JournalsrmiVol. 21, No. 2pp. 557–576

Extreme cases of weak type interpolation

  • Evgeniy Pustylnik

    Technion - Israel Institute of Technology, Haifa, Israel
Extreme cases of weak type interpolation cover

Abstract

We consider quasilinear operators TT of {\it joint weak type} (a,b;p,q)(a,b;p,q) (in the sense of [Bennett, Sharpley: Interpolation of operators, Academic Press, 1988]) and study their properties on spaces Lφ,EL_{\varphi,E} with the norm φ(t)f(t)E~\|\varphi(t)f^*(t) \|_{\tilde E}, where E~\tilde E is arbitrary rearrangement-invariant space with respect to the measure dt/tdt/t. A space Lφ,EL_{\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/a1/a or to 1/p1/p. For all possible kinds of such ``closeness", we give sharp estimates for the function ψ(t)\psi(t) so as to obtain that every T:Lφ,ELψ,ET:L_{\varphi,E}\to L_{\psi,E}.

Cite this article

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

DOI 10.4171/RMI/429