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Hardy inequality in variable exponent Lebesgue spaces. (English) Zbl 1132.26341
Summary: We prove the Hardy inequality $\left\| x^{\alpha (x)+\mu (x)-1} \int_0^x \frac{f(y)dy}{y^{\alpha(y)}}\right\|_{L^{q(\cdot)}(\mathbb{R}^1_+)}\leq C\| f\|_{L^{p(\cdot)}(\mathbb{R}^1_+)}$
and a similar inequality for the dual Hardy operator for variable exponent Lebesgue spaces, where $$0 \leq \mu(0) < \frac{1}{p(0)}$$, $$0 \leq \mu(\infty) < \frac{1}{p(\infty)}$$, $$\frac{1}{q(0)}= \frac{1}{p(0)}-\mu(0)$$, $$\frac{1}{q(\infty)}= \frac{1}{p(\infty)}-\mu(\infty)$$ and $$\alpha(0)< \frac{1}{p'(0)}$$, $$\alpha(\infty)<\frac{1}{p'(\infty)}$$, $$\beta(0)>-\frac{1}{p(0)}$$, $$\beta(\infty)>-\frac{1}{p(\infty)}$$, not requiring local log-condition on $$\mathbb{R}^1_+$$, but supposing that this condition holds for $$\alpha(x),\mu(x)$$ and $$p(x)$$ only at the points $$x = 0$$ and $$x =\infty$$. These Hardy inequalities are proved by means of the general result of independent interest stating that any convolution operator on $$\mathbb{R}^n$$ with the kernel $$k(x- y)$$ admitting the estimate $$|k(x)| \leq c(1 + |x|)^{-\nu}$$ with $$\nu > n(1-\frac{1}{p(\infty)}+\frac{1}{q(\infty)})$$, is bounded in $$L^{p(\cdot)}(\mathbb{R}^n)$$ without local log-condition on $$p(\cdot)$$, only under the decay log-condition at infinity.

MSC:
 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.) 47B38 Linear operators on function spaces (general)
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