Harjulehto, Petteri; Hästö, Peter; Koskenoja, Mika Hardy’s inequality in a variable exponent Sobolev space. (English) Zbl 1096.46017 Georgian Math. J. 12, No. 3, 431-442 (2005). 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. Reviewer: Stefan G. Samko (Faro) Cited in 32 Documents MSC: 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10 Inequalities involving derivatives and differential and integral operators Keywords:variable exponent; Sobolev spaces; Hardy inequality Citations:Zbl 1099.42021 PDFBibTeX XMLCite \textit{P. Harjulehto} et al., Georgian Math. J. 12, No. 3, 431--442 (2005; Zbl 1096.46017)