Distribution and rearrangement estimates of the maximal function and interpolation

Irina Asekritova; Natan Krugljak; Lech Maligranda; Lars-Erik Persson

Studia Mathematica (1997)

  • Volume: 124, Issue: 2, page 107-132
  • ISSN: 0039-3223

Abstract

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There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous methods allow us to obtain K-functional formulas in terms of the maximal function for couples of weighted L p -spaces.

How to cite

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Asekritova, Irina, et al. "Distribution and rearrangement estimates of the maximal function and interpolation." Studia Mathematica 124.2 (1997): 107-132. <http://eudml.org/doc/216401>.

@article{Asekritova1997,
abstract = {There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous methods allow us to obtain K-functional formulas in terms of the maximal function for couples of weighted $L_p$-spaces.},
author = {Asekritova, Irina, Krugljak, Natan, Maligranda, Lech, Persson, Lars-Erik},
journal = {Studia Mathematica},
keywords = {maximal functions; weights; weak type estimate; rearrangement; distribution functioni; inequalities; interpolation; K-functional; weighted spaces; distribution functions; Riesz inequality; Wiener inequality; Stein inequality; Herz inequality},
language = {eng},
number = {2},
pages = {107-132},
title = {Distribution and rearrangement estimates of the maximal function and interpolation},
url = {http://eudml.org/doc/216401},
volume = {124},
year = {1997},
}

TY - JOUR
AU - Asekritova, Irina
AU - Krugljak, Natan
AU - Maligranda, Lech
AU - Persson, Lars-Erik
TI - Distribution and rearrangement estimates of the maximal function and interpolation
JO - Studia Mathematica
PY - 1997
VL - 124
IS - 2
SP - 107
EP - 132
AB - There are given necessary and sufficient conditions on a measure dμ(x)=w(x)dx under which the key estimates for the distribution and rearrangement of the maximal function due to Riesz, Wiener, Herz and Stein are valid. As a consequence, we obtain the equivalence of the Riesz and Wiener inequalities which seems to be new even for the Lebesgue measure. Our main tools are estimates of the distribution of the averaging function f** and a modified version of the Calderón-Zygmund decomposition. Analogous methods allow us to obtain K-functional formulas in terms of the maximal function for couples of weighted $L_p$-spaces.
LA - eng
KW - maximal functions; weights; weak type estimate; rearrangement; distribution functioni; inequalities; interpolation; K-functional; weighted spaces; distribution functions; Riesz inequality; Wiener inequality; Stein inequality; Herz inequality
UR - http://eudml.org/doc/216401
ER -

References

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  1. [1] I. U. Asekritova, On the K-functional of the pair ( K Φ 0 ( X ) , K Φ 1 ( X ) ) , in: Theory of Functions of Several Real Variables, Yaroslavl’, 1980, 3-32 (in Russian). 
  2. [2] C. Bennett and R. Sharpley, Weak type inequalities for H p and BMO, in: Proc. Sympos. Pure Math. 35, Amer. Math. Soc., 1979, 201-229. Zbl0423.30026
  3. [3] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988. Zbl0647.46057
  4. [4] J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976. Zbl0344.46071
  5. [5] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. Zbl0047.10201
  6. [6] J. García-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. 
  7. [7] G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), 81-116. Zbl56.0264.02
  8. [8] C. Herz, The Hardy-Littlewood maximal theorem, in: Symposium on Harmonic Analysis, University of Warwick, 1968, 1-27. 
  9. [9] L. Maligranda, The K-functional for symmetric spaces, in: Lecture Notes in Math. 1070, Springer, 1984, 169-182. 
  10. [10] F. Riesz, Sur un théorème de maximum de MM. Hardy et Littlewood, J. London Math. Soc. 7 (1932), 10-13. 
  11. [11] P. Sjögren, A remark on the maximal function for measures in ℝ, Amer. J. Math. 105 (1983), 1231-1233. 
  12. [12] E. M. Stein, Note on the class LlogL, Studia Math. 32 (1969), 305-310. Zbl0182.47803
  13. [13] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
  14. [14] A. M. Vargas, On the maximal function for rotation invariant measures in ℝ, Studia Math. 110 (1994), 9-17. Zbl0818.42009
  15. [15] N. Wiener, The ergodic theorem, Duke Math. J. 5 (1939), 1-18. Zbl0021.23501
  16. [16] A. Zygmund, Trigonometric Series, Vol. I, Cambridge Univ. Press, Cambridge, 1959. Zbl0085.05601

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