Description of logarithmic interpolation spaces by means of the J-functional and applications

We work with logarithmic interpolation methods (A0,A1)θ,q,A(A0,A1)θ,q,A where θ=0θ=0 or 1. On the contrary to the case 0<θ<10<θ<1, we show that their description in terms of the J-functional changes depending on the relationship between q   and AA, and that there is no description in a certain range. Then we use these J  -descriptions to investigate the behavior of compact operators and weakly compact operators under logarithmic interpolation methods. In particular, we extend a recent compactness result of Edmunds and Opic for operators between LpLp-spaces over finite measure spaces to σ  -finite measure spaces. We also determine the dual of (A0,A1)θ,q,A(A0,A1)θ,q,A when θ=0θ=0 or 1.