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On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data
Let α , β be orientation-preserving diffeomorphism (shifts) of R + = ( 0 , ∞ ) onto itself with the only fixed points 0 and ∞ and U α , U β be the isometric shift operators on L p ( R + ) given by U α f = ( α ′ ) 1 / p ( f ∘ α ) , U β f = ( β ′ ) 1 / p ( f ∘ β ) , and P 2 ± = ( I ± S 2 ) / 2 where (...
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Published in: | Complex analysis and operator theory 2016-08, Vol.10 (6), p.1101-1131 |
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container_start_page | 1101 |
container_title | Complex analysis and operator theory |
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creator | Karlovich, Alexei Yu Karlovich, Yuri I. Lebre, Amarino B. |
description | Let
α
,
β
be orientation-preserving diffeomorphism (shifts) of
R
+
=
(
0
,
∞
)
onto itself with the only fixed points
0
and
∞
and
U
α
,
U
β
be the isometric shift operators on
L
p
(
R
+
)
given by
U
α
f
=
(
α
′
)
1
/
p
(
f
∘
α
)
,
U
β
f
=
(
β
′
)
1
/
p
(
f
∘
β
)
, and
P
2
±
=
(
I
±
S
2
)
/
2
where
(
S
2
f
)
(
t
)
:
=
1
π
i
∫
0
∞
t
τ
1
/
2
-
1
/
p
f
(
τ
)
τ
-
t
d
τ
,
t
∈
R
+
,
is the weighted Cauchy singular integral operator. We prove that if
α
′
,
β
′
and
c
,
d
are continuous on
R
+
and slowly oscillating at
0
and
∞
, and
lim sup
t
→
s
|
c
(
t
)
|
<
1
,
lim sup
t
→
s
|
d
(
t
)
|
<
1
,
s
∈
{
0
,
∞
}
,
then the operator
(
I
-
c
U
α
)
P
2
+
+
(
I
-
d
U
β
)
P
2
-
is Fredholm on
L
p
(
R
+
)
and its index is equal to zero. Moreover, its regularizers are described. |
doi_str_mv | 10.1007/s11785-015-0452-0 |
format | article |
fullrecord | <record><control><sourceid>crossref_sprin</sourceid><recordid>TN_cdi_crossref_primary_10_1007_s11785_015_0452_0</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>10_1007_s11785_015_0452_0</sourcerecordid><originalsourceid>FETCH-LOGICAL-c288t-98f8ba15e8a5791578a8ca1f2c3b24a982766e0c4dc869c06967008825fb57da3</originalsourceid><addsrcrecordid>eNp9kE1LAzEQhoMoWKs_wFv-wOok3c3HUepHC4UVqngM0zS73RJ3S5JS-u9NqXj0MMwc3meYeQi5Z_DAAORjZEyqqgCWq6x4ARdkxIRgheKCX_7NVXlNbmLcAgiQWo_Ie91TpF-uazfJremy69u9x0DnfXJtQE_rnQuYhkAPXdrQ5aZrUqTY56gfDv5I62g77zFlkD5jwlty1aCP7u63j8nn68vHdFYs6rf59GlRWK5UKrRq1ApZ5RRWUrNKKlQWWcPtZMVL1IpLIRzYcm2V0BaEFhJA5Q-aVSXXOBkTdt5rwxBjcI3Zhe4bw9EwMCcl5qzEZCXmpMRAZviZiTnbty6Y7bAPfT7zH-gH_YZjjg</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data</title><source>Springer Nature</source><creator>Karlovich, Alexei Yu ; Karlovich, Yuri I. ; Lebre, Amarino B.</creator><creatorcontrib>Karlovich, Alexei Yu ; Karlovich, Yuri I. ; Lebre, Amarino B.</creatorcontrib><description>Let
α
,
β
be orientation-preserving diffeomorphism (shifts) of
R
+
=
(
0
,
∞
)
onto itself with the only fixed points
0
and
∞
and
U
α
,
U
β
be the isometric shift operators on
L
p
(
R
+
)
given by
U
α
f
=
(
α
′
)
1
/
p
(
f
∘
α
)
,
U
β
