Summary
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.
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Research partially supported by N.S.E.R.C. operating grant number 4822.
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Fournier, J.J.F. Mixed norms and rearrangements: Sobolev's inequality and Littlewood's inequality. Annali di Matematica pura ed applicata 148, 51–76 (1987). https://doi.org/10.1007/BF01774283
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DOI: https://doi.org/10.1007/BF01774283