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Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality. (English) Zbl 0639.46034

Let \(N(f)=N_ 1(f)+...+N_ K(f)\) for a measurable function f over \(R^ K\), where \(N_ k(f)\) is the mixed norm of power (1,...,1,\(\infty,1,...,1)\), \(\infty\) being on the k-th place, \(k=1,2,...,K\), \(K\geq 2\). It is shown that if \(N(f)<\infty\), then f is rearrangeable. Moreover, if g is a measure-preserving rearrangement of f such that for every \(\lambda >0\) the set where \(| g| >\lambda\) is essentially a K-cube with edges parallel to the coordinate axes, then \(N(g)\leq N(f)\). Finally, supposing \(N(f)<\infty\), f belongs to the Lorentz space \(L(r,1)\) with \(r=K/(K-1)\) and \(\| f\|_{L(r,1)}\leq N(f)/K\). These results are applied to prove sharper forms of the Sobolev inequality
Reviewer: M.Abel

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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