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BELTRAMI EQUATIONS WITH COEFFICIENT IN THE FRACTIONAL SOBOLEV SPACE
In this paper, we look at quasiconformal solutions φ: C → C of Beltrami equations ∂ z ̄ φ ( z ) = μ ( z ) ∂ z φ ( z ) , where μ ∈ L∞(C) is compactly supported on D, and ‖μ‖∞ < 1 and belongs to the fractional Sobolev space W α , 2 α ( C ) . Our main result states that log ∂ z φ ∈ W α , 2 α ( C )...
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| Published in: | Proceedings of the American Mathematical Society 2017-01, Vol.145 (1), p.139-149 |
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| Language: | English |
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| container_end_page | 149 |
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| container_start_page | 139 |
| container_title | Proceedings of the American Mathematical Society |
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| creator | BAISÓN, ANTONIO L. CLOP, ALBERT OROBITG, JOAN |
| description | In this paper, we look at quasiconformal solutions φ: C → C of Beltrami equations
∂
z
̄
φ
(
z
)
=
μ
(
z
)
∂
z
φ
(
z
)
,
where μ ∈ L∞(C) is compactly supported on D, and ‖μ‖∞ < 1 and belongs to the fractional Sobolev space
W
α
,
2
α
(
C
)
. Our main result states that
log
∂
z
φ
∈
W
α
,
2
α
(
C
)
whenever
α
≥
1
2
. Our method relies on an n-dimensional result, which asserts the compactness of the commutator
[
b
,
(
−
Δ
)
β
2
]
:
L
n
p
n
−
β
p
(
R
n
)
→
L
p
(
R
n
)
between the fractional laplacian
(
−
Δ
)
β
2
and any symbol
b
∈
W
β
,
n
β
(
R
n
)
, provided that
1
<
p
<
n
β
. |
| format | article |
| fullrecord | <record><control><sourceid>jstor</sourceid><recordid>TN_cdi_jstor_primary_procamermathsoci_145_1_139</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><jstor_id>procamermathsoci.145.1.139</jstor_id><sourcerecordid>procamermathsoci.145.1.139</sourcerecordid><originalsourceid>FETCH-jstor_primary_procamermathsoci_145_1_1393</originalsourceid><addsrcrecordid>eNqVi9sKgjAAQEcUZJd_2A9Im5q5xzk2HJiWrnqUIUZKYmy-9PcZ9AM9HQ6cMwMORlHkhpEXzoGDEPJcQnyyBCtru0kxCQ4OYDFPVUGPEvLzhSqZZyW8SZVAlnMhJJM8U1BmUCUcioKyb0FTWOZxnvIrLE-U8Q1Y3PXTNtsf12AnuGKJ29lxMNXLtL0274lDrfvG9Hp82KFuKxzsK1xhn_j_Hx-cMj-n</addsrcrecordid><sourcetype>Publisher</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>BELTRAMI EQUATIONS WITH COEFFICIENT IN THE FRACTIONAL SOBOLEV SPACE</title><source>American Mathematical Society Publications (Freely Accessible)</source><source>JSTOR</source><creator>BAISÓN, ANTONIO L. ; CLOP, ALBERT ; OROBITG, JOAN</creator><creatorcontrib>BAISÓN, ANTONIO L. ; CLOP, ALBERT ; OROBITG, JOAN</creatorcontrib><description>In this paper, we look at quasiconformal solutions φ: C → C of Beltrami equations
∂
z
̄
φ
(
z
)
=
μ
(
z
)
∂
z
φ
(
z
)
,
where μ ∈ L∞(C) is compactly supported on D, and ‖μ‖∞ < 1 and belongs to the fractional Sobolev space
W
α
,
2
α
(
C
)
. Our main result states that
log
∂
z
φ
∈
W
α
,
2
α
(
C
)
whenever
α
≥
1
2
. Our method relies on an n-dimensional result, which asserts the compactness of the commutator
[
b
,
(
−
Δ
)
β
2
]
:
L
n
p
n
−
β
p
(
R
n
)
→
L
p
(
R
n
)
between the fractional laplacian
(
−
Δ
)
β
2
and any symbol
b
∈
W
β
,
n
β
(
R
n
)
, provided that
1
<
p
<
n
β
.</description><identifier>ISSN: 0002-9939</identifier><identifier>EISSN: 1088-6826</identifier><language>eng</language><publisher>American Mathematical Society</publisher><subject>B. ANALYSIS</subject><ispartof>Proceedings of the American Mathematical Society, 2017-01, Vol.145 (1), p.139-149</ispartof><rights>2016 American Mathematical Society</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://www.jstor.org/stable/pdf/procamermathsoci.145.1.139$$EPDF$$P50$$Gjstor$$H</linktopdf><linktohtml>$$Uhttps://www.jstor.org/stable/procamermathsoci.145.1.139$$EHTML$$P50$$Gjstor$$H</linktohtml><link.rule.ids>314,780,784,58238,58471</link.rule.ids></links><search><creatorcontrib>BAISÓN, ANTONIO L.</creatorcontrib><creatorcontrib>CLOP, ALBERT</creatorcontrib><creatorcontrib>OROBITG, JOAN</creatorcontrib><title>BELTRAMI EQUATIONS WITH COEFFICIENT IN THE FRACTIONAL SOBOLEV SPACE</title><title>Proceedings of the American Mathematical Society</title><description>In this paper, we look at quasiconformal solutions φ: C → C of Beltrami equations
∂
z
̄
φ
(
z
)
=
μ
(
z
)
∂
z
φ
(
z
)
,
where μ ∈ L∞(C) is compactly supported on D, and ‖μ‖∞ < 1 and belongs to the fractional Sobolev space
W
α
,
2
α
(
C
)
. Our main result states that
log
∂
z
φ
∈
W
α
,
2
α
(
C
)
whenever
α
≥
1
2
. Our method relies on an n-dimensional result, which asserts the compactness of the commutator
[
b
,
(
−
Δ
)
β
2
]
:
L
n
p
n
−
β
p
(
R
n
)
→
L
p
(
R
n
)
between the fractional laplacian
(
−
Δ
)
β
2
and any symbol
b
∈
W
β
,
n
β
(
R
n
)
, provided that
1
<
p
<
n
β
.</description><subject>B. ANALYSIS</subject><issn>0002-9939</issn><issn>1088-6826</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNqVi9sKgjAAQEcUZJd_2A9Im5q5xzk2HJiWrnqUIUZKYmy-9PcZ9AM9HQ6cMwMORlHkhpEXzoGDEPJcQnyyBCtru0kxCQ4OYDFPVUGPEvLzhSqZZyW8SZVAlnMhJJM8U1BmUCUcioKyb0FTWOZxnvIrLE-U8Q1Y3PXTNtsf12AnuGKJ29lxMNXLtL0274lDrfvG9Hp82KFuKxzsK1xhn_j_Hx-cMj-n</recordid><startdate>20170101</startdate><enddate>20170101</enddate><creator>BAISÓN, ANTONIO L.</creator><creator>CLOP, ALBERT</creator><creator>OROBITG, JOAN</creator><general>American Mathematical Society</general><scope/></search><sort><creationdate>20170101</creationdate><title>BELTRAMI EQUATIONS WITH COEFFICIENT IN THE FRACTIONAL SOBOLEV SPACE</title><author>BAISÓN, ANTONIO L. ; CLOP, ALBERT ; OROBITG, JOAN</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-jstor_primary_procamermathsoci_145_1_1393</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>B. ANALYSIS</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>BAISÓN, ANTONIO L.</creatorcontrib><creatorcontrib>CLOP, ALBERT</creatorcontrib><creatorcontrib>OROBITG, JOAN</creatorcontrib><jtitle>Proceedings of the American Mathematical Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>BAISÓN, ANTONIO L.</au><au>CLOP, ALBERT</au><au>OROBITG, JOAN</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>BELTRAMI EQUATIONS WITH COEFFICIENT IN THE FRACTIONAL SOBOLEV SPACE</atitle><jtitle>Proceedings of the American Mathematical Society</jtitle><date>2017-01-01</date><risdate>2017</risdate><volume>145</volume><issue>1</issue><spage>139</spage><epage>149</epage><pages>139-149</pages><issn>0002-9939</issn><eissn>1088-6826</eissn><abstract>In this paper, we look at quasiconformal solutions φ: C → C of Beltrami equations
∂
z
̄
φ
(
z
)
=
μ
(
z
)
∂
z
φ
(
z
)
,
where μ ∈ L∞(C) is compactly supported on D, and ‖μ‖∞ < 1 and belongs to the fractional Sobolev space
W
α
,
2
α
(
C
)
. Our main result states that
log
∂
z
φ
∈
W
α
,
2
α
(
C
)
whenever
α
≥
1
2
. Our method relies on an n-dimensional result, which asserts the compactness of the commutator
[
b
,
(
−
Δ
)
β
2
]
:
L
n
p
n
−
β
p
(
R
n
)
→
L
p
(
R
n
)
between the fractional laplacian
(
−
Δ
)
β
2
and any symbol
b
∈
W
β
,
n
β
(
R
n
)
, provided that
1
<
p
<
n
β
.</abstract><pub>American Mathematical Society</pub></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0002-9939 |
| ispartof | Proceedings of the American Mathematical Society, 2017-01, Vol.145 (1), p.139-149 |
| issn | 0002-9939 1088-6826 |
| language | eng |
| recordid | cdi_jstor_primary_procamermathsoci_145_1_139 |
| source | American Mathematical Society Publications (Freely Accessible); JSTOR |
| subjects | B. ANALYSIS |
| title | BELTRAMI EQUATIONS WITH COEFFICIENT IN THE FRACTIONAL SOBOLEV SPACE |
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