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

Abstract

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.