f
=
(
β
′
)
1
/
p
(
f
∘
β
)
, and
P
2
±
=
(
I
±
S
2
)
/
2
where
(
S
2
f
)
(
t
)
:
=
1
π
i
∫
0
∞
t
τ
1
/
2
-
1
/
p
f
(
τ
)
τ
-
t
d
τ
,
t
∈
R
+
,
is the weighted Cauchy singular integral operator. We prove that if
α
′
,
β
′
and
c
,
d
are continuous on
R
+
and slowly oscillating at
0
and
∞
, and
lim sup
t
→
s
|
c
(
t
)
|
<
1
,
lim sup
t
→
s
|
d
(
t
)
|
<
1
,
s
∈
{
0
,
∞
}
,
then the operator
(
I
-
c
U
α
)
P
2
+
+
(
I
-
d
U
β
)
P
2
-
is Fredholm on
L
p
(
R
+
)
and its index is equal to zero. Moreover, its regularizers are described.</description><identifier>ISSN: 1661-8254</identifier><identifier>EISSN: 1661-8262</identifier><identifier>DOI: 10.1007/s11785-015-0452-0</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Analysis ; Mathematics ; Mathematics and Statistics ; Operator Theory</subject><ispartof>Complex analysis and operator theory, 2016-08, Vol.10 (6), p.1101-1131</ispartof><rights>Springer Basel 2015</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c288t-98f8ba15e8a5791578a8ca1f2c3b24a982766e0c4dc869c06967008825fb57da3</citedby><cites>FETCH-LOGICAL-c288t-98f8ba15e8a5791578a8ca1f2c3b24a982766e0c4dc869c06967008825fb57da3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27923,27924</link.rule.ids></links><search><creatorcontrib>Karlovich, Alexei Yu</creatorcontrib><creatorcontrib>Karlovich, Yuri I.</creatorcontrib><creatorcontrib>Lebre, Amarino B.</creatorcontrib><title>On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data</title><title>Complex analysis and operator theory</title><addtitle>Complex Anal. Oper. Theory</addtitle><description>Let
α
,
β
be orientation-preserving diffeomorphism (shifts) of
R
+
=
(
0
,
∞
)
onto itself with the only fixed points
0
and
∞
and
U
α
,
U
β
be the isometric shift operators on
L
p
(
R
+
)
given by
U
α
f
=
(
α
′
)
1
/
p
(
f
∘
α
)
,
U
β
f
=
(
β
′
)
1
/
p
(
f
∘
β
)
, and
P
2
±
=
(
I
±
S
2
)
/
2
where
(
S
2
f
)
(
t
)
:
=
1
π
i
∫
0
∞
t
τ
1
/
2
-
1
/
p
f
(
τ
)
τ
-
t
d
τ
,
t
∈
R
+
,
is the weighted Cauchy singular integral operator. We prove that if
α
′
,
β
′
and
c
,
d
are continuous on
R
+
and slowly oscillating at
0
and
∞
, and
lim sup
t
→
s
|
c
(
t
)
|
<
1
,
lim sup
t
→
s
|
d
(
t
)
|
<
1
,
s
∈
{
0
,
∞
}
,
then the operator
(
I
-
c
U
α
)
P
2
+
+
(
I
-
d
U
β
)
P
2
-
is Fredholm on
L
p
(
R
+
)
and its index is equal to zero. Moreover, its regularizers are described.</description><subject>Analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operator Theory</subject><issn>1661-8254</issn><issn>1661-8262</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2016</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LAzEQhoMoWKs_wFv-wOok3c3HUepHC4UVqngM0zS73RJ3S5JS-u9NqXj0MMwc3meYeQi5Z_DAAORjZEyqqgCWq6x4ARdkxIRgheKCX_7NVXlNbmLcAgiQWo_Ie91TpF-uazfJremy69u9x0DnfXJtQE_rnQuYhkAPXdrQ5aZrUqTY56gfDv5I62g77zFlkD5jwlty1aCP7u63j8nn68vHdFYs6rf59GlRWK5UKrRq1ApZ5RRWUrNKKlQWWcPtZMVL1IpLIRzYcm2V0BaEFhJA5Q-aVSXXOBkTdt5rwxBjcI3Zhe4bw9EwMCcl5qzEZCXmpMRAZviZiTnbty6Y7bAPfT7zH-gH_YZjjg</recordid><startdate>20160801</startdate><enddate>20160801</enddate><creator>Karlovich, Alexei Yu</creator><creator>Karlovich, Yuri I.</creator><creator>Lebre, Amarino B.</creator><general>Springer International Publishing</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20160801</creationdate><title>On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data</title><author>Karlovich, Alexei Yu ; Karlovich, Yuri I. ; Lebre, Amarino B.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c288t-98f8ba15e8a5791578a8ca1f2c3b24a982766e0c4dc869c06967008825fb57da3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2016</creationdate><topic>Analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operator Theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Karlovich, Alexei Yu</creatorcontrib><creatorcontrib>Karlovich, Yuri I.</creatorcontrib><creatorcontrib>Lebre, Amarino B.</creatorcontrib><collection>CrossRef</collection><jtitle>Complex analysis and operator theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Karlovich, Alexei Yu</au><au>Karlovich, Yuri I.</au><au>Lebre, Amarino B.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data</atitle><jtitle>Complex analysis and operator theory</jtitle><stitle>Complex Anal. Oper. Theory</stitle><date>2016-08-01</date><risdate>2016</risdate><volume>10</volume><issue>6</issue><spage>1101</spage><epage>1131</epage><pages>1101-1131</pages><issn>1661-8254</issn><eissn>1661-8262</eissn><abstract>Let
α
,
β
be orientation-preserving diffeomorphism (shifts) of
R
+
=
(
0
,
∞
)
onto itself with the only fixed points
0
and
∞
and
U
α
,
U
β
be the isometric shift operators on
L
p
(
R
+
)
given by
U
α
f
=
(
α
′
)
1
/
p
(
f
∘
α
)
,
U
β
f
=
(
β
′
)
1
/
p
(
f
∘
β
)
, and
P
2
±
=
(
I
±
S
2
)
/
2
where
(
S
2
f
)
(
t
)
:
=
1
π
i
∫
0
∞
t
τ
1
/
2
-
1
/
p
f
(
τ
)
τ
-
t
d
τ
,
t
∈
R
+
,
is the weighted Cauchy singular integral operator. We prove that if
α
′
,
β
′
and
c
,
d
are continuous on
R
+
and slowly oscillating at
0
and
∞
, and
lim sup
t
→
s
|
c
(
t
)
|
<
1
,
lim sup
t
→
s
|
d
(
t
)
|
<
1
,
s
∈
{
0
,
∞
}
,
then the operator
(
I
-
c
U
α
)
P
2
+
+
(
I
-
d
U
β
)
P
2
-
is Fredholm on
L
p
(
R
+
)
and its index is equal to zero. Moreover, its regularizers are described.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s11785-015-0452-0</doi><tpages>31</tpages></addata></record> |
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identifier | ISSN: 1661-8254 |
ispartof | Complex analysis and operator theory, 2016-08, Vol.10 (6), p.1101-1131 |
issn | 1661-8254 1661-8262 |
language | eng |
recordid | cdi_crossref_primary_10_1007_s11785_015_0452_0 |
source | Springer Nature |
subjects | Analysis Mathematics Mathematics and Statistics Operator Theory |
title | On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data |
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