The multilinear Hörmander multiplier theorem with a Lorentz–Sobolev condition

• Published in: Annali di matematica pura ed applicata Vol. 201; no. 1; pp. 111 - 126
• Main Authors: Grafakos, Loukas, Park, Bae Jun
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-02-2022; Springer Nature B.V
• Subjects: Mathematics; Mathematics and Statistics; Sobolev space; Theorems; Hörmander’s multiplier theorem; Multilinear operators; 42B25; Primary 42B15
• ISSN: 0373-3114; 1618-1891
• DOI: 10.1007/s10231-021-01109-2

In this article, we provide a multilinear version of the Hörmander multiplier theorem with a Lorentz–Sobolev space condition. The work is motivated by the recent result of the first author and Slavíková [
12
] where an analogous version of classical Hörmander multiplier theorem was obtained; this version is sharp in many ways and reduces the number of indices that appear in the statement of the theorem. As a natural extension of the linear case, in this work, we prove that if
m
n
/
2
<
s
<
m
n
, then
‖
T
σ
(
f
1
,
⋯
,
f
m
)
‖
L
p
(
R
n
)
≲
sup
k
∈
Z
‖
σ
(
2
k
·
)
Ψ
(
m
)
^
‖
L
s
m
n
/
s
,
1
(
R
mn
)
‖
f
1
‖
L
p
1
(
R
n
)
⋯
‖
f
m
‖
L
p
m
(
R
n
)
for certain
p
,
p
1
,
⋯
,
p
m
with
1
/
p
=
1
/
p
1
+
⋯
+
1
/
p
m
. We also show that the above estimate is sharp, in the sense that the Lorentz–Sobolev space
L
s
m
n
/
s
,
1
cannot be replaced by
L
s
r
,
q
for
r
<
m
n
/
s
,
0
<
q
≤
∞
, or by
L
s
m
n
/
s
,
q
for
q
>
1
.