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ArticleOriginal scientific text

Title

Distribution and rearrangement estimates of the maximal function and interpolation

Authors 1, 2, 3, 3

Affiliations

  1. Department of Mathematics, Yaroslavl' Pedagogical University, Respublikanskaya 108, 150 000 Yaroslavl', Russia
  2. Department of Mathematics, Yaroslavl' State University, Sovetskaya 14, 150 00 Yaroslavl', Russia
  3. Department of Mathematics, Luleå University, S-971 87 Luleå, Sweden

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 Lp-spaces.

Keywords

ENG
maximal functions, weights, weak type estimate, rearrangement, distribution functioni, inequalities, interpolation, K-functional, weighted spaces

Bibliography

  1. I. U. Asekritova, On the K-functional of the pair (KΦ_0(X),KΦ_1(X)), in: Theory of Functions of Several Real Variables, Yaroslavl', 1980, 3-32 (in Russian).
  2. C. Bennett and R. Sharpley, Weak type inequalities for Hp and BMO, in: Proc. Sympos. Pure Math. 35, Amer. Math. Soc., 1979, 201-229.
  3. C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
  4. J. Bergh and J. Löfström, Interpolation Spaces, Springer, Berlin, 1976.
  5. A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139.
  6. J. García-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.
  7. G. H. Hardy and J. E. Littlewood, A maximal theorem with function-theoretic applications, Acta Math. 54 (1930), 81-116.
  8. C. Herz, The Hardy-Littlewood maximal theorem, in: Symposium on Harmonic Analysis, University of Warwick, 1968, 1-27.
  9. L. Maligranda, The K-functional for symmetric spaces, in: Lecture Notes in Math. 1070, Springer, 1984, 169-182.
  10. F. Riesz, Sur un théorème de maximum de MM. Hardy et Littlewood, J. London Math. Soc. 7 (1932), 10-13.
  11. P. Sjögren, A remark on the maximal function for measures in ℝ, Amer. J. Math. 105 (1983), 1231-1233.
  12. E. M. Stein, Note on the class LlogL, Studia Math. 32 (1969), 305-310.
  13. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
  14. A. M. Vargas, On the maximal function for rotation invariant measures in ℝ, Studia Math. 110 (1994), 9-17.
  15. N. Wiener, The ergodic theorem, Duke Math. J. 5 (1939), 1-18.
  16. A. Zygmund, Trigonometric Series, Vol. I, Cambridge Univ. Press, Cambridge, 1959.
Studia Mathematica
1997/124/2
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Pages:
107-132
Main language of publication
English
Received
1995-05-10
Accepted
1996-12-23
Published
1997
Exact and natural sciences