$L^1$ and $L^2$ norm inequalities for product of functions

Question

I am given functions $f_1,f_2,f_3 \in L^2(\mathbb{R}^2)$. And a function $g:\mathbb{R}^3 \to \mathbb{R}, g(x,y,z) = f_1(y,z)\cdot f_2(x,z)\cdot f_3(x,y)$. I am supposed to show that: $ \|g\|_{L^1} \leq \|f_1 \|_{L^2} \cdot \|f_2 \|_{L^2}\cdot \|f_3 \|_{L^2}$. However, I am doubting that the inequality even holds since if we have equal spaces (i.e everything in $\mathbb{R}^3$ we would have the inequality only if $ \frac{1}{2}+\frac{1}{2}+\frac{1}{2}=1$ which is cleary not the case here.) I was not able to find a counterexample after playing around for a while, so I was considering the fact that it is true after all. So I was trying to apply Hölder yet without much success since the dimensions do not add up nicely and I got stuck with some even crazier integrals and products of integrals.

I would be very grateful I someone could point out a hint to me.

Answer 1

Hint: Divide the integration of $|g|$ over $\mathbb R^3$ into three steps: first integrate in $x\in\mathbb R$ then integate in $y\in\mathbb R$ and then integate in $z\in\mathbb R$. In each step, use the Cauchy inequality. For example: \begin{align*} \int_\mathbb R|g(x,y,z)|\,dx&=\int_\mathbb R|f_1(y,z)||f_2(x,z)||f_3(x,y)|\,dx\\ &=|f_1(y,z)|\int_\mathbb R|f_2(x,z)||f_3(x,y)|\,dx\\ &\leq |f_1(y,z)|\left(\int_\mathbb R|f_2(x,z)|^2\,dx\right)^{1/2}\left(\int_\mathbb R|f_3(x,y)|^2\,dx\right)^{1/2}. \end{align*}

The second step:

Denote $$F_2(z)=\left(\int_\mathbb R|f_2(x,z)|^2\,dx\right)^{1/2}, \qquad F_3(y)=\left(\int_\mathbb R|f_3(x,y)|^2\,dx\right)^{1/2},$$then $$\int_\mathbb R|g(x,y,z)|\,dx\leq |f_1(y,z)|F_2(z)F_3(y).$$Integrating with respect to $y\in\mathbb R$ gives \begin{align*} \int_\mathbb R\int_\mathbb R|g(x,y,z)|\,dx\,dy&\leq \int_\mathbb R|f_1(y,z)|F_2(z)F_3(y)\,dy\\&=F_2(z)\int_\mathbb R|f_1(y,z)|F_3(y)\,dy\\&\leq F_2(z)\left(\int_\mathbb R|f_1(y,z)|^2\,dy\right)^{1/2}\left(\int_\mathbb R|F_3(y)|^2\,dy\right)^{1/2}\\&=F_2(z)\left(\int_\mathbb R|f_1(y,z)|^2\,dy\right)^{1/2}\|f_3\|_{L^2}.\end{align*}

The third step:

Denote $$F_1(z)=\left(\int_\mathbb R|f_1(y,z)|^2\,dy\right)^{1/2},$$then $$\int_\mathbb R\int_\mathbb R|g(x,y,z)|\,dx\,dy\leq F_2(z)F_1(z)\|f_3\|_{L^2}.$$ Integrating with respect to $z$ gives that \begin{align*} \int_\mathbb R\int_\mathbb R\int_\mathbb R|g(x,y,z)|\,dx\,dy\,dz&\leq \int_\mathbb RF_2(z)F_1(z)\|f_3\|_{L^2}\,dz\\&=\|f_3\|_{L^2}\int_\mathbb RF_2(z)F_1(z)\,dz\\&\leq \|f_3\|_{L^2}\left(\int_\mathbb R|F_2(z)|^2\,dz\right)^{1/2}\left(\int_\mathbb R|F_1(z)|^2\,dz\right)^{1/2}\\&=\|f_1\|_{L^2}\|f_2\|_{L^2}\|f_3\|_{L^2}.\end{align*}