Petteri Harjulehto, Peter Hästö, Mika Koskenoja
We show that a norm version of Hardy's inequality holds in a variable exponent Sobolev space provided the maximal operator is bounded. Our proof uses recent local versions of the inequality for a fixed exponent. We give an example to show that our assumptions on the exponent are essentially sharp. In the one-dimensional case, we derive a necessary and a sufficient condition for Hardy's inequality to hold.
Variable exponent, Sobolev space, Hardy's inequality
MSC 2000: Primary: 46E35. Secondary: 26D10