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Hardy’s inequality in a variable exponent Sobolev space. (English) Zbl 1096.46017
The authors prove the following version of the Hardy inequality for variable exponent Lebesgue spaces: there exists an \(a_0>0\) such that \[ \left\|\delta^{a-1}u\right\|_{p(\cdot)}\leq C\left\|\delta^a\nabla u\right\|_{p(\cdot)}, \quad u\in W^{1.p(\cdot)}_0\tag{1} \] for all \(0\leq a<a_0\), where \( \Omega\) is a bounded open set in \(\mathbb{R}^n\), \( \delta(x)= \text{dist}(x,\partial \Omega)\) and \(1<\inf_{x\in\Omega}p(x)\) and \(\sup_{x\in\Omega}p(x)<\infty\), and either the boundary satisfies a certain condition or \(\inf_{x\in\Omega}p(x)>n\).
They also give a version of the one-dimensional Hardy inequality for variable exponents, close to what was proved by V. Kokilashivili and the reviewer [Rev. Mat. Iberoam. 20, No. 2, 493–515 (2004; Zbl 1099.42021)] and provide a counterexample of an exponent \(p(x)\) which does not satisfy the log-condition at the origin, for which the one-dimensional Hardy inequality does not hold.

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators