On Sharp Olsen’s and Trace Inequalities for Multilinear Fractional Integrals

• Published in: Potential analysis Vol. 59; no. 3; pp. 1039 - 1050
• Main Authors: Grafakos, Loukas, Meskhi, Alexander
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-10-2023; Springer Nature B.V
• Subjects: Fractional calculus; Functional Analysis; Geometry; Homogeneity; Integrals; Mathematics; Mathematics and Statistics; Operators (mathematics); Partial differential equations; Potential Theory; Probability Theory and Stochastic Processes; Weighting functions; Morrey spaces; Multilinear fractional integrals; 45P05; Olsen’s inequality; 46E30; 26A33; Trace inequality
• ISSN: 0926-2601; 1572-929X
• DOI: 10.1007/s11118-022-09991-y

We establish a sharp Olsen type inequality
g
I
α
(
f
1
,
…
,
f
m
)
L
r
q
≤
C
g
L
ℓ
q
∏
j
=
1
m
f
j
L
s
j
p
j
for multilinear fractional integrals
I
α
(
f
→
)
(
x
)
=
∫
(
ℝ
n
)
m
f
1
(
y
1
)
⋯
f
m
(
y
m
)
(
|
x
−
y
1
|
+
⋯
+
|
x
−
y
m
|
)
m
n
−
α
d
y
→
,
x
∈
ℝ
n
, 0 <
α
<
m
n
, where
L
r
q
,
L
ℓ
q
,
L
s
j
p
j
,
j
= 1,…,
m
, are Morrey space with indices satisfying certain homogeneity conditions. This inequality is sharp because it gives necessary and sufficient condition on a weight function
V
for which the inequality
I
α
(
f
1
,
…
,
f
m
)
L
r
q
(
V
)
≤
C
∏
j
=
1
m
f
j
L
s
j
p
j
holds. Morrey spaces play an important role in relation to regularity problems of solutions of partial differential equations. They describe the integrability more precisely than Lebesgue spaces. We also derive a characterization of the trace inequality
B
α
(
f
1
,
f
2
)
L
r
q
(
d
μ
)
≤
C
∏
j
=
1
2
f
j
L
s
j
p
j
(
ℝ
n
)
,
in terms of a Borel measure
μ
, where
B
α
is the bilinear fractional integral operator given by the formula
B
α
(
f
1
,
f
2
)
(
x
)
=
∫
ℝ
n
f
1
(
x
+
t
)
f
2
(
x
−
t
)
|
t
|
n
−
α
d
t
,
0
<
α
<
n
,
Some of our results are new even in the linear case, i.e. when
m
= 1.