# A Generalized Version of Maximal Correlation and Hypercontractivity of Conditional Expectation Operator

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as $$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||g||_2},$$ where supremum is taken over all pair of functions $(f,g)$ such that $\mathbb{E}[f(X)]=\mathbb{E}[g(Y)]=0$ and $f\in L^2(\mathcal{X})$ and $g\in L^2(\mathcal{Y})$.

Renyi showed that that $\rho_2(X;Y)=0$ if and only if $X$ and $Y$ are independent and $\rho_2(X;Y)=1$ if there exists a pair of functions $f$ and $g$ such that $f(X)=g(Y)$ with probability one.

Having seen this definition, I was thinking about the following generalization for a $p\geq 1$ $$\rho_p(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_p||g||_q},$$ where supremum is taken over all pair of functions $(f,g)$ such that $\mathbb{E}[f(X)]=\mathbb{E}[g(Y)]=0$ and $f\in L^p(\mathcal{X})$ and $g\in L^q(\mathcal{Y})$ and $\frac{1}{p}+\frac{1}{q}=1$.

Holder inequality and a straightforward manipulation can show that $$\rho_p(X;Y):=\sup\frac{||Tf||_p}{||f||_p},$$ where the supremum is taken over all $f\in L^p(\mathcal{X})$ and $\mathbb{E}[f(X)]=0$ and conditional expectation operator $T:L^p(\mathcal{X})\to L^p(\mathcal{Y})$ is defined by $f\mapsto \mathbb{E}[f(X)|Y]$.

Note that we still have $\rho_p(X;Y)=0$ if and only if $X$ and $Y$ are independent and $\rho_p(X;Y)=1$ if there exists a pair of functions $f$ and $g$ such that $f(X)=g(Y)$ with probability one.

I am interested in properties of this measure, in particular the following:

1. The monotonicity of map $p\mapsto \rho_p(X;Y)$,
2. The connection between $\rho_p(X;Y)$ and the well-known hypercontractivity of operator $T$,
3. What is the value of $\rho_p(X;Y)$ if $X$ and $Y$ are bivariate Gaussian random variables.

I am sure that this definition appears somewhere in literature but I failed to find any. I will appreciate it if you know any reference about this.