Siberian Mathematical Journal

, Volume 49, Issue 2, pp 322–338 | Cite as

Interpolation of operators of weak type (ϕ, ϕ)

  • B. I. PeleshenkoEmail author


Considering the measurable and nonnegative functions ϕ on the half-axis [0, ∞) such that ϕ(0) = 0 and ϕ(t) → ∞ as t → ∞, we study the operators of weak type (ϕ, ϕ) that map the classes of ϕ-Lebesgue integrable functions to the space of Lebesgue measurable real functions on ℝn. We prove interpolation theorems for the subadditive operators of weak type (ϕ0, ϕ0) bounded in L (ℝn) and subadditive operators of weak types (ϕ0, ϕ0) and (ϕ1, ϕ1) in L ϕ(ℝ n ) under some assumptions on the nonnegative and increasing functions ϕ(x) on [0, ∞). We also obtain some interpolation theorems for the linear operators of weak type (ϕ0, ϕ0) bounded from L (ℝn) to BMO(ℝ n). For the restrictions of these operators to the set of characteristic functions of Lebesgue measurable sets, we establish some estimates for rearrangements of moduli of their values; deriving a consequence, we obtain a theorem on the boundedness of operators in rearrangement-invariant spaces.


interpolation of operators ϕ-integrable function operator of weak type rearrangement-invariant space modular inequality 


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  1. 1.
    Stampacchia G., “The spaces L (p, λ), N (p, λ) and interpolation,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. Ser. III, 19, No. 3, 443–462 (1965).zbMATHMathSciNetGoogle Scholar
  2. 2.
    Campanato S., “Su un teorema di interpolazione di G. Stampacchia,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 20, No. 3, 649–652 (1966).zbMATHMathSciNetGoogle Scholar
  3. 3.
    Spanne S., “Sur l’interpolation entre les espaces L kp, ϕ,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 20, No. 3, 625–648 (1966).zbMATHMathSciNetGoogle Scholar
  4. 4.
    Riviere N. M., “Interpolation a la Marcinkiewicz,” Rev. Mat. Argentina, 25, 363–377 (1971).zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bennett C. and Rudnick K., On Lorentz-Zygmund Spaces, Panstw. Wydawn. Nauk, Warszawa (1980).Google Scholar
  6. 6.
    Bergh J. and Lofstrom J., Interpolation Spaces. An Introduction [Russian translation], Mir, Moscow (1980).Google Scholar
  7. 7.
    Stein E. and Weiss G., Introduction to Harmonic Analysis on Euclidean Spaces [Russian translation], Mir, Moscow (1974).Google Scholar
  8. 8.
    Hunt R. A., “On L(p, q) spaces,” Enseign. Math., 12, 249–276 (1966).zbMATHGoogle Scholar
  9. 9.
    Calderón A. P., “Spaces between L 1 and L and the theorem of Marcinkiewicz,” Studia Math., 26, No. 3, 273–299 (1966).zbMATHMathSciNetGoogle Scholar
  10. 10.
    Riviere N. M., “Singular integrals and multiplier operators,” Ark. Mat., 9, No. 2, 243–278 (1971).zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Dmitriev V. I. and Krein S. G., “Interpolation of operators of weak type,” Anal. Math., 4, No. 2, 83–99 (1978).zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Krein S. G., Petunin Yu. I., and Semenov E. M., Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).Google Scholar
  13. 13.
    de Guzmán M., Differentiation of Integrals in ℝn, Springer-Verlag, Berlin (1975).Google Scholar
  14. 14.
    Zygmund A., Trigonometric Series. Vol. 1 [Russian translation], Mir, Moscow (1965).zbMATHGoogle Scholar
  15. 15.
    John F. and Nirenberg L., “On functions of bounded mean oscillation,” Comm. Pure Appl. Math., 14, No. 3, 415–426 (1961).zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J. (1970).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Dnepropetrovsk Agricultural State UniversityDnepropetrovskthe Ukraine

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