Appendix A. Proof of Lemma 2.2

We begin analyzing

by following the same arguments of Example 6.24 in Ref. [58]. The key idea is to extract the dependence on |ξ| from the integral and show, by symmetry, that the remaining factor does not depend on ξ. First, note that

and likewise

so that

Next, write ξ = |ξ| ν, where |ν| = 1. Then

Therefore, the imaginary part of the integral (A.1) is proportional to the integral

which is zero by symmetry. More precisely, the change-of-variables

shows the integral is the negative of itself. Thus, the integral (A.1) is real-valued and given by

Next we argue that the integral above does not depend on ξ (note that this is not obvious at first glance, because ν = ξ/|ξ|). Denote the integral in the above equation by F(ν). We seek to show that F(ν) is the constant D_n,s. Let T be an orthonormal rotation; then

The change of variable θ → Tθ, which has Jacobian determinant 1 in absolute value, then gives

Since Ta ⋅ Tb = a ⋅ b,

Therefore F(ν) does not depend on ν; for example F(ν) = F(e₁), by choosing an orthonormal T such that T(ν) = e₁. We can then write the above constant as

For the first term above, consider the transformation θ′ ↦ −θ′. Under this transformation, θ ⋅ θ′ ↦ − θ ⋅ θ′. This transformation also maps the unit sphere {|θ′| = 1} to itself and has Jacobian determinant 1 in absolute value. Therefore

which proves the first term in (A.2) is zero. Therefore

We write

This can be simplified to

In the third and fourth integrals, consider the change-of-variables (θ, θ′) ↦ − (θ, θ′). This maps the range of integration in the third term to that of the first term, the range of integration in the fourth term to that of the second term, leaves the integrands invariant, and has Jacobian determinant 1 in absolute value. Therefore, the above can be written

Finally, for the second term above, consider the transformation θ′ ↦ -θ′. This maps the range of integration to that of the first term, flips the sign of θ′ ⋅ θ, and has Jacobian determinant 1 in absolute value. The above difference of integrals can therefore be written

Now write

Then we can write (A.3) as

For the second integral here, consider the transformation

This maps the set {|θ′| = 1, θ1′ ≥ 0} to itself, since it leaves both |θ′| = (θ1′)2+(θ2′)2+...(θn′)2 and θ'₁ invariant. It also has Jacobian determinant 1 in absolute value. But it flips the sign of the second integrand in (A.4). Therefore, the second integral is zero. So, we are left with

We see that the integral above is positive, since the integrand is positive over the region of integration. In fact, by Fubini’s theorem, we have

Appendix B The unweighted Green identity for fractional operators

In Section 3.3, we showed how (2.7) for the fractional interaction kernel (3.3) yielded the unweighted fractional Green’s identity (3.6), i.e.,

The second term on the right-hand side can be split as

Meanwhile, the first term on the right-hand side can be split as:

The identity

implies, with B = ℝⁿ, A = ℝⁿ ∖ Ω and therefore B ∖ A = Ω, that

Therefore, the first three integrals can be combined into

The fourth term can be written as

This cancels with the first term in (B.1), so (3.6) reduces to

This is precisely the fractional Green’s identity (3.5) reported by Ref. [34]. Thus, the latter can be viewed as a special case of the unweighted nonlocal Green’s identity of Ref. [38] with the fractional interaction kernel (3.3).

Note that Ref. [38] also defines a nonlocal interaction operator for a two-point vector field w(x, y) and for x ∈ ℝⁿ ∖ Ω,

which they use to introduce a notion of nonlocal flux,

With the same fractional interaction kernel α given by (3.3), using (2.3) we have

The identity (3.6) can therefore be rewritten using the notation 𝓝 [𝓖 u] (x), just as (3.5) uses the notation 𝓝_s u(x). The two operators are not equivalent, as

Appendix C Proof of Theorem 3.4

Theorem 3.4 is a direct consequence of the following two lemmas.

Appendix D Proof of Theorem 3.5

Theorem 3.5 is a direct consequence of the following two lemmas.