## A necessary and sufficient condition for Hardy’s operator in the variable Lebesgue space.(English)Zbl 1470.42041

Summary: The variable exponent Hardy inequality $$\|x^{\beta(x) - 1} \int_0^x f(t) d t\|_{L^{p(.)}(0, l)} \leq C \|x^{\beta(x)} f\|_{L^{p(.)}(0, l)}$$, $$f \geq 0$$ is proved assuming that the exponents $$p :(0, l) \rightarrow(1, \infty)$$, $$\beta :(0, l) \rightarrow \mathbb{R}$$ not rapidly oscilate near origin and $$1 / p'(0) - \beta > 0$$. The main result is a necessary and sufficient condition on $$p$$, $$\beta$$ generalizing known results on this inequality.

### MSC:

 42B30 $$H^p$$-spaces 26D10 Inequalities involving derivatives and differential and integral operators 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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### References:

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