Extreme cases of weak type interpolation

Pustylnik, Evgeniy

Rev. Mat. Iberoamericana, Tome 21 (2005) no. 2, p. 557-576 / Harvested from Project Euclid

Résumé

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

Détail

Publié le : 2005-03-14
Classification:  rearrangement invariant spaces,  Boyd indices,  weak interpolation,  46B70,  46E30