Consider the resolvent operator $ R(z) := (-\Delta - z)^{-1}$ of the Laplace operator on $L^2(\mathbb R^d)$, where $z\in \rho(-\Delta) = \mathbb C \setminus \mathopen [0, \infty)$.


For $p \geq 1$, let $\lVert \cdot \rVert\_p$ denote the Schatten $p$-norm on the space of compact operators and let $1\_{\Gamma\_n}(x)$ denote multiplication by the indicator function of some cube $\Gamma\_n := n + [0,1]^d, n \in \mathbb Z^d$. For a given $E>0$, I am interested whether
$$\sup\_{\substack{z = E+i\epsilon \\ -1 \leq \epsilon \leq 1}}\lVert 1\_{\Gamma\_n}(x) R(z) 1\_{\Gamma\_n}(x) \rVert\_p < \infty \tag{1}$$
for some suitable $p=p(d)$, presumably all $p>d/2$.
So far I could not find any reference proving this, the problem being the supremum in front: In the book *Trace Ideals and their Applications* by Barry Simon, one can find the bound $$\lVert f(x) g(-i\nabla) \rVert\_p \leq C \lVert f \rVert\_p \lVert g \rVert\_p.$$
Noting that $\lVert 1\_{\Gamma\_n}(x) R(z) 1\_{\Gamma\_n}(x) \rVert\_p \leq \lVert 1\_{\Gamma\_n}(x) R(z) \rVert\_p$ and applying the above inequality to $f(x) := 1\_{\Gamma\_n}(x)$ and $g\_z(x):= \frac{1}{\lvert x \rvert^2 - z}$ yields
$$\lVert 1\_{\Gamma\_n}(x) R(z) 1\_{\Gamma\_n}(x) \rVert\_p \leq C \lVert g\_z \rVert\_p,$$
which is finite if $p>d/2$. Unfortunately, this is not enough for $(1)$ since the expression blows up as $\operatorname{Im }z = \epsilon \to 0$.


Still, I think $(1)$ should be true. For example, if $d=3$, we know that the kernel of $R(z)$ is given by
$$R(x,y;z) = \frac{1}{4\pi \lvert x - y \rvert} e^{-\sqrt{-z} \lvert x -y \rvert}$$
so that we can explicitly compute the Hilbert-Schmidt norm
$$\lVert 1\_{\Gamma\_n}(x) R(z) 1\_{\Gamma\_n}(x) \rVert\_2^2 = \int\_{\Gamma\_n} \int\_{\Gamma\_n} \lvert R(x,y;z) \rvert^2 \, dx dy \leq C \int\_{\Gamma\_n} \int\_{\Gamma\_n - y} \frac{1}{\lvert x \rvert^2} \, dx dy < \infty$$
uniformly in $z$. However, this does not work in $d>3$, where a higher $p$-norm would be needed (I assume $p>d/2$ as suggested by the above). Any help is appreciated!



Finiteness of Schatten $p$-norm of truncated free resolvent
I found a positive answer to this question in this [paper](https://arxiv.org/abs/1404.2817) of Frank and Sabin: *Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates*, Theorem 12.


