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The reverse Hölder inequality, the Muckenhoupt condition, and equimeasurable rearrangements of functions. (English. Russian original) Zbl 0802.42017
Russ. Acad. Sci., Dokl., Math. 45, No. 2, 301-304 (1992); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 323, No. 2, 229-232 (1992).
A general result of the following type for the rearrangement \(f^*\) of a nonnegative function \(f\) is proved: \(F(f^*)\leq F(f)\), where \(F\) is some special functional.
It allows to prove the following theorem. Suppose that the nonnegative function \(f\in L^ q([a,b])\), \(q>1\), satisfies the reverse Hölder inequality \[ \left({1\over \beta-\alpha} \int^ \beta_ \alpha f^ q(x)dx\right)^{1/q}\leq B\int^ \beta_ \alpha f(x)dx,\quad [\alpha,\beta]\subset [a,b]. \] Then \(f\in L^ p([a,b])\) for all \(p\geq q\) such that \[ G(p)\equiv 1+ {q-p\over p} \left({Bp\over p-1}\right)^ q> 0. \] The assertion ceases to be valid if \(p\geq q\) is such that \(G(p)\leq 0\).
Another useful corollary is as follows. Suppose that the nonnegative weight \(\omega\) on \([a,b]\) satisfies the \(A_ p\)-condition with some constant \(B\). Then the rearrangement \(\omega^*\) satisfies the \(A_ p\)- condition on \((0,b-a]\) with the same constant \(B\).
42B25 Maximal functions, Littlewood-Paley theory
26D15 Inequalities for sums, series and integrals