Riesz transforms on a solvable Lie group of polynomial growth

We study Riesz transforms associated with a sublaplacian H on a solvable Lie group G, where G has polynomial volume growth. It is known that the standard second order Riesz transforms corresponding to H are generally unbounded in L^p(G). In this paper, we establish boundedness in L^p for modified second order Riesz transforms, which are defined using derivatives on a nilpotent group G_N associated with G. Our method utilizes a new algebraic approach which associates a distinguished choice of Cartan subalgebra with the sublaplacian H. We also obtain estimates for higher derivatives of the heat kernel of H, and give a new proof (without the use of homogenization theory) of the boundedness of first order Riesz transforms. Our results can be generalized to an arbitrary (possibly non-solvable) Lie group of polynomial growth.