A Modified Morrey-Kohn-Hörmander Identity and Applications to the ∂¯-Problem
We prove a modified form of the classical Morrey-Kohn-Hörmander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in C n , where the inner domain has C 1 , 1 boundary, we show that the L 2 Dolbeault cohomology group in bidegree...
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| Veröffentlicht in: | The Journal of Geometric Analysis 2021-10, Vol.31 (10), p.9639-9676 |
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| container_title | The Journal of Geometric Analysis |
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| creator | Chakrabarti, Debraj Harrington, Phillip S. |
| description | We prove a modified form of the classical Morrey-Kohn-Hörmander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in
C
n
, where the inner domain has
C
1
,
1
boundary, we show that the
L
2
Dolbeault cohomology group in bidegree (
p
,
q
) vanishes if
1
≤
q
≤
n
-
2
and is Hausdorff and infinite-dimensional if
q
=
n
-
1
, so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the
L
2
Sobolev space
W
1
on any pseudoconvex domain with
C
1
,
1
boundary. We also generalize our results to annuli between domains which are weakly
q
-convex in the sense of Ho for appropriate values of
q
. |
| doi_str_mv | 10.1007/s12220-021-00623-2 |
| format | Article |
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C
n
, where the inner domain has
C
1
,
1
boundary, we show that the
L
2
Dolbeault cohomology group in bidegree (
p
,
q
) vanishes if
1
≤
q
≤
n
-
2
and is Hausdorff and infinite-dimensional if
q
=
n
-
1
, so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the
L
2
Sobolev space
W
1
on any pseudoconvex domain with
C
1
,
1
boundary. We also generalize our results to annuli between domains which are weakly
q
-convex in the sense of Ho for appropriate values of
q
.</description><identifier>ISSN: 1050-6926</identifier><identifier>EISSN: 1559-002X</identifier><identifier>DOI: 10.1007/s12220-021-00623-2</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Abstract Harmonic Analysis ; Annuli ; Convex and Discrete Geometry ; Differential Geometry ; Domains ; Dynamical Systems and Ergodic Theory ; Fourier Analysis ; Geometry ; Global Analysis and Analysis on Manifolds ; Homology ; Mathematics ; Mathematics and Statistics ; Sobolev space</subject><ispartof>The Journal of Geometric Analysis, 2021-10, Vol.31 (10), p.9639-9676</ispartof><rights>Mathematica Josephina, Inc. 2021</rights><rights>COPYRIGHT 2021 Springer</rights><rights>Mathematica Josephina, Inc. 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-g1812-ca32bce8d117291d046a14fb26611537936de94ceb5a2a29788eda77db8987b3</cites><orcidid>0000-0002-0473-8542 ; 0000-0002-1398-0162</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s12220-021-00623-2$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s12220-021-00623-2$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Chakrabarti, Debraj</creatorcontrib><creatorcontrib>Harrington, Phillip S.</creatorcontrib><title>A Modified Morrey-Kohn-Hörmander Identity and Applications to the ∂¯-Problem</title><title>The Journal of Geometric Analysis</title><addtitle>J Geom Anal</addtitle><description>We prove a modified form of the classical Morrey-Kohn-Hörmander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in
C
n
, where the inner domain has
C
1
,
1
boundary, we show that the
L
2
Dolbeault cohomology group in bidegree (
p
,
q
) vanishes if
1
≤
q
≤
n
-
2
and is Hausdorff and infinite-dimensional if
q
=
n
-
1
, so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the
L
2
Sobolev space
W
1
on any pseudoconvex domain with
C
1
,
1
boundary. We also generalize our results to annuli between domains which are weakly
q
-convex in the sense of Ho for appropriate values of
q
