Riesz transforms in the absence of a preservation condition, with applications to the Dirichlet Laplacian

Let L be a linear operator defined such that −L generates an associated heat semigroup e^−tL. Suppose this semigroup does not satisfy a preservation condition. A proof that a generalised Riesz transform Dg(L) satisfies L^p bounds for some range 2<p<q is provided based on two new estimates. One is a Hardy type inequality for L, the other a bound regarding how the gradient of the heat semigroup acts on a characteristic function. Applications are to cases where L is the Dirichlet Laplacian Δ_Ω on inner uniform subsets of Rⁿ.