I am desperately trying to solve the following problem:
Let $ f \in C^1_c(\mathbb{R}^2)$ (continuously differentiable with compact support), show that $\int_{\mathbb{R}^2} |f(x_1,x_2)|^2 \ d\lambda_2(x_1,x_2) \leq (\int_{\mathbb{R}^2} |\nabla f(x_1,x_2)| \ d\lambda_2(x_1,x_2))^2$.
So far I was able to show that: $ f(x_1,x_2) = \int_{-\infty}^{x_1} \frac{\partial}{\partial x_1} f(t,x_2) \ d\lambda_1(t)$. Furthermore, I was trying to use the well known inequality $ \sqrt{\left|ab\right|} \leq \sqrt{a^2 + b^2} $ in order to change $|\nabla f(x_1,x_2)|$ into a product of two partial derivatives and try to substitute them with $ f(x_1,x_2) = \int_{-\infty}^{x_1} f(t,x_2) \ d\lambda_1(t)$. I also tried applying Fubini and Hölder, without any luck. I do not see how I can make further progress.
I would very much appreciate further hints.
You are almost done! If you use that $f(x,y) = ∫_{-∞}^x ∂_tf(t,y) \,\mathrm d t$, and similarly for $\overline{f(x,y)}$, then you get four integrals in the LHS. You also have four integrals in the RHS. All that remains is to make proper use of absolute values (and vector norms) and inequalities.