Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents
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| Format: | Preprint |
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2024
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| _version_ | 1866912084124499968 |
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| author | Dao, Nguyen Anh Do, Anh Xuan Lam, Nguyen Lu, Guozhen |
| author_facet | Dao, Nguyen Anh Do, Anh Xuan Lam, Nguyen Lu, Guozhen |
| contents | We establish a new family of the critical higher order Sobolev interpolation inequalities for radial functions as well as for non-radial functions. These Sobolev interpolation inequalities are sharp in the sense that they use the optimal quadratic forms of the sharp Hardy-Rellich inequalities and cover the Sobolev critical exponents. Our results extend those studied by Dietze and Nam in [15] for the first order derivative case to higher order setting. The well-known Pólya-Szegö symmetrization principle and the nonlinear ground state representation play an important role in the work of [15]. To overcome the absence of the Pólya-Szegö principle and the nonlinear ground state representation in the higher order case, our proofs rely on the Fourier analysis and a higher order verion of the Talenti comparison principle. We also study a new version of the critical Hardy-Sobolev interpolation inequality involving the critical quadratic form of the Hardy inequality and Lorentz norms. Our critical Hardy-Sobolev interpolation inequality complements the result of Dietze and Nam in [15]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_18335 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents Dao, Nguyen Anh Do, Anh Xuan Lam, Nguyen Lu, Guozhen Analysis of PDEs Classical Analysis and ODEs We establish a new family of the critical higher order Sobolev interpolation inequalities for radial functions as well as for non-radial functions. These Sobolev interpolation inequalities are sharp in the sense that they use the optimal quadratic forms of the sharp Hardy-Rellich inequalities and cover the Sobolev critical exponents. Our results extend those studied by Dietze and Nam in [15] for the first order derivative case to higher order setting. The well-known Pólya-Szegö symmetrization principle and the nonlinear ground state representation play an important role in the work of [15]. To overcome the absence of the Pólya-Szegö principle and the nonlinear ground state representation in the higher order case, our proofs rely on the Fourier analysis and a higher order verion of the Talenti comparison principle. We also study a new version of the critical Hardy-Sobolev interpolation inequality involving the critical quadratic form of the Hardy inequality and Lorentz norms. Our critical Hardy-Sobolev interpolation inequality complements the result of Dietze and Nam in [15]. |
| title | Sobolev interpolation inequalities with optimal Hardy-Rellich inequalities and critical exponents |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2410.18335 |