Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.

Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$ is concave and increasing and $\int_{B(0,\delta)} \rho^\delta(z)dz =1$.

Is it true that

$f^{-1}\left(\int_{B(0,h)} f\left(\int_{B(0,\delta)} |u(x,z)| \rho^\delta(z)dz\right)\;dx\right)\leq \int_{B(0,\delta)} f^{-1}\left(\int_{B(0,h)} f\left(|u(x,z)| \right)\;dx\right)\rho^\delta(z)dz$?

When $f(t)=t^p$, then $f^{-1}(t)=t^\frac{1}{p}$ and this is precisely Minkowski's Inequality for integrals, as the title suggests, and the proof uses duality in $L^p$. Does this theorem stretch to some class of $N$-functions $f$ ($\Delta_2$, etc.)?

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.