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To show Sobolev inequalities, we can use potential theory (Riesz potential), which is a way of using integral representations, or the Gagliardo–Nirenberg inequality, \(\displaystyle \mbox{for } u\in C_c^\infty (\mathbb {R}^N)\qquad |u|_{L^{\frac N{N-1}}(\mathbb {R}^N)}\leqslant \prod _{j=1}^N|\partial _ju|_{L^1(\mathbb {R}^N)}^{\frac 1N}. \) The proofs are often very technical and are difficult to adapt to spaces that are invariant under rearrangement.

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Motivations and Generalities on the Monotone Rearrangement

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  • 发表日期：

    2025年

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  • Jean Michel Rakotoson

摘要

To show Sobolev inequalities, we can use potential theory (Riesz potential), which is a way of using integral representations, or the Gagliardo–Nirenberg inequality, \(\displaystyle \mbox{for } u\in C_c^\infty (\mathbb {R}^N)\qquad |u|_{L^{\frac N{N-1}}(\mathbb {R}^N)}\leqslant \prod _{j=1}^N|\partial _ju|_{L^1(\mathbb {R}^N)}^{\frac 1N}. \) The proofs are often very technical and are difficult to adapt to spaces that are invariant under rearrangement.