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A Sharp Weak-Type (∞,∞) Inequality for the Hilbert Transform
The paper is devoted to sharp weak type ( ∞ , ∞ ) estimates for H T and H R , the Hilbert transforms on the circle and real line, respectively. Specifically, it is proved that H T f W ( T ) ≤ ‖ f ‖ L ∞ ( T ) and H R f W ( R ) ≤ ‖ f ‖ L ∞ ( R ) , where W ( T ) and W ( R ) stand for the weak- L ∞ spac...
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| Published in: | Complex analysis and operator theory 2016-08, Vol.10 (6), p.1133-1143, Article 1133 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
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| Summary: | The paper is devoted to sharp weak type
(
∞
,
∞
)
estimates for
H
T
and
H
R
, the Hilbert transforms on the circle and real line, respectively. Specifically, it is proved that
H
T
f
W
(
T
)
≤
‖
f
‖
L
∞
(
T
)
and
H
R
f
W
(
R
)
≤
‖
f
‖
L
∞
(
R
)
,
where
W
(
T
)
and
W
(
R
)
stand for the weak-
L
∞
spaces introduced by Bennett, DeVore and Sharpley. In both estimates, the constant
1
on the right is shown to be the best possible. |
|---|---|
| ISSN: | 1661-8254 1661-8262 |
| DOI: | 10.1007/s11785-015-0454-y |