A GENERALIZATION OF A LOCALIZATION PROPERTY OF BESOV SPACES

The notion of a localization property of Besov spaces is introduced by G. Bourdaud, where he has provided that the Besov spaces B-p,q(s) (R-n), with s is an element of R and p, q is an element of [1, +infinity] such that p not equal q, are not localizable in the l(p) norm. Further, he has provided that the Besov spaces B-p,q(s) are embedded into localized Besov spaces (B-p,q(s))(lp) (i.e., B-p,q(s) hooked right arrow (B-p,q(s))(lp), for p >= q). Also, he has provided that the localized Besov spaces (B-p,q(s))(lp) are embedded into the Besov spaces B-p,q(s) (i.e., (B-p,q(s))(lp) hooked right arrow B-p,q(s), for p <= q). In particular, B-p,q(s) is localizable in the l(p) norm, where l(p) is the space of sequences (a(k))(k) such that parallel to(a(k))parallel to(lp) < infinity. In this paper, we generalize the Bourdaud theorem of a localization property of Besov spaces B-p,q(s)(R-n) on the l(r) space, where r is an element of[1, +infinity]. More precisely, we show that any Besov space B-p,q(s) is embedded into the localized Besov space (B-p,q(s))(lr) (i.e., B-p,q(s) hooked right arrow (B-p,q(s))(lr), for r >= max (p, q)). Also we show that any localized Besov space (B-p,q(s))(lr) is embedded into the Besov space B-p,q(s) (i.e., (B-p,q(s))(lr) hooked right arrow B-p,q(s), for r <= min (p, q)). Finally, we show that the Lizorkin-Triebel spaces F-p,q(s)(R-n), where s is an element of R and p, q is an element of[1, +infinity] are localizable in the l(p) norm (i.e., F-p,q(s) = (F-p,q(s))(lp)).