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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 $$B$$.
##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 26D15 Inequalities for sums, series and integrals