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^s_p,q(R^n), with sR and p,q[1,+∞] such that p eq q, are not localizable in the ^p norm. Further, he has provided that the Besov spaces B^s_p,q are embedded into localized Besov spaces (B^s_p,q)_^p (i.e., B^s_p,q(B^s_p,q)_^p , for p≥ q). Also, he has provided that the localized Besov spaces (B^s_p,q)_^p are embedded into the Besov spaces B^s_p,q (i.e., (B^s_p,q)_^p B^s_p,q, for p≤ q). In particular, B_p,p^s is localizable in the ^p norm, where ^p is the space of sequences (a_k)_k such that \|(a_k)\|_^p <∞. In this paper, we generalize the Bourdaud theorem of a localization property of Besov spaces B^s_p,q(R^n) on the ^r space, where r[1,+∞]. More precisely, we show that any Besov space B^s_p,q is embedded into the localized Besov space (B^s_p,q)_^r (i.e., B^s_p,q(B^s_p,q)_^r , for r≥(p,q)). Also we show that any localized Besov space (B^s_p,q)_^r is embedded into the Besov space B^s_p,q (i.e., (B^s_p,q)_^r B^s_p,q, for r≤(p,q)). Finally, we show that the Lizorkin-Triebel spaces F^s_p,q(R^n), where sR and p[1,+∞) and q[1,+∞] are localizable in the ^p norm (i.e., F^s_p,q=(F^s_p,q)_^p ). , number=1, journal=Carpathian Mathematical Publications, author=Ferahtia, N. and Allaoui, S.E., year=2018, month=Jul., pages=71–78