Markov-semigroup Sobolev inequality

Question

I have a question about the following definition:

A probability measure $\mu$, such that the Markov semigroup $e^{Lt} \in \mathcal{L}(L^2)$ exists and is symmetric, satisfies the Sobolev inequality iff for some $p \in (2, \infty)$ and two finite constants $(a,b) \in [0,\infty)^2$, we have

$$\|f\|_p^2 \le a \|\Gamma_1(f,f)\|_1 + b \|f\|_2^2$$ for any measurable function $f$ such that the right-hand side exists ($\Gamma_1$ is the carré du champ) and, in case that $b=0$ so that $\int f d\mu=0$. The carré du champ is basically defined as $||\Gamma_1(f,f)||_1 = - \langle f,Lf \rangle$ where $L$ is the negative generator of the semigroup, as written above.

I took this definition from the following paper [1, Definition 3.1] and I was wondering whether $a,b$ can depend on $p$ (maybe you have a guess, my intuition says yes.) and where this strange condition $b=0$ then $\int f d\mu=0$ comes from?

The reason why I am asking is that I could nowhere else find (exactly) this equation and I thought that this equation might be too localized for MSE.

Answer 1

You can find many details and references on this inequality (and related ones) in Chapter 6 of the recent book Analysis and Geometry of Markov diffusion operators by Bakry, Gentil and Ledoux. The constants $a$ and $b$ do indeed depend on $p$. In this book the inequalities are stated without the recentering, so that if $\mu$ is a probability measure, then $b$ is always larger than $1$. When $b=1$ the inequality is said to be tight. In that case it implies a Poincaré inequality with an explicit constant $C_P(a,p)$ (see proposition 6.2.2 in the book), so for centered functions one can write $$ \|f\|_p^2 \leq (a+C_P) \| \Gamma(f,f)\|_1 $$ and recover the form you mentioned ($b=0$ and centered functions).