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End-point maximal \(L^1\)-regularity for the Cauchy problem to a parabolic equation with variable coefficients

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Abstract

We consider maximal \(L^1\)-regularity for the Cauchy problem to a parabolic equation in the Besov space \(\dot{B}_{p,1}^0(\mathbb {R}^n)\) with \(1\le p\le \infty \). The estimate obtained here is not available by abstract theory of the class of unconditional martingale differences, because the end-point Besov space is included. We consider the end-point estimate and show that the optimality of maximal regularity in \(L^1\) for the linear parabolic equation with variable coefficients.

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Notes

  1. This was noticed by Iwabuchi [25].

  2. The fractional derivative is defined by \(|\nabla |^{\alpha }u\equiv \mathcal {F}^{-1}\big [|\xi |^{\alpha }\widehat{u(\xi )}\big ]\). If \(\alpha =2\) then this can be taken as any second derivatives \(\partial _k\partial _{\ell }\) with \(1\le k,\ell \le n\).

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Acknowledgments

The authors would like to thank the anonymous referees for pointing out mistakes in the first version of the manuscript and providing many helpful suggestions. All the comments have improved the present paper considerably. Thanks are also due to Professor Masashi Misawa and Professor Jan Prüss for their helpful comments on the variable coefficient case, and to Professor Tsukasa Iwabuchi for discussions on maximal regularity in the modulation spaces [25]. The work of the first author is partially supported by JSPS, Grant-in-Aid for Scientific Research S #25220702. The work of the second author is partially supported by JSPS, Grant-in-Aid for Scientific Research B #24340025 and the Alexander von Humboldt Foundation.

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Authors and Affiliations

  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Takayoshi Ogawa

  2. Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501, Japan

    Senjo Shimizu

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  1. Takayoshi Ogawa
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  2. Senjo Shimizu
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Correspondence to Senjo Shimizu.

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Ogawa, T., Shimizu, S. End-point maximal \(L^1\)-regularity for the Cauchy problem to a parabolic equation with variable coefficients. Math. Ann. 365, 661–705 (2016). https://doi.org/10.1007/s00208-015-1279-8

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  • DOI: https://doi.org/10.1007/s00208-015-1279-8

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