Averaging lemmas deduce smoothness of velocity averages, such as
\[
f
¯
(
x
)
:=
∫
Ω
f
(
x
,
v
)
d
v
,
Ω
⊂
R
d
,
\bar f(x):=\int _\Omega f(x,v)\, dv ,\quad \Omega \subset \mathbb {R}^d,
\]
from properties of
f
f
. A canonical example is that
f
¯
\bar f
is in the Sobolev space
W
1
/
2
(
L
2
(
R
d
)
)
W^{1/2}(L_2(\mathbb {R}^d))
whenever
f
f
and
g
(
x
,
v
)
:=
v
⋅
∇
x
f
(
x
,
v
)
g(x,v):=v\cdot \nabla _xf(x,v)
are in
L
2
(
R
d
×
Ω
)
L_2(\mathbb {R}^d\times \Omega )
. The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet decompositions, and interpolation can be used to prove
L
p
L_p
versions of the averaging lemma. For example, it is shown that
f
,
g
∈
L
p
(
R
d
×
Ω
)
f,g\in L_p(\mathbb {R}^d\times \Omega )
implies that
f
¯
\bar f
is in the Besov space
B
p
s
(
L
p
(
R
d
)
)
B_p^s(L_p(\mathbb {R}^d))
,
s
:=
min
(
1
/
p
,
1
/
p
′
)
s:=\min (1/p,1/p^\prime )
. Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint
p
=
1
p=1
.