Fubini's Theorem and General Minowski Inequality

Question

Suppose that $f$ is a positive function and $1\leq p<\infty$. Consider $$A=\int (\int f(x,y)^p\ dx)^{1/p}dy,\ B=(\int (\int f(x,y)\ dy)^p dx)^{1/p}.$$ Suppose all quantities are integrable and finite. Prove that $A \geq B$.

I think I'm supposed to use Fubini's Theorem alongside with the Minowski inequality, but I'm not sure how to apply it in the right way. How do I prove this?

This is called Minkowski's inequality for integrals. You can find a proof in Wikipedia. — geetha290krm, Dec 06 '23 at 07:11
See also: https://math.stackexchange.com/questions/4292693/why-do-we-need-fubinis-theorem-in-this-proof-of-minkowskis-inequality-for-inte?rq=1 — geetha290krm, Dec 06 '23 at 07:12

score 0 · Answer 1 · answered Dec 06 '23 at 07:25

0

$$B=\sup_{g\ge 0\|g\|_q=1} \int g(x)\int f(x,y)dydx=\int\left(\int g(x)f(x,y)dx\right)dy\le \int \|f(\cdot, y)\|_pdy$$

answered Dec 06 '23 at 07:25

Pelota

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Fubini's Theorem and General Minowski Inequality

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