Spaces on sets

This paper deals with spaces $\mathfrak B^s_{pq}$ and $\mathfrak F^s_{pq}$ of positive smoothness $s>0$, based on $L_p$-spaces with $0<p\leqslant\infty$ and reproducing formulae for smooth functions. These spaces are compared with other $B$-spaces and $F$-spaces obtained by different means.