On Besov spaces of logarithmic smoothness and Lipschitz spaces

We compare Besov spaces Bp,q0,b with zero classical smoothness and logarithmic smoothness b   defined by using the Fourier transform with the corresponding spaces Bp,q0,b defined by means of the modulus of smoothness. In particular, we show that B2,20,b+1/2=B2,20,b for b>−1/2b>−1/2. We also determine the dual of Bp,q0,b with the help of logarithmic Lipschitz spaces Lipp,q(1,−α). Finally we show embeddings between spaces Lipp,q(1,−α) and Bp,q1,b which complement and improve embeddings established by Haroske (2000) [28].