Distribution and rearrangement estimates of the maximal function and interpolation

• Autorzy: Irina U. Asekritova , Natan Ya. Krugljak , Lech Maligranda , Lars-Erik Persson
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 26D15: Inequalities for sums, series and integrals
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46M35: Abstract interpolation of topological vector spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1997
• Tom: 124
• Numer: 2
• Strony: 107-132

Daty

wydano

1997

otrzymano

1995-05-10

poprawiono

1996-12-23

Twórcy

autor

Irina U. Asekritova

• Department of Mathematics, Yaroslavl' Pedagogical University, Respublikanskaya 108, 150 000 Yaroslavl', Russia,

autor

Natan Ya. Krugljak

• Department of Mathematics, Yaroslavl' State University, Sovetskaya 14, 150 00 Yaroslavl', Russia,

autor

Lech Maligranda

• Department of Mathematics, Luleå University, S-971 87 Luleå, Sweden,

autor

Lars-Erik Persson

• Department of Mathematics, Luleå University, S-971 87 Luleå, Sweden,

Bibliografia

• [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] 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.
• [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.