Universal Bounds for the Hardy–Littlewood Inequalities on Multilinear Forms
The Hardy–Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers m , n ≥ 2 and all m -linear forms T : ℓ p 1 n × ⋯ × ℓ p m n → K ( K = R or C ) there are constants C m ≥ 1 (not depending on n ) such that ∑ j 1 , … , j m = 1 n T ( e j 1 , … , e j m ) ρ 1...
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| Published in: | Resultate der Mathematik Vol. 73; no. 3 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cham
Springer International Publishing
01-09-2018
|
| Subjects: | |
| Online Access: | Get full text |
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| Summary: | The Hardy–Littlewood inequalities for multilinear forms on sequence spaces state that for all positive integers
m
,
n
≥
2
and all
m
-linear forms
T
:
ℓ
p
1
n
×
⋯
×
ℓ
p
m
n
→
K
(
K
=
R
or
C
) there are constants
C
m
≥
1
(not depending on
n
) such that
∑
j
1
,
…
,
j
m
=
1
n
T
(
e
j
1
,
…
,
e
j
m
)
ρ
1
ρ
≤
C
m
sup
x
1
,
…
,
x
m
≤
1
T
(
x
1
,
…
,
x
m
)
,
where
ρ
=
2
m
m
+
1
-
2
1
p
1
+
⋯
+
1
p
m
if
0
≤
1
p
1
+
⋯
+
1
p
m
≤
1
2
or
ρ
=
1
1
-
1
p
1
+
⋯
+
1
p
m
if
1
2
≤
1
p
1
+
⋯
+
1
p
m
<
1
. Good estimates for the Hardy–Littlewood constants are, in general, associated to applications in Mathematics and even in Physics, but the exact behavior of these constants is still unknown. In this note we give some new contributions to the behavior of the constants in the case
1
2
≤
1
p
1
+
⋯
+
1
p
m
<
1
. As a consequence of our main result, we present a generalization and a simplified proof of a result due to Aron et al. on certain Hardy–Littlewood type inequalities. |
|---|---|
| ISSN: | 1422-6383 1420-9012 |
| DOI: | 10.1007/s00025-018-0886-6 |