Sobolev contractivity of gradient flow maximal functions
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866910344358658048 |
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| author | Bortz, Simon Egert, Moritz Saari, Olli |
| author_facet | Bortz, Simon Egert, Moritz Saari, Olli |
| contents | We prove that the energy dissipation property of gradient flows extends to the semigroup maximal operators in various settings. In particular, we show that the vertical maximal function relative to the $p$-parabolic extension does not increase the $\dot{W}^{1,p}$ norm of $\dot{W}^{1,p}(\mathbb{R}^n) \cap L^{2}(\mathbb{R}^n)$ functions when $p > 2$. We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable, real and symmetric coefficients, where the solutions do not have a representation formula via a convolution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1910_13150 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Sobolev contractivity of gradient flow maximal functions Bortz, Simon Egert, Moritz Saari, Olli Classical Analysis and ODEs Analysis of PDEs We prove that the energy dissipation property of gradient flows extends to the semigroup maximal operators in various settings. In particular, we show that the vertical maximal function relative to the $p$-parabolic extension does not increase the $\dot{W}^{1,p}$ norm of $\dot{W}^{1,p}(\mathbb{R}^n) \cap L^{2}(\mathbb{R}^n)$ functions when $p > 2$. We also obtain analogous results in the setting of uniformly parabolic and elliptic equations with bounded, measurable, real and symmetric coefficients, where the solutions do not have a representation formula via a convolution. |
| title | Sobolev contractivity of gradient flow maximal functions |
| topic | Classical Analysis and ODEs Analysis of PDEs |
| url | https://arxiv.org/abs/1910.13150 |