We work with spaces (A 0 , A 1) θ,q,A which are logarithmic perturbations of the real interpolation spaces. We determine the dual of (A 0 , A 1) θ,q,A when 0 < q < 1. As we show, if θ = 0 or 1 then the dual space depends on the relationship between q and A. Furthermore we apply the abstract results to compute the dual space of Besov spaces of logarithmic smoothness and the dual space of spaces of compact operators in a Hilbert space which are close to the Macaev ideals.

We work with Triebel-Lizorkin spaces F s q L p,r (R n ) and Besov spaces B s q L p,r (R n ) with Lorentz smoothness. Using their characterizations by real interpolation we show how to transfer a number of properties of the usual Triebel-Lizorkin and Besov spaces to the spaces with Lorentz smoothness. In particular, we give results on diffeomorphisms, extension operators, multipliers and we also show sufficient conditions on parameters for F s q L p,r (R n ) and B s q L p,r (R n ) to be multiplication algebras.

Working in the setting of quasi-Banach couples, we establish a formula for the measure of non-compactness of bilinear operators interpolated by the general real method. The result applies to the real method and to the real method with a function parameter.

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