Mixed norms and rearrangements: Sobolev's inequality and Littlewood's inequality

Abstract

In the 1930's, J. E. Littlewood and S. L. Sobolev each found useful estimates for Lp-norms. These results are usually not regarded as similar, because one of them is set in a discrete context and the other in a continuous setting. We show, however, that certain basic facts about mixed norms can be used to simplify proofs of both of these estimates. The same method yields a proof of a form of the isoperimetric inequality. We consider the effect of measure-preserving rearrangement on certain sums of permuted mixed norms of functions on RK, and show these sums are minimal when the rearranged function, f∼ say, has the property that, for each positive real number λ, the set on which ¦f∼¦>λ is a cube with edges parallel to the coordinate axes. Finally, we use the fact about rearrangements to prove sharper forms of the estimates of Littlewood and Sobolev.