A variant of the Λ(p)-set problem in Orlicz spaces
We introduce Λ ( Φ ) -sets as generalizations of Λ ( p ) -sets. These sets are defined in terms of Orlicz norms. We consider Λ ( Φ ) -sets when the Matuszewska-Orlicz index of Φ is larger than 2. When S is a Λ ( Φ ) -set, we establish an estimate of the size of S ∩ [ - N , N ] where N ∈ N . Next, we...
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| Veröffentlicht in: | Mathematische Zeitschrift 2022-12, Vol.302 (4), p.2545-2566 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | eng |
| Schlagworte: | |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | We introduce
Λ
(
Φ
)
-sets as generalizations of
Λ
(
p
)
-sets. These sets are defined in terms of Orlicz norms. We consider
Λ
(
Φ
)
-sets when the Matuszewska-Orlicz index of
Φ
is larger than 2. When
S
is a
Λ
(
Φ
)
-set, we establish an estimate of the size of
S
∩
[
-
N
,
N
]
where
N
∈
N
. Next, we construct a
Λ
(
Φ
1
)
-set which is not a
Λ
(
Φ
2
)
-set for any
Φ
2
such that
sup
u
≥
1
Φ
2
(
u
)
/
Φ
1
(
u
)
=
∞
by using a probabilistic method. With an additional assumption about a subset
E
of
Z
, we can construct such a
Λ
(
Φ
1
)
-set contained in
E
. These statements extend known results on the structure of
Λ
(
p
)
-sets to
Λ
(
Φ
)
-sets. |
|---|---|
| ISSN: | 0025-5874 1432-1823 |
| DOI: | 10.1007/s00209-022-03139-9 |