## Revista Matemática Iberoamericana

### Extreme cases of weak type interpolation

Evgeniy Pustylnik

#### Abstract

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

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

Dates
First available in Project Euclid: 11 August 2005

https://projecteuclid.org/euclid.rmi/1123766806

Mathematical Reviews number (MathSciNet)
MR2174916

Zentralblatt MATH identifier
1092.46016

#### Citation

Pustylnik, Evgeniy. Extreme cases of weak type interpolation. Rev. Mat. Iberoamericana 21 (2005), no. 2, 557--576. https://projecteuclid.org/euclid.rmi/1123766806

#### References

• 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.