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.

In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

We investigate the dissipativity properties of a class of scalar secondorder parabolic partial differential equations with time-dependentcoefficients. We provide explicit condition on the drift term which ensurethat the relative entropy of one particular orbit with respect to some otherone decreases to zero. The decay rate is obtained explicitly by the use ofa Sobolev logarithmic inequality for the associated semigroup, which is derived by an adaptation of Bakry's Γ-calculus.As a byproduct, the systematic method for constructing entropieswhich we propose here also yields the well-known intermediate asymptotics for the heat equation in a very quick way, and without having to rescale the original equation.