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