Petteri Harjulehto, Peter Hästö, Mika Koskenoja

Hardy's Inequality in a Variable Exponent Sobolev Space

Abstract:
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.

Keywords:
Variable exponent, Sobolev space, Hardy's inequality

MSC 2000: Primary: 46E35. Secondary: 26D10