We deal with solutions to transport equations

The averaging lemmas, [13], [12], [11],
state that in
the generic non-degenerate case, averaging over the velocity space, , yields a gain of *spatial* regularity.
The prototype statement reads

Variants of the averaging lemmas were used by DiPerna and Lions to
construct global weak (renormalized) solutions of Boltzmann,
Vlasov-Maxwell and related
kinetic systems, [9], [10]; in Bardos et. al.,
[1],
averaging lemmas were used to construct solutions of the incompressible
Navier-Stokes equations. We turn our attention to
their use in the context of
nonlinear conservation laws and related equations.

*Proof.*(Sketch). We shall sketch the proof in the
particular case, *p*=*q* which will suffice to demonstrate the
general case.

Let
denote the set where the symbol is 'small',

and decompose the average, accordingly:

Here, represents the usual *smooth* partitioning
relative to and its complement, .
On , the symbol is 'bounded away' from zero, so we gain
one derivative:

On - along the `non uniformly elliptic' rays,
we have no gain of regularity, but instead, our non-degeneracy
assumption implies that is a 'small' set and therefore

Both (2.2.12) and (2.2.13)
are straightforward for *p*=2 and by estimating the
corresponding multipliers, the case follows
by interpolation.
Finally, we consider the *K*-functional

The behavior of this functional, ,
characterize the smoothness of in the intermediate
space between and : more precisely,
belongs to Besov space
with 'intermediate' smoothness of order .

Now set , then
with appropriately scaled we find
that with
. This means that
belongs to Besov space,
and (2.2.9) (with *p*=*q*=*r*) follows.

*Remark.*
In the limiting case of in (2.2.8),
one finds that if

then averaging is a compact mapping,
.
The case *p*=2 follows from Gèrard's results [12].

Mon Dec 8 17:34:34 PST 1997