Bernstein Inequality for Multivariate Functions with Smooth Fourier Images

• Published in: Ukrainian mathematical journal Vol. 74; no. 11; pp. 1780 - 1794
• Main Authors: Bang, Ha Huy, Huy, Vu Nhat
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-04-2023; Springer; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Fourier transforms; Geometry; Mathematics; Mathematics and Statistics; Statistics
• ISSN: 0041-5995; 1573-9376
• DOI: 10.1007/s11253-023-02170-1

Let
K
be a compact set in ℝ
n
with (
O
)-property and let 1 ≤
p
≤ ∞
.
Then there exists a constant
C
K
<
∞ independent of
f
and
α
and such that
‖
D
α
f
‖
p
≤
C
K
sup
ξ
∈
K
ξ
α
‖
f
‖
H
p
,
K
,
3
for
all
α
∈
Z
+
n
and
f
∈
H
p
,
K
,
3
,
where
H
p
,
K
,
3
=
f
∈
L
p
R
n
:
supp
f
^
⊂
K
,
D
3
,
3
,
⋯
,
3
f
^
∈
C
R
n
,
‖
f
‖
H
p
,
K
,
3
=
‖
D
3
,
3
,
⋯
,
3
f
^
‖
∞
,
and
f
^
is the Fourier transform of
f.
Note that
K
is said to have the (
O
)-property if there exists a constant
C >
0 such that
sup
x
∈
K
x
α
+
e
j
≥
C
sup
x
∈
K
x
α
for
all
α
∈
Z
+
n
and
j
=
1
,
2
,
⋯
,
n
.