Let $X$ and $Y$ be a Banach spaces of functions on $\mathbb R^2.$ Define operator $$Tf(x,y)= f(x,x)$$ for $x,y \in \mathbb R, f\in X.$
Questions: (1) Dose there exists examples (or known theorems) of functions spaces $X$ and $Y$ so that operator $T$ maps $X$ to $Y$ with $\|Tf\|_{Y} \leq C \|f\|_{X}$ for some constant $C>0.$
(2) Specifically, I'm curious to see Banach spaces like, Sobolev spaces $H^s(\mathbb R^2) (s\in\mathbb R)$, and Fourier-Lebesgue spaces $\mathcal{F}L^p(\mathbb R^2) (1\leq p \leq \infty).$
Any references and comments would be helpful to me.
Edit: Using the comment below first we recall following.
We first define trace operator: $T_rf(x,y)=f(x,0)$ $(x,y \in \mathbb R).$ Trace Theorem: Let$s>\frac{1}{2}$. Then $T_r:H^{s}(\mathbb R^2) \to H^{s-\frac{1}{2}}(\mathbb R)$ with the norm inequality $$\|T_rf(x,y)\|_{H^{s-\frac{1}{2}}(\mathbb R)} \leq C \|f(x,y)\|_{H^{s}(\mathbb R^2)}$$
From this, Can we derive $$\|f(x,x)\|_{H^{s-\frac{1}{2}}(\mathbb R)} \leq C \|f\|_{H^{s}(\mathbb R^2)}?$$
Or I'm not sure flowing make sense or we can expect: $$\|f(x,x)\|_{H^{s-\frac{1}{2}}(\mathbb R^2)} \leq C \|f\|_{H^{s}(\mathbb R^2)}?$$