Sharp inequalities for Riesz transforms
Studia Mathematica, Tome 222 (2014) no. 1, p. 1.
Voir la notice de l'article dans European Digital Mathematics Library
We establish the following sharp local estimate for the family
R
j
j
=
1
d
of Riesz transforms on
ℝ
d
. For any Borel subset A of
ℝ
d
and any function
f
:
ℝ
d
→
ℝ
,
∫
A
|
R
j
f
(
x
)
|
d
x
≤
C
p
|
|
f
|
|
L
p
(
ℝ
d
)
|
A
|
1
/
q
, 1 < p < ∞.
Here q = p/(p-1) is the harmonic conjugate to p,
C
p
=
[
2
q
+
2
Γ
(
q
+
1
)
/
π
q
+
1
∑
k
=
0
∞
(
-
1
)
k
/
(
2
k
+
1
)
q
+
1
]
1
/
q
, 1 < p < 2,
and
C
p
=
[
4
Γ
(
q
+
1
)
/
π
q
∑
k
=
0
∞
1
/
(
2
k
+
1
)
q
]
1
/
q
, 2 ≤ p < ∞.
This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.
Classification :
60G44, 46E30, 42B10, 42B20
Mots-clés : Riesz transform, Hilbert transform, weak-type inequality, martingale inequality, sharp constant
Mots-clés : Riesz transform, Hilbert transform, weak-type inequality, martingale inequality, sharp constant
@article{STUMA_2014__222_1_285404,
author = {Adam Os\k{e}kowski},
title = {Sharp inequalities for {Riesz} transforms},
journal = {Studia Mathematica},
pages = {1},
publisher = {mathdoc},
volume = {222},
number = {1},
year = {2014},
zbl = {1305.42011},
language = {en},
url = {https://geodesic-test.mathdoc.fr/item/STUMA_2014__222_1_285404/}
}
Adam Osękowski. Sharp inequalities for Riesz transforms. Studia Mathematica, Tome 222 (2014) no. 1, p. 1. https://geodesic-test.mathdoc.fr/item/STUMA_2014__222_1_285404/