Limiting reiteration for real interpolation with logarithmic functions

The real interpolation method X¯¯¯¯TH,q,b=(X0,X1)TH,q,b involving iterated logarithms with any number of iterations is considered. Reiteration relations of the types (X0,X¯¯¯¯0,q,a)TH,r,b and (X¯¯¯¯1,q,aX1)TH,r,b (0<TH<1) are investigated. Using any number of iterations allows in particular obtaining effects where the resulting space includes three iterated logarithms although the initial scale includes only the uniterated logarithm. Application to Lorentz-Zygmund spaces is given.