A GENERALIZATION OF A LOCALIZATION PROPERTY OF BESOV SPACES - FERAHTIA N; ALLAOUI S.E (Université de M'sila, 2021)
The notion of a localization property of Besov spaces is introduced by G. Bourdaud, where he has provided that the Besov spaces Bs p,q(Rn), with s 2 R and p, q 2 [1,+¥] such that p 6= q, are not localizable in the `p norm. Further, he has provided that the Besov spaces Bs p,q are embedded into localized Besov spaces (Bs p,q)`p (i.e., Bs p,q ,! (Bs p,q)`p , for p q). Also, he has provided that the localized Besov spaces (Bs p,q)`p are embedded into the Besov spaces Bs p,q (i.e., (Bs p,q)`p ,! Bs p,q, for p q). In particular, Bs p,p is localizable in the `p norm, where `p is the space of sequences (ak)k such that k(ak)k`p < ¥. In this paper, we generalize the Bourdaud theorem of a localization property of Besov spaces Bs p,q(Rn) on the `r space, where r 2 [1,+¥]. More precisely, we show that any Besov space Bs p,q is embedded into the localized Besov space (Bs p,q)`r (i.e., Bs p,q ,! (Bs p,q)`r , for r max(p, q)). Also we show that any localized Besov space (Bs p,q)`r is embedded into the Besov space Bs p,q (i.e., (Bs p,q)`r ,! Bs p,q, for r min(p, q)). Finally, we show that the Lizorkin-Triebel spaces Fsp ,q(Rn), where s 2 R and p, q 2 [1,+¥] are localizable in the `p norm (i.e., Fsp ,q = (Fsp,q)`p )
Keywords: Besov spaces, Lizorkin-Triebel spaces, Localization property.