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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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