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***]](http://cermics.enpc.fr/~roussetm/cours_m2/guionnet_logsob_course.pdf) 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.



Markov-semigroup Sobolev inequality
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](http://doi.org/10.1007/978-3-319-00227-9) 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). 


