## Integral operators on the cone of monotone functions.(English)Zbl 0837.26011

Let $$1< p< \infty$$, let $$f$$ and $$v$$ be measurable nonnegative functions on $$[0, \infty)$$ with $$v$$ locally integrable and let $I(f)= \sup_{0\leq g\downarrow} \Biggl( \int^\infty_0 fg dx\Biggr) \Biggl(\int^\infty_0 \Biggl({1\over x} \int^x_0 g dy\Biggr)^p v(x) dx\Biggr)^{-1/p}.$ The main result is the sharp two-sided estimate of $$I(f)$$.
It allows to reduce some inequalities for non-increasing functions to modified inequalities for arbitrary measurable functions. In particular, this approach allows to find for the case in which $$0< p\leq q< \infty$$, $$q\geq 1$$, necessary and sufficient conditions on nonnegative measurable functions $$w$$ and $$v$$ for the following inequality $\Biggl(\int^\infty_0 \Biggl({1\over x} \int^x_0 g dy\Biggr)^q w(x)dx\Biggr)^{1/q}\leq C\Biggl(\int^\infty_0 \Biggl({1\over x} \int^x_0 g dy\Biggr)^p v(x)dx\Biggr)^{1/p}$ to be valid for all non-increasing nonnegative functions $$g$$ with $$C$$ independent of $$g$$.

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 42B25 Maximal functions, Littlewood-Paley theory
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