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We consider $K$-interpolation spaces involving slowly varying functions, andderive necessary and sufficient conditions for a Holmstedt-type formula to beheld in the limiting case $\theta_0=\theta_1\in\{0,1\}.$ We also study the case$\theta_0=\theta_1\in (0,1).$ Applications are given to Lorentz-Karamataspaces, generalized gamma spaces and Besov spaces.

Holmstedt's formula for the $K$-functional: the limit case $\theta_0=\theta_1$