Bounded composition operators with closed range on the Dirichlet space

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator C-phi on the Dirichet space D is d nu(phi) = n(phi) dA, where n(phi)(z) is the cardinality of the preimage phi(-1) (z). The composition operator will have closed range if and only if the corresponding measure satisfies a "reverse Carleson measure" theorem: parallel to f parallel to(D)(2) less than or equal to integral \f'\(2) d nu(phi) for all f is an element of D. Assuming C-phi is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) nu(phi) (S) greater than or equal to c\S\ for all Carleson squares S. It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures nu(phi). That is, we construct a function phi such that C phi is bounded and nu(phi) satisfies (C) but the composition operator C-phi does not have closed range.