Prove that $\int_{\mathbb{R}^3} f(x, y)g(y, z)h(z, x) dλ_3(x, y, z) ≤ ∥f∥_{L_2(\mathbb{R}^2 )}∥g∥_{L_2(\mathbb{R}^2 )}∥h∥_{L_2(\mathbb{R}^2 )}$

Question

Hey I have this problem where I am stuck on solving it.

I Think it is very easy but I dont know how to proceed.

The Exercise is

Let $f, g, h ∈ L_2(\mathbb{R}^2 )$.

To show is:

$\int_{\mathbb{R}^3} f(x, y)g(y, z)h(z, x) dλ_3(x, y, z) ≤ ∥f∥_{L_2(\mathbb{R}^2 )}∥g∥_{L_2(\mathbb{R}^2 )}∥h∥_{L_2(\mathbb{R}^2 )}$

So what I thought is $\int_{\mathbb{R}^3} f(x, y)g(y, z)h(z, x) dλ_3(x, y, z) ≤|\int_{\mathbb{R}^3} f(x, y)g(y, z)h(z, x) dλ_3(x, y, z)|\\ \leq\int_{\mathbb{R}^3} |f(x, y)g(y, z)h(z, x)| dλ_3(x, y, z) \\ =∥f(x, y)g(y, z)h(z, x)∥_{L_1(\mathbb{R}^2 )}$

and with Cauchy Schwarz we have

$∥f(x, y)g(y, z)h(z, x)∥_{L_1(\mathbb{R}^2 )} \leq ∥f∥_{L_2(\mathbb{R}^2 )}∥g∥_{L_2(\mathbb{R}^2 )}∥h∥_{L_2(\mathbb{R}^2 )}$

So the problem is that Cauchy Schwarz works only with two functions.

What I thought to do is than take $m(x,y,z):=f(x,y) \cdot g(y,z)$

and do

$∥f(x, y)g(y, z)h(z, x)∥_{L_1(\mathbb{R}^2 )}=∥m(x,y,z) \cdot h(z, x)∥_{L_1(\mathbb{R}^2 )} \\ \leq ∥m∥_{L_2(\mathbb{R}^2 )}∥h∥_{L_2(\mathbb{R}^2 )}=∥f \cdot g∥_{L_2(\mathbb{R}^2 )}∥h∥_{L_2(\mathbb{R}^2 )} \\ \leq ∥f∥_{L_2(\mathbb{R}^2 )}∥g∥_{L_2(\mathbb{R}^2 )}∥h∥_{L_2(\mathbb{R}^2 )}$

I don't know if what I have done is correct. Can someone confirm it? If I have done any mistake can someone correct me?

Answer 1

Using Funibi's theorem you can write \begin{align} \int_{\mathbb{R}^3} f(x, y)g(y, z)h(z, x) \lambda_3(dx,dy,dz)= \int_{\mathbb{R}^2} f(x,y)\Big(\int_{\mathbb{R}}g(y,z)h(z,x)\,dz\Big)\,\lambda_2(dx, dy) \end{align}

Set $p(x,y)=\int_{\mathbb{R}}|g(y,z)h(z,x)|\,dz$. An application of Holder's inequality yields

$$ \Big|\int_{\mathbb{R}^3} f(x, y)g(y, z)h(z, x) \lambda_3(dx,dy,dz)\Big|\leq \|f\|_{L^2(\lambda_2)}\|p\|_{L^2(\lambda_2)} $$

By the generalized Minkowski's inequality (GMI), Fubibi's theorem (F) and Holder's inequality (HI) \begin{align} \|p\|_{L^2(\lambda_2)}&=\left(\int_{\mathbb{R}^2}\Big(\int_{\mathbb{R}}|g(y,z)||h(z,x)|\,dz\Big)^2\,\lambda_2(dx,dy)\right)^{1/2}\\ &\stackrel{GMI}{\leq}\int_{\mathbb{R}}\Big(\int_{\mathbb{R}^2}|g(y,z)|^2|h(z,x)|^2\,\lambda_2(dx,dy)\Big)^{1/2}\,dz\\ &\stackrel{F}{=}\int_{\mathbb{R}}\Big(\int_{\mathbb{R}} |g(y,z)|^2\,dy\Big)^{1/2}\Big(\int_{\mathbb{R}} |h(z,x)|^2\,dx\Big)^{1/2}\,dz\\ &\stackrel{HI}{\leq} \Big(\int_{\mathbb{R}}\Big(\int_{\mathbb{R}}|g(y,z)|^2\,dy\Big)\,dz\Big)^{1/2} \Big(\int_{\mathbb{R}}\Big(\int_{\mathbb{R}}|h(z,x)|^2\,dx\Big)\,dz\Big)^{1/2}\\ &\stackrel{F}{=}\|g\|_{L^2(\lambda_2)}\|h\|_{L^2(\lambda_2)} \end{align}