What is the effect of squeezing on the Husimi phase space representation or Q-function? The effect of the squeezing operator
\begin{equation}
S = e^{- r (a^2 + a^{\dagger 2}) / 2}
\end{equation}
on a Wigner phase space representation or W-function of a system with density matrix $\rho$
\begin{equation}
W(x, p) = \frac{1}{\pi \hbar} \int_{\mathbb{R}} dy e^{- 2 i p y / \hbar} \langle x + y | \rho | x - y \rangle
\end{equation}
is
\begin{equation}
S \circ W(x, p) = W(x e^r, p e^{-r})
\end{equation}
(I am using the convention that $x = 2^{-1/2} (a + a^{\dagger})$ and $p = - i 2^{-1/2} (a - a^{\dagger})$).  Is there a simple relation for the effect of the squeezing operator on the Husimi phase space representation or Q-function
\begin{equation}
Q(x, p) = \frac{2}{\pi} \int_{\mathbb{R}^2} du dv e^{- 2 (x - u)^2 - 2 (p - v)^2} W(u, v) ,
\end{equation}
i.e.,
\begin{equation}
S \circ Q(x, p) = ?
\end{equation}
More generally, I'd also like to know if there is a simple relation for the effect of squeezing on a generalised phase space representation or R-function (sometimes also called an $s$-parametrised W-function)
\begin{equation}
R(x, p, \tau) = \frac{1}{\pi \tau} \int_{\mathbb{R}} du dv e^{- \tau^{-1} (x - u)^2 - \tau^{-1} (p - v)^2} P(u, v)
\end{equation}
where $P(x, p)$ is the Glauber-Sudarshan phase space representation or P-function.  $P(x, p)$ satisfies
\begin{equation}
\rho = \int_{\mathbb{C}} d\alpha P(\alpha) |\alpha\rangle \langle \alpha |
\end{equation}
where $|\alpha\rangle$ are the coherent states of $a$.  The reason I ask is that I have developed some numerical algorithms that calculate a generalised phase space representation for certain states very accurately and I'd now like to see the effect of squeezing, preferably without having to perform numerical integration etc.
 A: Isn't this question a bit ambiguous?
More directly, skipping the superfluous creating and annihilation operators, the symplectic dilation operator on phase-space variables is just
$$ S= e^{r(x\partial_x- p \partial_p)}    ,$$
so that
$$ S W(x,p)= W(x e^r , p e^{-r} ) $$
Further note $S\partial_x= e^{-r} \partial_x$ and $S \partial_p= e^r \partial_p $.
The Husimi function is the Weierstrass transform of the WF,
$$Q(x,p)=T W(x,p),        $$
where
$$ T\equiv e^{\frac{\hbar}{4} (\partial_x^2+\partial_p^2)}  , $$
eqn (122) of Ref 1, equivalent to the integral kernel you provide (in which you have implicitly
chosen/absorbed $\hbar=1/2$.)
Thus, purely formally as you are asking,
$$SQ(x,p)=Q(x e^r,p e^{-r})=  e^{\frac{\hbar}{4} (e^{-2r}\partial_x^2+e^{2r}\partial_p^2)} W(xe^{r},pe^{-r})    .$$
I assume, however, you are asking the question because you are interested, instead, in the Husimi function of a squeezed state, (ref. 2 Chapter 12.2 and prob 12.1): that is, low-pass-filtering a squeezed WF, SW, with an unsqueezed Gaussian, that is leaving the Weierstrass transform operator T unsqueezed. 
Your choices are dictated by the application you are interested in, for example, whether you are fussing with the characteristic star Husimi-Voros product or not, etc.
References:


*

*Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos,  A Concise Treatise on Quantum Mechanics in Phase Space, World Scientific, 2014. The PDF file is available here.  

*Wolfgang P. Schleich, Quantum Optics in Phase Space, Wiley ,  2011.  
