I have a question about the following definition:

A probability measure $\mu$, such that the Markov semi-group $e^{Lt} \in L(L^2)$ 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 semi-group, 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.