Revista Matemática Iberoamericana

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Volume 21, Issue 2, 2005, pp. 557–576
DOI: 10.4171/RMI/429

Published online: 2005-08-31

Extreme cases of weak type interpolation

Evgeniy Pustylnik[1]

(1) Technion - Israel Institute of Technology, Haifa, Israel

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 $T:L_{\varphi,E}\to L_{\psi,E}$.

Keywords: Rearrangement invariant spaces, Boyd indices, weak interpolation

Pustylnik Evgeniy: Extreme cases of weak type interpolation. Rev. Mat. Iberoam. 21 (2005), 557-576. doi: 10.4171/RMI/429