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Let $(X,B,\mu)$ and $(Y,C,\nu)$ be probability spaces, and let $m$ be the product measure. Let $f:X \times Y \rightarrow [0,\infty )$ be a $B \otimes C $ measurable function, $1 < p < \infty$, and define $f_x(y):=f(x,y)$.

What is the most general condition on $f$ to make sure that: $||f||_{L_p(m)}=\int{||f_x||_{L_p(\nu)}}d\mu(x)$ ?

I know that if $\int{|f_x(y)|^p}d\nu(y)$ is constant there is equality. But is it the only case?

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up vote 7 down vote accepted

yes, it is the only case. Denote $g(x)=(\int{|f(x,y)|^pd\nu(y))}^{1/p}$, then your condition is $||g||_1=||g||_p$ which is known to imply that $g$ is a constant function, see here.

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Thank you. I wonder if under the same conditions as in the original question, is it always true that $||f||_{L_p(m)} \gt ∫||f_x||_{L_p(ν)}dμ(x)$? I think it can be proven using Tonelli's theorem and concave Jensen's inequality, but I'm not sure they are applicable here. –  Shlomi May 2 '11 at 14:56

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