Extrapolation of $L^p$ Data from a Modular Inequality

http://dx.doi.org/10.4153/CMB-2002-003-x
Canad. Math. Bull. 45(2002), 25-35

  Published:2002-03-01
 Printed: Mar 2002

If an operator $T$ satisfies a modular inequality on a rearrangement invariant space $L^\rho (\Omega,\mu)$, and if $p$ is strictly between the indices of the space, then the Lebesgue inequality $\int |Tf|^p \leq C \int |f|^p$ holds. This extrapolation result is a partial converse to the usual interpolation results. A modular inequality for Orlicz spaces takes the form $\int \Phi (|Tf|) \leq \int \Phi (C |f|)$, and here, one can extrapolate to the (finite) indices $i(\Phi)$ and $I(\Phi)$ as well.