Hardy Type Inequalities for Aharonov-bohm Magnetic Potentials with Multiple Singularities

In this paper we are interested in similar inequalities for magnetic Dirichlet forms with Aharonov-Bohm vector potentials that have multiple singularities. Let P1 = (x1, y1), . . . , Pn = (xn, yn) be n different points in R . We can identify R with C by the correspondence (x, y) 7→ z = x + iy and the points P1, . . . , Pn then correspond to the complex numbers z1 = x1 + iy1, . . . , zn = xn + iyn. Consider a smooth vector potential A = (A1(x, y), A2(x, y)) in the punctured plane M = R \ {P1, . . . , Pn} with zero magnetic field (1.2) and denote by ωA the differential 1-form A1(x, y)dx+A2(x, y)dy. Then (1.2) says that ωA is a closed differential form in M , i.e. dωA = 0. Such a vector potential A is known as a magnetic vector potential of