On the Necessity of Logarithmic Estimates for Hypoellipticity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916013306544128 |
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| author | Akhunov, Timur Korobenko, Lyudmila |
| author_facet | Akhunov, Timur Korobenko, Lyudmila |
| contents | This paper is focused on necessary conditions for hypoellipticity of an operator $L$ of the form $L=L_1(x)+g(x)L_2(y)$, where the operator $L_1$ is either elliptic or parabolic, $L_2$ is degenerately elliptic and $g(x)$ may itself vanish adding further degeneracy. First, we establish a logarithmic criterion: if the operator $L$ above is hypoelliptic and $L_1$ has a family of spectral solutions we define in the paper, then the remaining part $L_2$ must gain a power of a logarithm of a derivative. Such a property can be thought of as a restriction on degeneracy of the operator $L_2$. We then use this criterion to examine degenerate elliptic and parabolic operators closing gaps between sufficiency and necessity that have been open since 1980s in three and higher dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15070 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Necessity of Logarithmic Estimates for Hypoellipticity Akhunov, Timur Korobenko, Lyudmila Analysis of PDEs 35B50, 35B51, 35B65, 35H10, 35H20, 35J15, 35K10, 47B25 This paper is focused on necessary conditions for hypoellipticity of an operator $L$ of the form $L=L_1(x)+g(x)L_2(y)$, where the operator $L_1$ is either elliptic or parabolic, $L_2$ is degenerately elliptic and $g(x)$ may itself vanish adding further degeneracy. First, we establish a logarithmic criterion: if the operator $L$ above is hypoelliptic and $L_1$ has a family of spectral solutions we define in the paper, then the remaining part $L_2$ must gain a power of a logarithm of a derivative. Such a property can be thought of as a restriction on degeneracy of the operator $L_2$. We then use this criterion to examine degenerate elliptic and parabolic operators closing gaps between sufficiency and necessity that have been open since 1980s in three and higher dimensions. |
| title | On the Necessity of Logarithmic Estimates for Hypoellipticity |
| topic | Analysis of PDEs 35B50, 35B51, 35B65, 35H10, 35H20, 35J15, 35K10, 47B25 |
| url | https://arxiv.org/abs/2605.15070 |