A GENERALIZATION OF A LOCALIZATION PROPERTY OF BESOV SPACES
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Date
2021
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Université de M'sila
Abstract
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 )
Description
Keywords
Besov spaces, Lizorkin-Triebel spaces, Localization property.