A RECURRENCE CONDITION FOR SOME SUBORDINATED STRONGLY LOCAL DIRICHLETFORMS

We give a recurrence condition for subordinated strongly local Dirichl et forms. Let L be the associated negative definite self adjoint diffu sion operator. An outcome of our result is that if the derivative v(r) of the volume growth function satisfies v(r)less than or equal to cr( alpha-1-epsilon) for some 0<epsilon less than or equal to alpha-1 and 1<alpha<2 then -(-L)(alpha/2) is recurrent. In the one-dimensional cas e our hypotheses are weak, but in general we need a strong ''radial sy mmetry'' condition.