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On the Sharp Estimates for Convolution Operators with Oscillatory Kernel
In this article, we studied the convolution operators M k with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator M k is associated to the characteristic hypersurfaces Σ ⊂ R 3 of a hyperbolic equation and smooth amplitude...
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| Published in: | The Journal of fourier analysis and applications 2024-06, Vol.30 (3), Article 29 |
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| creator | Ikromov, Isroil A. Ikromova, Dildora I. |
| description | In this article, we studied the convolution operators
M
k
with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator
M
k
is associated to the characteristic hypersurfaces
Σ
⊂
R
3
of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order
-
k
for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point
v
∈
Σ
at which, exactly one of the principal curvatures of the surface
Σ
does not vanish. Such surfaces exhibit singularities of the type
A
in the sense of Arnold’s classification. Denoting by
k
p
the minimal number such that
M
k
is
L
p
↦
L
p
′
-bounded for
k
>
k
p
,
we showed that the number
k
p
depends on some discrete characteristics of the surface
Σ
. |
| doi_str_mv | 10.1007/s00041-024-10085-z |
| format | article |
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M
k
with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator
M
k
is associated to the characteristic hypersurfaces
Σ
⊂
R
3
of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order
-
k
for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point
v
∈
Σ
at which, exactly one of the principal curvatures of the surface
Σ
does not vanish. Such surfaces exhibit singularities of the type
A
in the sense of Arnold’s classification. Denoting by
k
p
the minimal number such that
M
k
is
L
p
↦
L
p
′
-bounded for
k
>
k
p
,
we showed that the number
k
p
depends on some discrete characteristics of the surface
Σ
.</description><identifier>ISSN: 1069-5869</identifier><identifier>EISSN: 1531-5851</identifier><identifier>DOI: 10.1007/s00041-024-10085-z</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Abstract Harmonic Analysis ; Amplitudes ; Approximations and Expansions ; Cauchy problems ; Convolution ; Fourier Analysis ; Hyperspaces ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Operators ; Partial Differential Equations ; Signal,Image and Speech Processing</subject><ispartof>The Journal of fourier analysis and applications, 2024-06, Vol.30 (3), Article 29</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-180d8c006fd54bbc2171a2f46d3c468e2dc4cda1d1f1f06452e1e8153b34fa7a3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,777,781,27905,27906</link.rule.ids></links><search><creatorcontrib>Ikromov, Isroil A.</creatorcontrib><creatorcontrib>Ikromova, Dildora I.</creatorcontrib><title>On the Sharp Estimates for Convolution Operators with Oscillatory Kernel</title><title>The Journal of fourier analysis and applications</title><addtitle>J Fourier Anal Appl</addtitle><description>In this article, we studied the convolution operators
M
k
with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator
M
k
is associated to the characteristic hypersurfaces
Σ
⊂
R
3
of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order
-
k
for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point
v
∈
Σ
at which, exactly one of the principal curvatures of the surface
Σ
does not vanish. Such surfaces exhibit singularities of the type
A
in the sense of Arnold’s classification. Denoting by
k
p
the minimal number such that
M
k
is
L
p
↦
L
p
′
-bounded for
k
>
k
p
,
we showed that the number
k
p
depends on some discrete characteristics of the surface
Σ
.</description><subject>Abstract Harmonic Analysis</subject><subject>Amplitudes</subject><subject>Approximations and Expansions</subject><subject>Cauchy problems</subject><subject>Convolution</subject><subject>Fourier Analysis</subject><subject>Hyperspaces</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operators</subject><subject>Partial Differential Equations</subject><subject>Signal,Image and Speech Processing</subject><issn>1069-5869</issn><issn>1531-5851</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9UMtOwzAQtBBIlMIPcLLEObBrO48eUVUoolIOwNlyHZumCnGwXVD79RiCxI3Tzq5mZjVDyCXCNQKUNwEABGbARJb2Ks8OR2SCOccsr3I8ThiKWcLF7JSchbAFYMhLPiHLuqdxY-jTRvmBLkJs31Q0gVrn6dz1H67bxdb1tB6MV9H5QD_buKF10G3XfR_29NH43nTn5MSqLpiL3zklL3eL5_kyW9X3D_PbVaZZCTHDCppKAxS2ycV6rRmWqJgVRcO1KCrDGi10o7BBixYKkTODpkpJ1lxYVSo-JVej7-Dd-86EKLdu5_v0UnLIgfGUvkwsNrK0dyF4Y-XgUzK_lwjyuzE5NiZTY_KnMXlIIj6KQiL3r8b_Wf-j-gJq7m7l</recordid><startdate>20240601</startdate><enddate>20240601</enddate><creator>Ikromov, Isroil A.</creator><creator>Ikromova, Dildora I.</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20240601</creationdate><title>On the Sharp Estimates for Convolution Operators with Oscillatory Kernel</title><author>Ikromov, Isroil A. ; Ikromova, Dildora I.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-180d8c006fd54bbc2171a2f46d3c468e2dc4cda1d1f1f06452e1e8153b34fa7a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Abstract Harmonic Analysis</topic><topic>Amplitudes</topic><topic>Approximations and Expansions</topic><topic>Cauchy problems</topic><topic>Convolution</topic><topic>Fourier Analysis</topic><topic>Hyperspaces</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operators</topic><topic>Partial Differential Equations</topic><topic>Signal,Image and Speech Processing</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Ikromov, Isroil A.</creatorcontrib><creatorcontrib>Ikromova, Dildora I.</creatorcontrib><collection>CrossRef</collection><jtitle>The Journal of fourier analysis and applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ikromov, Isroil A.</au><au>Ikromova, Dildora I.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the Sharp Estimates for Convolution Operators with Oscillatory Kernel</atitle><jtitle>The Journal of fourier analysis and applications</jtitle><stitle>J Fourier Anal Appl</stitle><date>2024-06-01</date><risdate>2024</risdate><volume>30</volume><issue>3</issue><artnum>29</artnum><issn>1069-5869</issn><eissn>1531-5851</eissn><abstract>In this article, we studied the convolution operators
M
k
with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator
M
k
is associated to the characteristic hypersurfaces
Σ
⊂
R
3
of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order
-
k
for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point
v
∈
Σ
at which, exactly one of the principal curvatures of the surface
Σ
does not vanish. Such surfaces exhibit singularities of the type
A
in the sense of Arnold’s classification. Denoting by
k
p
the minimal number such that
M
k
is
L
p
↦
L
p
′
-bounded for
k
>
k
p
,
we showed that the number
k
p
depends on some discrete characteristics of the surface
Σ
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00041-024-10085-z</doi></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 1069-5869 |
| ispartof | The Journal of fourier analysis and applications, 2024-06, Vol.30 (3), Article 29 |
| issn | 1069-5869 1531-5851 |
| language | eng |
| recordid | cdi_proquest_journals_3050230857 |
| source | Springer Nature |
| subjects | Abstract Harmonic Analysis Amplitudes Approximations and Expansions Cauchy problems Convolution Fourier Analysis Hyperspaces Mathematical Methods in Physics Mathematics Mathematics and Statistics Operators Partial Differential Equations Signal,Image and Speech Processing |
| title | On the Sharp Estimates for Convolution Operators with Oscillatory Kernel |
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