further to investigate its relations to the Hardy–Littlewood maximalfunction M, and to the Lebesgue points. We surmount the intricate issue of measurability of T f by approaching itwith a sequence of carefully constructed auxiliary functions for which measurability is easier to prove. After this, wegive analogues of the recent results on the discrete frequency function. We then connect the points of discontinuity ofMf for f simple to the zeros of T f , and to the non-Lebesgue points of f .
Anahtar Kelime:

Konular: Matematik
Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
  • [1] Folland GB. Real Analysis. 2nd ed. New York, NY, USA: Wiley, 1999.
  • [2] Grahl J. Exceptional Lebesgue densities and random Riemann sums. PhD, University College London, London, UK, 2011.
  • [3] Hajłasz P, Onninen J. On boundedness of maximal functions in Sobolev spaces. Annales Academiæ Scientiarum Fennicæ Mathematica 2004; 29 (1): 167-176.
  • [4] Kurka O. On the variation of the Hardy–Littlewood maximal function. Annales Academiæ Scientiarum Fennicæ- Mathematica 2015; 40: 100-133.
  • [5] Kurka O. Optimal quality of exceptional points for the Lebesgue density theorem. Acta Mathematica Hungarica 2012; 134 (3): 209-268.
  • [6] Lin F, Yang X. Geometric Measure Theory. Beijing, China: Science press, 2002.
  • [7] Lacey MT. The bilinear maximal functions map into Lp for 2/3 < p 1. Annals of Mathematics, second series 2001; 151 (1): 35-57.
  • [8] Luiro L. Continuity of the maximal operator in Sobolev spaces. Proceedings of American Mathematical Society 2007; 135 (1): 243-251.
  • [9] Luiro H. Regularity properties of maximal functions. PhD, University of Jyväskylä, Jyväskylä, Finland, 2008.
  • [10] Rudin W. Real Analysis. 3rd edition, USA: McGraw-Hill, 1987.
  • [11] Temur F. Level set estimates for the discrete frequency function. Journal of Fourier Analysis and Applications 2018. doi: 10.1007/500041-018-9595-5
  • [12] Stein E. Harmonic analysis. Princeton, NJ, USA: Princeton University Press, 1993.
  • [13] Steinerberger S. A rigidity phenomenon for the Hardy–Littlewood maximal function. Studia Mathematica 2015; 229: 263-278.
  • [14] Szenes A. Exceptional points for Lebesgue’s density theorem on the real line. Advances in Mathematics 2011; 226 (1): 764-778.
APA Temur F (2019). The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. , 1755 - 1769. 10.3906/mat-1901-41
Chicago Temur Faruk The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. (2019): 1755 - 1769. 10.3906/mat-1901-41
MLA Temur Faruk The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. , 2019, ss.1755 - 1769. 10.3906/mat-1901-41
AMA Temur F The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. . 2019; 1755 - 1769. 10.3906/mat-1901-41
Vancouver Temur F The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. . 2019; 1755 - 1769. 10.3906/mat-1901-41
IEEE Temur F "The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function." , ss.1755 - 1769, 2019. 10.3906/mat-1901-41
ISNAD Temur, Faruk. "The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function". (2019), 1755-1769. https://doi.org/10.3906/mat-1901-41
APA Temur F (2019). The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. Turkish Journal of Mathematics, 43(3), 1755 - 1769. 10.3906/mat-1901-41
Chicago Temur Faruk The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. Turkish Journal of Mathematics 43, no.3 (2019): 1755 - 1769. 10.3906/mat-1901-41
MLA Temur Faruk The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. Turkish Journal of Mathematics, vol.43, no.3, 2019, ss.1755 - 1769. 10.3906/mat-1901-41
AMA Temur F The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. Turkish Journal of Mathematics. 2019; 43(3): 1755 - 1769. 10.3906/mat-1901-41
Vancouver Temur F The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function. Turkish Journal of Mathematics. 2019; 43(3): 1755 - 1769. 10.3906/mat-1901-41
IEEE Temur F "The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function." Turkish Journal of Mathematics, 43, ss.1755 - 1769, 2019. 10.3906/mat-1901-41
ISNAD Temur, Faruk. "The frequency function and its connections to the Lebesgue points and the Hardy–Littlewood maximal function". Turkish Journal of Mathematics 43/3 (2019), 1755-1769. https://doi.org/10.3906/mat-1901-41