Sharp estimates for convolution operators associated to hypersurfaces in $\mathbb{R}^3$ with height $h\le2$
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866914707232784384 |
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| author | Akramov, Ibrokhimbek Ikromov, Isroil A. |
| author_facet | Akramov, Ibrokhimbek Ikromov, Isroil A. |
| contents | In this article, we study the convolution operator $M_k$ with oscillatory kernel, which is related with solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurface $Σ\subset \mathbb{R}^3$ of the equation and the smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the support of the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in Σ$ at which the height of the surface is less or equal to two. Such class contains surfaces related to simple and the $X_9, \, J_{10}$ type singularities in the sense of Arnol'd's classification. Denoting by $k_p$ the minimal exponent such that $M_k$ is $L^p\mapsto L^{p'}$-bounded for $k>k_p,$ we show that the number $k_p$ depends on some discrete characteristics of the Newton polygon of a smooth function constructed in an appropriate coordinate system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04413 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sharp estimates for convolution operators associated to hypersurfaces in $\mathbb{R}^3$ with height $h\le2$ Akramov, Ibrokhimbek Ikromov, Isroil A. Analysis of PDEs Classical Analysis and ODEs In this article, we study the convolution operator $M_k$ with oscillatory kernel, which is related with solutions to the Cauchy problem for the strictly hyperbolic equations. The operator $M_k$ is associated to the characteristic hypersurface $Σ\subset \mathbb{R}^3$ of the equation and the smooth amplitude function, which is homogeneous of order $-k$ for large values of the argument. We study the convolution operators assuming that the support of the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point $v\in Σ$ at which the height of the surface is less or equal to two. Such class contains surfaces related to simple and the $X_9, \, J_{10}$ type singularities in the sense of Arnol'd's classification. Denoting by $k_p$ the minimal exponent such that $M_k$ is $L^p\mapsto L^{p'}$-bounded for $k>k_p,$ we show that the number $k_p$ depends on some discrete characteristics of the Newton polygon of a smooth function constructed in an appropriate coordinate system. |
| title | Sharp estimates for convolution operators associated to hypersurfaces in $\mathbb{R}^3$ with height $h\le2$ |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2403.04413 |