Sobolev trace-type inequalities via time-space fractional heat equations
Faculty Advisor
Date
2025
Keywords
Sobolev inequality, logarithmic Sobolev inequality, Hardy inequality, heat kernel
Abstract (summary)
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 W˙ ν/2p (Rn) in terms of the solution u(x, t). Additionally, fractional logarithmic Gagliardo–Nirenberg inequalities are proven, leading to Lp−logarithmic Sobolev inequalities for W˙ ν/2p (Rn). As a byproduct, Sobolev affine trace-type inequalities for H˙ −ν/2(Rn) and local Sobolev-type trace inequalities for Qν/2(Rn) are established.
Publication Information
Tang, Y., Li, P., Hu, R., & Zhai, Z. (2025). Sobolev trace-type inequalities via time-space fractional heat equations. Canadian Journal of Mathematics, 77(4), 1093–1134. https://doi.org/10.4153/S0008414X24000269
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