Consider the manifold $M := mathbb{R}^3 setminus B_1$ where $B_1$ is the unit ball. Equip $M$ with an asymptotically flat metric $g$ of high order. Let $gamma$ be the induced metric on $partial M$.

The Riesz transform is the bounded linear map $R: L^2(partial M) to L^2 (Lambda^1 (T^* partial M))$ defined by

$$R (f) = d (- Delta_{gamma})^{-frac{1}{2}}$$

Define another operator $R^* : L^2(partial M) to L^2 (Lambda^1 (T^* partial M))$ by

$$R^*(f) = d N_g^{-1} (f)$$

Where $N_g: H^1(partial M) to L^2(partial M)$ is the Dirichlet to Neumann map, which is defined by the following: if $h = N_g(f)$, then $h = nu cdot nabla u $ where $nu$ is the unit normal vector field on $partial M$ and $u$ is the unique function on $M$ that goes to $0$ at infinity and satisfies $Delta_gu=0$ and $left. u right|_{partial M} = f$.

It is well known that $N_g$ is a pesudo differential operator of order 1 with principal part $- (-Delta_{gamma})^{frac{1}{2}}$. Also, $N_g$ is invertible (that’s not true on bounded domains).

My question will not be very specific. I want to understand $R^*$ more as well as the relationship between $R$ and $R^*$.

Is $R^*$ also a bounded operator? If so how does $Vert R^*Vert$ depend on $g$? What do we know about $R-R^*$? I would imagine this is a “small” operator in some sense because $N_g$ is “close” to $- (-Delta_{gamma})^{frac{1}{2}}$. Maybe $R-R^*$ is a smoothing operator (which is a psuedo differential operator of order $-infty$.)? Is there anything in the literature about $R^*$?

You can assume $g$ is the euclidean metric and so $gamma$ is the round metric on the sphere if that makes the question easier. In that case, I think $N_g = – sqrt{Delta_{gamma} + frac{1}{4}} + frac{3}{2}$

Any help or references will really be appreciated.