We present reiteration formulae with limiting values θ = 0 and θ = 1 for a real interpolation method involving slowly varying functions. Applications to the Lorentz-Karamata spaces, the Fourier transform and the Riesz potential are given. In particular, our results yield improvements of limiting Sobolev-type embeddings due to Trudinger, Hansson, Brézis and Wainger, Edmunds, Gurka and Opic, Fusco, Lions and Sbordone, et al., and they are related to those of Edmunds, Kerman and Pick, or Pustylnik.

Abstract. We study generalized Lorentz-Zygmund spaces with broken logarithmic functions. We derive necessary and sufficient conditions for embeddings between them. We give a complete characterization of their associate spaces. We establish necessary and sufficient conditions for a generalized Lorentz-Zygmund space to be a Banach function space and to have absolutely continuous (quasi-)norm. We describe completely relations between these spaces and Orlicz spaces.Mathematics subject classification (1991): 46E30, 26D15.

Abstract. We present "reiteration theorems" with limiting values θ = 0 and θ = 1 for a real interpolation method involving broken-logarithmic functors. The resulting spaces lie outside of the original scale of spaces and to describe them new interpolation functors are introduced. For an ordered couple of (quasi-) Banach spaces similar results were presented without proofs by Doktorskii in [D].

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We present embedding theorems for certain logarithmic Bessel potential spaces modelled upon generalized Lorentz Zygmund spaces and clarify the role of the logarithmic terms involved in the norms of the space mentioned. In particular, we get refinements of the Sobolev embedding theorems, Trudinger's limiting embedding as well as embeddings of Sobolev spaces into space of *-Ho lder-continuous functions including the result of Bre zis and Wainger.1997 Academic Press