In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $L$ be a full lattice in $\mathbb{R}^N$ with fundamental parallelepid $F$. Then, for $t>0$, $$\#(L\cap tD)=\frac{\mathrm{vol}(D)}{\mathrm{vol}(F)}t^N+O(t^{N-1}).$$ Here, $\#(L\cap tD)$ is the number of lattice points in the dilation $tD$ of $D$. Also, the constant in $O$ depends on $L$, $N$ and the Lipschitz constants.

I have sketched a (somewhat tedious, perhaps wrong) proof of the following generalization. Let all as above, and let $h\colon\mathbb{R}^N\to\mathbb{R}$ be a polynomial of degree $d$. Then
$$\sum_{\lambda\in L\cap tD}h(\lambda)=\frac{t^{d+N}}{\mathrm{vol(F)}}\int_Dh+O(t^{d+N-1}),$$ where now the constant in $O$ also depends on $h$. This is certainly true if $D$ is a polytope, which follows, for example, by the work of Berline and Vergne on local Euler-Malaurin formulas. If $h=1$, one recovers the original result.

If true, the above generalization must be written somewhere but I'm unable to find a reference.

Does someone knows where to find this result (or something similar in generality)?

In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $L$ be a full lattice in $\mathbb{R}^N$ with fundamental parallelepid $F$. Then, for $t>0$, $$\#(L\cap tD)=\frac{\mathrm{vol}(D)}{\mathrm{vol}(F)}t^N+O(t^{N-1}).$$ Here, $\#(L\cap tD)$ is the number of lattice points in the dilation $tD$ of $D$. Also, the constant in $O$ depends on $L$, $N$ and the Lipschitz constants.

I have sketched a (somewhat tedious, perhaps wrong) proof of the following generalization. Let all as above, and let $h\colon\mathbb{R}^N\to\mathbb{R}$ be a polynomial of degree $d$. Then
$$\sum_{\lambda\in L\cap tD}h(\lambda)=\frac{t^{d+N}}{\mathrm{vol(F)}}\int_Dh+O(t^{d+N-1}),$$ where now the constant in $O$ also depends on $h$. This is certainly true if $D$ is a polytope, which follows, for example, by the work of Berline and Vergne on local Euler-Maclaurin formulas. If $h=1$, one recovers the original result.

If true, the above generalization must be written somewhere but I'm unable to find a reference.

Does someone knows where to find this result (or something similar in generality)?