A class of weighted Hardy type inequalities in RN

In the paper we prove the weighted Hardy type inequality 1 integral V-RN phi(2)mu(x)dx <= integral(RN) vertical bar del phi vertical bar(2)mu(x)dx + K integral(RN)phi(2)mu(x)dx, (1) for functions. in a weighted Sobolev space H-mu(1), for a wider class of potentials V than inverse square potentials and for weight functions mu of a quite general type. The case mu = 1 is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators Lu = Delta u + del mu/mu center dot del u perturbed by singular potentials.