Abstract: 
We prove an interpolation theorem for weak-type operators. This is closely
related to interpolation between weak-type classes. Weak-type classes at the
ends of interpolation scales play a similar role to that played by ${\rm BMO}$
with respect to the $L^{p}$ interpolation scale. We also clarify the roles
of some of the parameters appearing in the definition of the weak-type
classes. The interpolation theorem follows from a $K$-functional inequality
for the operators, involving the Calder\'{o}n operator. The inequality was
inspired by a $K$-$J$ inequality approach developed by Jawerth and Milman.
We show that the use of the Calder\'{o}n operator is necessary. We use a new
version of the strong fundamental lemma of 
interpolation theory that does
not require the interpolation couple to be mutually closed.

1. Department of Mathematics
   Lule\aa University of Technology
   SE-971 87 Lule\aa, Sweden
   
2. Department of Mathematical Sciences
   Florida Atlantic University
   Boca Raton, FL 33431-0991, U.S.A.
   
3. Department of Mathematics
   Technion–Israel Institute of Technology
   32000 Haifa, Israel