.</description><subject>Abstract Harmonic Analysis</subject><subject>Annuli</subject><subject>Convex and Discrete Geometry</subject><subject>Differential Geometry</subject><subject>Domains</subject><subject>Dynamical Systems and Ergodic Theory</subject><subject>Fourier Analysis</subject><subject>Geometry</subject><subject>Global Analysis and Analysis on Manifolds</subject><subject>Homology</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Sobolev space</subject><issn>1050-6926</issn><issn>1559-002X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNpFULFOwzAQtRBIlMIPMEVidvHZie2MUQW0oogOHdgiJ3ZaV2kcnHToysTfsPEB8Cd8CYYgoRvu3um9d6eH0CWQCRAirjuglBJMKGBCOGWYHqERJEkaIH06DjNJCOYp5aforOu2hMScxWKElln04LStrNFh8N4c8L3bNHj2-e53qtHGR3Ntmt72hyjAKGvb2paqt67pot5F_cZEX68vH2946V1Rm905OqlU3ZmLvz5Gq9ub1XSGF49382m2wGuQQHGpGC1KIzWAoCno8I-CuCoo5wAJEynj2qRxaYpEUUVTIaXRSghdyFSKgo3R1WDbeve8N12fb93eN-FiThMeQwIE4sCaDKy1qk1um8r1XpWhtNnZ0jWmsmGfCSIYk5LLIGCDoGu9bdbG_9sCyX-izoeo8xB1_ht1Ttk3Yd9x-w</recordid><startdate>20211001</startdate><enddate>20211001</enddate><creator>Chakrabarti, Debraj</creator><creator>Harrington, Phillip S.</creator><general>Springer US</general><general>Springer</general><general>Springer Nature B.V</general><scope>IAO</scope><orcidid>https://orcid.org/0000-0002-0473-8542</orcidid><orcidid>https://orcid.org/0000-0002-1398-0162</orcidid></search><sort><creationdate>20211001</creationdate><title>A Modified Morrey-Kohn-Hörmander Identity and Applications to the ∂¯-Problem</title><author>Chakrabarti, Debraj ; Harrington, Phillip S.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-g1812-ca32bce8d117291d046a14fb26611537936de94ceb5a2a29788eda77db8987b3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Abstract Harmonic Analysis</topic><topic>Annuli</topic><topic>Convex and Discrete Geometry</topic><topic>Differential Geometry</topic><topic>Domains</topic><topic>Dynamical Systems and Ergodic Theory</topic><topic>Fourier Analysis</topic><topic>Geometry</topic><topic>Global Analysis and Analysis on Manifolds</topic><topic>Homology</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Sobolev space</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Chakrabarti, Debraj</creatorcontrib><creatorcontrib>Harrington, Phillip S.</creatorcontrib><collection>Gale Academic OneFile</collection><jtitle>The Journal of Geometric Analysis</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Chakrabarti, Debraj</au><au>Harrington, Phillip S.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A Modified Morrey-Kohn-Hörmander Identity and Applications to the ∂¯-Problem</atitle><jtitle>The Journal of Geometric Analysis</jtitle><stitle>J Geom Anal</stitle><date>2021-10-01</date><risdate>2021</risdate><volume>31</volume><issue>10</issue><spage>9639</spage><epage>9676</epage><pages>9639-9676</pages><issn>1050-6926</issn><eissn>1559-002X</eissn><abstract>We prove a modified form of the classical Morrey-Kohn-Hörmander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in
C
n
, where the inner domain has
C
1
,
1
boundary, we show that the
L
2
Dolbeault cohomology group in bidegree (
p
,
q
) vanishes if
1
≤
q
≤
n
-
2
and is Hausdorff and infinite-dimensional if
q
=
n
-
1
, so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the
L
2
Sobolev space
W
1
on any pseudoconvex domain with
C
1
,
1
boundary. We also generalize our results to annuli between domains which are weakly
q
-convex in the sense of Ho for appropriate values of
q
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s12220-021-00623-2</doi><tpages>38</tpages><orcidid>https://orcid.org/0000-0002-0473-8542</orcidid><orcidid>https://orcid.org/0000-0002-1398-0162</orcidid></addata></record> |
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| identifier | ISSN: 1050-6926 |
| ispartof | The Journal of Geometric Analysis, 2021-10, Vol.31 (10), p.9639-9676 |
| issn | 1050-6926 1559-002X |
| language | eng |
| recordid | cdi_proquest_journals_2564151014 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Abstract Harmonic Analysis Annuli Convex and Discrete Geometry Differential Geometry Domains Dynamical Systems and Ergodic Theory Fourier Analysis Geometry Global Analysis and Analysis on Manifolds Homology Mathematics Mathematics and Statistics Sobolev space |
| title | A Modified Morrey-Kohn-Hörmander Identity and Applications to the ∂¯-Problem |
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