Holder norm estimates for elliptic operators on finite and infinite-dimensional spaces

We introduce a new method for proving the estimate Equation Omitted. Where u solves the equation Δu − λu = f. The method can be applied to the Laplacian on R^∞. It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations