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Extremal of Log-Sobolev Functionals and Li-Yau Estimate on \(\textrm{RCD}^*(K,N)\) Spaces

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Abstract

In this work, we study the extremal functions of the log-Sobolev functional on compact metric measure spaces satisfying the \(\textrm{RCD}^*(K,N)\) condition for K in \(\mathbb {R}\) and N in \((2,\infty )\). We show the existence, regularity and positivity of non-negative extremal functions. Based on these results, we prove a Li-Yau type estimate for the logarithmic transform of any non-negative extremal functions of the log-Sobolev functional. As applications, we show a Harnack type inequality as well as lower and upper bounds for the non-negative extremal functions.

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Acknowledgements

National Science Foundation of China, Grants Numbers: 11971310 and 11671257 are gratefully acknowledged.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China

    Samuel Drapeau & Liming Yin

  2. Shanghai Advanced Institute for Finance, Shanghai, China

    Samuel Drapeau

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  1. Samuel Drapeau
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  2. Liming Yin
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Correspondence to Samuel Drapeau.

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Drapeau, S., Yin, L. Extremal of Log-Sobolev Functionals and Li-Yau Estimate on \(\textrm{RCD}^*(K,N)\) Spaces. Potential Anal 60, 935–964 (2024). https://doi.org/10.1007/s11118-023-10075-8

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