CORE
πΊπ¦Β
Β make metadata, not war
Services
Services overview
Explore all CORE services
Access to raw data
API
Dataset
FastSync
Content discovery
Recommender
Discovery
OAI identifiers
OAI Resolver
Managing content
Dashboard
Bespoke contracts
Consultancy services
Support us
Support us
Membership
Sponsorship
Community governance
Advisory Board
Board of supporters
Research network
About
About us
Our mission
Team
Blog
FAQs
Contact us
research
L
p
β
L
q
L^p-L^q
L
p
β
L
q
estimates for maximal operators associated to families of finite type curves
Authors
Ramesh Manna
Publication date
21 April 2020
Publisher
View
on
arXiv
Abstract
We study the boundedness problem for maximal operators
M
\mathbb{M}
M
associated to averages along families of finite type curves in the plane, defined by
M
f
(
x
)
β
:
=
β
sup
β‘
1
β€
t
β€
2
β£
β«
C
f
(
x
β
t
y
)
β
Ο
(
y
)
β
d
Ο
(
y
)
β£
,
\mathbb{M}f(x) \, := \, \sup_{1 \leq t \leq 2} \left|\int_{\mathbb{C}} f(x-ty) \, \rho(y) \, d\sigma(y)\right|,
M
f
(
x
)
:=
1
β€
t
β€
2
sup
β
β
β«
C
β
f
(
x
β
t
y
)
Ο
(
y
)
d
Ο
(
y
)
β
,
where
d
Ο
d\sigma
d
Ο
denotes the normalised Lebesgue measure over the curves
C
\mathbb{C}
C
. Let
β³
\triangle
β³
be the closed triangle with vertices
P
=
(
2
5
,
1
5
)
,
Β
Q
=
(
1
2
,
1
2
)
,
Β
R
=
(
0
,
0
)
.
P=(\frac{2}{5}, \frac{1}{5}), ~ Q=(\frac{1}{2}, \frac{1}{2}), ~ R=(0, 0).
P
=
(
5
2
β
,
5
1
β
)
,
Β
Q
=
(
2
1
β
,
2
1
β
)
,
Β
R
=
(
0
,
0
)
.
In this paper, we prove that for
(
1
p
,
1
q
)
β
(
β³
β
{
P
,
Q
}
)
β©
{
(
1
p
,
1
q
)
:
q
>
m
}
(\frac{1}{p}, \frac{1}{q}) \in (\triangle \setminus \{P, Q\}) \cap \left\{(\frac{1}{p}, \frac{1}{q}) :q > m \right\}
(
p
1
β
,
q
1
β
)
β
(
β³
β
{
P
,
Q
})
β©
{
(
p
1
β
,
q
1
β
)
:
q
>
m
}
, there is a constant
B
B
B
such that
β₯
M
f
β₯
L
q
(
R
2
)
β€
β
B
β
β₯
f
β₯
L
p
(
R
2
)
\|\mathbb{M}f\|_{L^q(\mathbb{R}^2)} \leq \, B \, \|f\|_{L^p(\mathbb{R}^2)}
β₯
M
f
β₯
L
q
(
R
2
)
β
β€
B
β₯
f
β₯
L
p
(
R
2
)
β
. Furthermore, if
m
<
5
,
m <5,
m
<
5
,
then we have
β₯
M
f
β₯
L
5
,
β
(
R
2
)
β€
B
β₯
f
β₯
L
5
2
,
1
(
R
2
)
.
\|\mathbb{M}f\|_{L^{5, \infty}(\mathbb{R}^2)} \leq B \|f\|_{L^{\frac{5}{2} ,1} (\mathbb{R}^2)}.
β₯
M
f
β₯
L
5
,
β
(
R
2
)
β
β€
B
β₯
f
β₯
L
2
5
β
,
1
(
R
2
)
β
.
We shall also consider a variable coefficient version of maximal theorem and we obtain the
L
p
β
L
q
L^p-L^q
L
p
β
L
q
boundedness result for
(
1
p
,
1
q
)
β
β³
β
β©
{
(
1
p
,
1
q
)
:
q
>
m
}
,
(\frac{1}{p}, \frac{1}{q}) \in \triangle^{\circ} \cap \left\{(\frac{1}{p}, \frac{1}{q}) :q > m \right\},
(
p
1
β
,
q
1
β
)
β
β³
β
β©
{
(
p
1
β
,
q
1
β
)
:
q
>
m
}
,
where
β³
β
\triangle^{\circ}
β³
β
is the interior of the triangle with vertices
(
0
,
0
)
,
Β
(
1
2
,
1
2
)
,
Β
(
2
5
,
1
5
)
.
(0,0), ~(\frac{1}{2}, \frac{1}{2}), ~(\frac{2}{5}, \frac{1}{5}).
(
0
,
0
)
,
Β
(
2
1
β
,
2
1
β
)
,
Β
(
5
2
β
,
5
1
β
)
.
An application is given to obtain
L
p
β
L
q
L^p-L^q
L
p
β
L
q
estimates for solution to higher order, strictly hyperbolic pseudo-differential operators.Comment: 16 pages. revised version of the file. Several references have been modified. arXiv admin note: text overlap with arXiv:1510.08649, arXiv:1609.0814
Similar works
Full text
Available Versions
arXiv.org e-Print Archive
See this paper in CORE
Go to the repository landing page
Download from data provider
oai:arXiv.org:1702.06754
Last time updated on 28/02/2017