Revista Matemática Iberoamericana

Extreme cases of weak type interpolation

Evgeniy Pustylnik

Full-text: Open access


We consider quasilinear operators $T$ of {\it joint weak type} $(a,b;p,q)$ (in the sense of [Bennett, Sharpley: Interpolation of operators, Academic Press, 1988]) and study their properties on spaces $L_{\varphi,E}$ with the norm $\|\varphi(t)f^*(t) \|_{\tilde E}$, where $\tilde E$ is arbitrary rearrangement-invariant space with respect to the measure $dt/t$. A space $L_{\varphi,E}$ is said to be ``close" to one of the endpoints of interpolation if the corresponding Boyd index of this space is equal to $1/a$ or to $1/p$. For all possible kinds of such ``closeness", we give sharp estimates for the function $\psi(t)$ so as to obtain that every $T:L_{\varphi,E}\to L_{\psi,E}$.

Article information

Rev. Mat. Iberoamericana, Volume 21, Number 2 (2005), 557-576.

First available in Project Euclid: 11 August 2005

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B70: Interpolation between normed linear spaces [See also 46M35] 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

rearrangement invariant spaces Boyd indices weak interpolation


Pustylnik, Evgeniy. Extreme cases of weak type interpolation. Rev. Mat. Iberoamericana 21 (2005), no. 2, 557--576.

Export citation


  • Bennett, C. and Rudnick, K.: On Lorentz-Zygmund spaces. Dissertationes Math. 175 (1980), 5-67.
  • Bennett, C. and Sharpley, R.: Interpolation of operators. Pure and Applied Mathematics 129. Academic Press, Boston, MA, 1988.
  • Boyd, D.: Indices of function spaces and their relationship to interpolation. Canad. J. Math. 21 (1969), 1245-1254.
  • Calderón, A.P.: Spaces between $L^1$ and $L^\i$ and the theorem of Marcinkiewicz. Studia Math. 26 (1966), 273-299.
  • Cwikel, M. and Pustylnik, E.: Weak type interpolation near ``endpoint" spaces. J. Funct. Anal. 171 (2000), no. 2, 235-277.
  • Evans, W.D., Opic, B. and Pick, L.: Interpolation of operators on scales of generalized Lorentz-Zygmund spaces. Math. Nachr. 182 (1996), 127-181.
  • Kre\u in, S. G.; Petunin, J.I. and Semenov, E.M.: Interpolation of linear operators. Translations of Mathematical Monographs 54. American Mathematical Society, Providence R.I., 1982.
  • Lorentz, G.G.: Relations between function spaces. Proc. Amer. Math. Soc. 12 (1961), 127-132.
  • Pustylnik, E.: Optimal interpolation in spaces of Lorentz-Zygmund type. J. Anal. Math. 79 (1999), 113-157.
  • Pustylnik, E.: Real interpolation for non-distant Marcinkiewicz spaces. Rev. Mat. Complut. 14 (2001), no. 1, 127-143.
  • Pustylnik, E.: Ultrasymmetric spaces. J. London Math. Soc.(2) 68 (2003), no. 1, 165-182.