We have well known inequality for convolution: $\|g\ast f\|_{X} \leq \|g\|_{Y} \|f\|_{Z}$ where $Y, X, Z$ are suitable Lebesgue spaces, e.g., see Young's inequality.
Let $f:\mathbb R^{2}\to \mathbb C, K:\mathbb R \to \mathbb C$ ( assume that $f\in \mathcal{S}(\mathbb R^2)$ (Schwartz space), $K\in \mathcal{S}(\mathbb R)$), We define convolution in the following sense: $$F(x,y) =\int_{\mathbb R} [K(x-z)-K(y-z)] f(z,z) dz$$
Can we expect to find norm linear spaces $(X, \|\cdot\|_{X}), (Y, \|\cdot\|_{Y}$ consists of functions on $\mathbb R^2$ and norm space $(Z, \|\cdot\|)$ consists of functions on $\mathbb R$ so that $$\|F\|_{X} \leq C \|K\|_{Z} \|f\|_{Y}$$?
Your question is a little vague, but only the values of $f$ on the diagonal is used. Its more natural therefore to use $g(z) = f(z,z)$ and look for results for the one variable function $g$. Then your $F(x,y)$ is just
$$ F(x,y) = K*g(x) - K*g(y)$$
and for example, we have the crude estimate $$ |K*g(x) - K*g(y)| = \left| \int_y^x K'*g \right| \le \int_{\mathbb R }|K'*g|$$
This implies $$ \|F\|_{L^\infty(\mathbb R^2)} \le \|K'*g\|_{L^1} \le \|K\|_{W^{1,p}} \|g\|_{L^q}$$ for suitable $p,q$ via Young's inequality.