On various approaches to Besov-type spaces of variable smoothness

The paper is concerned with Besov spaces of variable smoothness Bp,qφ0(Rn,{tk}), in which the norms are defined in terms of convolutions with smooth functions. A relation is found between the spaces Bp,qφ0(Rn,{tk}) and the spaces B˜p,q,rl(Rn,{tk}), which were introduced earlier by the author. As an application an equivalent description of the trace space of the weighted Sobolev space with Muckenhoupt weight is given.