4 1. INTRODUCTION
When the fundamental solution G is smooth, as in the case of the heat equation,
one uses Burkholder’s inequality to see that
E(|u(t, x) u(t,
y)|p)
(1.6)
C E
t
0
ds
Rd
du
Rd
dv (G(t s, x u) G(t s, y u))Z(s, u)
× f(u v)Z(s, v)(G(t s, x v) G(t s, y v))
p/2
.
Then, the smoothness of G, together with integrability of Z, leads to regularity of
u(t, ·). For the wave equation, G(t) is singular with respect to Lebesgue measure
(see (1.3)), so this kind of approach is not feasible.
A different idea is to pass the increments on G in (1.6) onto the factor
Z(s, u)f(u v)Z(s, v),
using a change of variables; the right-hand side of (1.6) becomes the sum of

t
0
ds
R3
G(s, du)
R3
G(s, dv)
D2f(v
u, x y) (1.7)
× E(Z(t s, x u)Z(t s, x v)),
where
D2f(u,
x) = f(u + x) 2f(u) + f(u x), and of three other terms of similar
form (see the proof of Proposition 3.5 for details). Focussing on the term (1.7), one
checks that in the case where f(x) =
|x|−β
is a Riesz kernel,
(1.8)
|D2f(u,
x)| c|f
(u)||x|2

c|u|−(β+2)|x|2,
where f denotes the second order differential of f. This would lead to the following
bound for (1.7):
(1.9) |x
y|2
t
0
ds
R3
G(s, du)
R3
G(s, dv) |v
u|−(β+2).
The factor |x
y|2
looks too good to be true, and it is! Indeed, the triple integral
is equal to the left-hand side of (1.4), and we have already pointed out that this is
finite if and only if the exponent β + 2 is less than 2. However, this is not the case
since β ]0, 2[.
Even though the bound (1.9) equals +∞, this approach contains the premises of
our argument. Indeed, instead of differentiating f twice as in (1.8), we shall estimate
D2f by using a fractional derivative of order γ, where γ 2 β. It turns out that
for f(u) = |u|−β, the fractional derivative f (γ) is such that f (γ)(u) |u|−(β+γ).
This leads to the following bound for (1.7):
|x
y|γ
t
0
ds
R3
G(s, du)
R3
G(s, dv) |v
u|−(β+γ).
The triple integral is now finite since β + γ 2 and this gives the correct order of
regularity for u(t, ·).
The precise properties of Riesz kernels and rigorous use of their fractional
derivatives (or rather, their fractional Laplacians) are given in Lemma 2.6 (with
a there replaced by 3 β γ, b replaced by γ, and d = 3).
In short, there are mainly three ideas which have been central to obtaining
the results of this paper. First, the smoothing in space of the fundamental solu-
tion by means of a regularisation procedure based on time-scaled approximations of
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