MULTIPOLAR POTENTIALS AND WEIGHTED HARDY INEQUALITIES

In this paper we state the following weighted Hardy type inequality for any functions phi in a weighted Sobolev space and for weight functions mu of a quite general type J J J cN,mu RN V phi 2 mu(x)dx <= RN | backward difference phi|2 mu(x)dx + C mu RNW phi 2 mu(x)dx, where V is a multipolar potential and W is a bounded function from above depending on mu. Our method is based on introducing a suitable vector-valued function and an integral identity that we state in the paper. We prove that the constant cN,mu in the estimate is optimal by building a suitable sequence of functions.