In the paper we prove the weighted Hardy type inequality 1∫RNVφ2μ(x)dx≤∫RN|∇φ|2μ(x)dx+K∫RNφ2μ(x)dx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{{{\\mathbb {R}}}^N}V\\varphi ^2 \\mu (x)dx\\le \\int _{\\mathbb {R}^N}|\ abla \\varphi |^2\\mu (x)dx +K\\int _{\\mathbb {R}^N}\\varphi ^2\\mu (x)dx, \\end{aligned}$$\\end{document}for functions φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} in a weighted Sobolev space Hμ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1_\\mu $$\\end{document}, for a wider class of potentials V than inverse square potentials and for weight functions μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} of a quite general type. The case μ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu =1$$\\end{document} is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators Lu=Δu+∇μμ·∇u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} Lu=\\varDelta u+\\frac{\ abla \\mu }{\\mu }\\cdot \ abla u \\end{aligned}$$\\end{document}perturbed by singular potentials.