Minkowski's Inequality for Integrals in Orlicz spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T05:14:25Z http://mathoverflow.net/feeds/question/116167 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116167/minkowskis-inequality-for-integrals-in-orlicz-spaces Minkowski's Inequality for Integrals in Orlicz spaces Daniel Spector 2012-12-12T11:50:21Z 2012-12-13T08:38:50Z <p>EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces.</p> <p>Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, $f^{-1}:[0,\infty)\to[0,\infty)$ is concave and increasing and $\int_{B(0,\delta)} \rho^\delta(z)dz =1$.</p> <p>Is it true that </p> <p>$f^{-1}\left(\int_{B(0,h)} f\left(\int_{B(0,\delta)} |u(x,z)| \rho^\delta(z)dz\right)\;dx\right)\leq \int_{B(0,\delta)} f^{-1}\left(\int_{B(0,h)} f\left(|u(x,z)| \right)\;dx\right)\rho^\delta(z)dz$?</p> <p>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.)?</p>