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On the reverse Hölder inequality. (English. Russian original) Zbl 1132.26009
Math. Notes 81, No. 3, 318-328 (2007); translation from Mat. Zametki 81, No. 3, 361-373 (2007).
If \(\alpha< \beta,\, \alpha\beta\neq 0\), and \(B>1\), let \(RH_{d \mu(x)}^{\alpha, \beta }(B)\) denote the class of all non-negative functions defined on the parallelepiped \(R_0\subset \mathbb R^d, d>1\), satisfying the reverse Hölder inequality
\[ \Biggl\{{1\over \mu(R)}\int_R f^{ \beta}d \mu\Biggr\}^{ \beta}\leq \Biggl\{{1\over \mu(R)}\int_R f^{ \alpha}d \mu\Biggr\}^{ \alpha} \] uniformly on all parallelepipeds \(R\subset R_0\). The main result of this paper is: if \(f\in RH_{d \mu(x)}^{\alpha, \beta }(B)\) and \( \gamma \in\, ]-\infty,\min\{0,\alpha\}[\, \cup\, ]\max\{0, \beta\}, \infty[\) is such that
\[ \Bigl(1-{ \alpha\over \gamma}\Bigr)^{1/ \alpha} >B\Bigl(1-{ \beta\over \gamma} \Bigr)^{1/ \beta},\tag{1} \] then there exist \(B'= B'(\alpha, \beta, B, \gamma), B''= B''(\alpha, \beta, B, \gamma)\), positive, such that for any \(R\subset R_0\):
\[ {1\over B'}\Biggl\{{1\over \mu(R)}\int_R f^{ \alpha}d \mu\Biggr\}^{ \alpha}\leq \Biggl\{{1\over \mu(R)}\int_R f^{ \gamma}d \mu\Biggr\}^{ \gamma}\leq{1\over B''} \Biggl\{{1\over \mu(R)}\int_R f^{ \beta}d \mu\Biggr\}^{ \beta}.\tag{2} \] If \( d \mu(x) = d(x)\) and \( \gamma \in\, ]-\infty,\min\{0,\alpha\}[\, \cup\, ]\max\{0, \beta\}, \infty[\) but does not satisfy (1) then at least one of the two inequalities in (2) fails. The method of proof depends on using a weighted analogue of Hardy’s inequalities [see L. D’Apuzzo and C. Sbordone, Rend. Mat. Appl., VII. Ser. 10, No. 2, 357–366 (1990; Zbl 0711.42027); A. Popoli, Matematiche 52, No. 1, 159–170 (1997; Zbl 0909.42010)].

MSC:
26D15 Inequalities for sums, series and integrals
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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