Generalized Besov spaces and Triebel-Lizorkin spaces

In this paper the classical Besov spaces B _p,q ^s and Triebel-Lizorkin spaces F _p,q ^s for s are generalized in an isotropy way with the smoothness weights {|2j|-ln α}∞-j = 0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by B-p,q α and F-p,q α for α^k and k , respectively, keep many interesting properties, such as embedding theorems (with scales property for all smoothness weights), lifting properties for all parameters \vec α, and duality for index 0 < p < ∞. By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between B _p,q/s and _t>s B _p,q ^t , and between F _p,q ^s and _t>s F _p,q ^t , respectively.