Geodesic
Sobolev trace-type inequalities via time-space fractional heat equations
Canadian journal of mathematics, Tome 77 (2025) no. 4, pp. 1093-1134

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DOI

This article aims to establish fractional Sobolev trace inequalities, logarithmic Sobolev trace inequalities, and Hardy trace inequalities associated with time-space fractional heat equations. The key steps involve establishing dedicated estimates for the fractional heat kernel, regularity estimates for the solution of the time-space fractional equations, and characterizing the norm of $\dot {W}^{\nu /2}_p(\mathbb {R}^n)$ in terms of the solution $u(x,t)$. Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to $L^p-$logarithmic Sobolev inequalities for $\dot {W}^{\nu /2}_{p}(\mathbb R^{n})$. As a byproduct, Sobolev affine trace-type inequalities for $\dot {H}^{-\nu /2}(\mathbb {R}^n)$ and local Sobolev-type trace inequalities for $Q_{\nu /2}(\mathbb {R}^n)$ are established.
DOI : 10.4153/S0008414X24000269
Mots-clés : Sobolev inequality, logarithmic Sobolev inequality, Hardy inequality, heat kernel 
Tang, Yongrui; Li, Pengtao; Hu, Rui; Zhai, Zhichun. Sobolev trace-type inequalities via time-space fractional heat equations. Canadian journal of mathematics, Tome 77 (2025) no. 4, pp. 1093-1134. doi: 10.4153/S0008414X24000269
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     title = {Sobolev trace-type inequalities via time-space fractional heat equations},
     journal = {Canadian journal of mathematics},
     pages = {1093--1134},
     year = {2025},
     volume = {77},
     number = {4},
     doi = {10.4153/S0008414X24000269},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X24000269/}
}
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