Sobolev trace-type inequalities via time-space fractional heat equations

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 Ẇν /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 Ẇν /2ₚ( Rⁿ) . As a byproduct, Sobolev affine trace-type inequalities for Ḣ-ν /2(Rⁿ) and local Sobolev-type trace inequalities for Qν /2(Rⁿ) are established.