For Schwartz $f$ on $\mathbb{R}^2$ with $f(x,x)=0$, is the integral $\int_{\mathbb{R}^2} \frac{f(x,y)}{\lvert x-y \rvert^{n}}dxdy$ finite?

Question

Let $f$ be a Schwartz function on $\mathbb{R}^2$ such that $f(x,x)=0$ for all $x \in \mathbb{R}$.

Then, I wonder if the integral \begin{equation} \int_{\mathbb{R}^2} \frac{f(x,y)}{\lvert x-y \rvert^{n}}dxdy \end{equation} is finite for each $n \in \mathbb{N}$.

I have tried to find a bound for the denominator $\lvert x-y \rvert^{n}$ but it is a bit tricky due to $x$ and $y$ being together. At least, I somehow guess that $f(x,y)$ converges to zero faster than any $\lvert x-y \rvert^{n}$ as $(x,y) \to (x,x)$.

Could anyone please help me?