10 2. FUNDAMENTAL SOLUTION AND COVARIANCE FUNCTION

Example 2.5. (a) The basic case is when ϕ ≡ 1. In this case, f ≡ kβ is termed

a Riesz kernel. We recall that kβ = cd,β Fkd−β [36, Chapter V].

(b) Another possibility is ϕ(x) =

exp(−σ2|x|2/2).

In this case, f(x) is indeed

a covariance function, since f = F(kd−β ∗ ψ), where

ψ(ξ) =

(2πσ2)−3/2 exp(−|ξ|2/(2σ2)).

The parameter δ in Assumption 2.4 can be set equal to 1. Condition (1.4) is satisfied

since β ∈ ]0, 2]. Indeed,

Rd

dξ (kd−β ∗ ψ)(ξ)

|FG(t)(ξ)|2

=

Rd

dη ψ(η)

Rd

dξ kd−β (ξ) |FG(t)(ξ −

η)|2

≤ sup

η∈Rd Rd

dξ kd−β (ξ) |FG(t)(ξ −

η)|2,

and the right-hand side is finite when β ∈ ]0, 2]: see [9, Lemma 8].

The Riesz potentials Ia associated with the function kβ (x) are defined by

(Iaϕ)(x) =

1

γ(a)

Rd

|x −

y|−d+aϕ(y)dy,

for ϕ ∈

S(Rd),

a ∈ ]0, d[ and γ(a) =

πd/22aΓ( a

2

)/Γ(

d−a

2

). Riesz potentials can be

interpreted as fractional integrals and have the semigroup property

Ia+b ϕ = Ia(Ibϕ), ϕ ∈

S(Rd),

a + b ∈ ]0, d[

(see [36, p.118]). This property implies in particular that

(2.11) |x −

y|−d+(a+b)

=

Rd

dz |x −

z|−d+a|z

−

y|−d+b,

provided a + b ∈ ]0, d[. This equality can be informally interpreted by saying that

| ·

|−d+(a+b)

is the fractional integral of order a of | ·

|−d+b,

which is natural since

(−∆)a/2(|·|−d+(a+b)) = |·|−d+b, as can be checked by taking Fourier transforms.

We will make heavy use of properties of first and second order increments of

Riesz kernels. For a function f : Rd → R, we set

Df(u, x) = f(u + x) − f(u),

D2f(u,

x) = f(u − x) − 2f(u) + f(u + x),

¯

D

2f(u,

x, y) = f(u + x + y) − f(u + x) − f(u + y) + f(u). (2.12)

Notice that D2f(u, x) =

¯

D 2f(u − x, x, x).

Lemma 2.6. Fix u, x, y ∈

Rd,

a + b ∈ ]0, d[. The following properties hold.

(a) For any c ∈ R,

Dkd−a−b(u, cx) =

|c|b

Rd

dw kd−a(u − c w)Dkd−b(w, x).

(b) For any b ∈ ]0, 1[ and any vector e ∈

Rd

with |e| = 1,

Rd

dw |Dkd−b(w, e)| ∞.