In a problem I am currently working on, an extension of the noise operator arises naturally, and I was curious whether there has been prior work. First let me revise the basic noise operator $T\_{\varepsilon}$ on real-valued Boolean functions. Given a function $f: \{0,1\}^n \to \mathbb{R}$ and $\varepsilon$, $p$ s.t $0 \leq \varepsilon \leq 1$, $\varepsilon = 1 - 2p$, 
we define $T\_{\varepsilon} \to \mathbb{R}$ as
$T\_{\varepsilon} f(x) = E\_{y \sim \mu\_p} [f(x+y)]$

$\mu\_p$ is the distribution on $y$ obtained by setting each bit of a $n$-bit vector to be $1$ independently with probability $p$ and $0$ otherwise. Equivalently, we can think of this process as flipping each bit of $x$ with independent probability $p$. Now this noise operator has many useful properties, including being multiplicative $T\_{\varepsilon\_1} T\_{\varepsilon\_2} =
T\_{\varepsilon\_1 \varepsilon\_2}$ and having nice eigenvalues and eigenvectors($T\_{\varepsilon}(\chi\_S) = \varepsilon^{|S|} \chi\_S$ where $\chi\_S$ belongs to the parity basis).

Let me now define my extension of $T\_\varepsilon$, which I denote as $R\_{(p\_1,p\_2)}$. $R\_{(p\_1,p\_2)} \to \mathbb{R}$ is given by $R\_{(p\_1,p\_2)} f(x) = E\_{y \sim \mu\_{p,x}} [f(x+y)]$. But here our distribution $\mu\_{p,x}$ is such that we flip the $1$ bits of $x$ to $0$ with probability $p\_1$ and $0$ bits of $x$ to $1$ with probability $p\_2$. ( $\mu\_{p,x}$ is now clearly a distribution dependent on the $x$ where the function is evaluated, and if $p\_1 = p\_2$ then $R\_{(p\_1,p\_2)}$ reduces to the 'regular' noise operator.)

I was wondering, has this operator $R\_{(p\_1,p\_2)}$ already been well-studied somewhere in the literature? Or are the basic properties of it obvious? I am just starting with Boolean analysis, so this might be straightforward to someone more familiar with the theory than I am. In particular I am interested in if the eigenvectors and eigenvalues have some nice characterization, or whether there is any multiplicative property.