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***]][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.



  [1]: http://cermics.enpc.fr/~roussetm/cours_m2/guionnet_logsob_course.pdf