For
1
≤
p
>
∞
1 \leq p > \infty
,
Ω
\Omega
an open and bounded subset of
R
n
{R^n}
, and a nonincreasing and nonnegative function
φ
\varphi
defined in
(
0
,
ρ
0
]
(0,{\rho _0}]
,
ρ
0
=
diam
Ω
{\rho _0} = \operatorname {diam} \Omega
, we introduce the space
M
φ
,
0
p
(
Ω
)
\mathcal {M}_{\varphi ,0}^p(\Omega )
of locally integrable functions satisfying
\[
inf
c
∈
C
{
∫
B
(
x
0
,
ρ
)
∩
Ω
|
f
(
x
)
−
c
|
p
d
x
}
≤
A
|
B
(
x
0
,
ρ
)
|
φ
p
(
ρ
)
{\inf _{c \in C}}\left \{ {\int \limits _{B({x_0},\rho ) \cap \Omega } {|f(x) - c{|^p}dx} } \right \} \leq A|B({x_0},\rho )|{\varphi ^p}(\rho )
\]
for every
x
0
∈
Ω
{x_0} \in \Omega
and
0
>
ρ
≤
ρ
0
0 > \rho \leq {\rho _0}
, where
|
B
(
x
0
,
ρ
)
|
|B({x_0},\rho )|
denotes the volume of the ball centered in
x
0
{x_0}
and radius
ρ
\rho
. The constant
A
>
0
A > 0
does not depend on
(
x
0
,
ρ
)
({x_0},\rho )
. (i) We list some results on the structure, regularity, and density properties of the space so defined. (ii)
M
φ
,
0
p
\mathcal {M}_{\varphi ,0}^p
is represented as the dual of an atomic space.