<s>
In	O
harmonic	O
analysis	O
in	O
mathematics	O
,	O
a	O
function	O
of	O
bounded	B-Algorithm
mean	I-Algorithm
oscillation	I-Algorithm
,	O
also	O
known	O
as	O
a	O
BMO	O
function	O
,	O
is	O
a	O
real-valued	O
function	O
whose	O
mean	B-Algorithm
oscillation	I-Algorithm
is	O
bounded	B-Algorithm
(	O
finite	O
)	O
.	O
</s>
<s>
The	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
of	O
bounded	B-Algorithm
mean	I-Algorithm
oscillation	I-Algorithm
(	O
BMO	O
)	O
,	O
is	O
a	O
function	B-Algorithm
space	I-Algorithm
that	O
,	O
in	O
some	O
precise	O
sense	O
,	O
plays	O
the	O
same	O
role	O
in	O
the	O
theory	O
of	O
Hardy	O
spaces	O
Hp	O
that	O
the	O
space	O
L∞	O
of	O
essentially	O
bounded	B-Algorithm
functions	O
plays	O
in	O
the	O
theory	O
of	O
Lp-spaces	O
:	O
it	O
is	O
also	O
called	O
John	O
–	O
Nirenberg	O
space	O
,	O
after	O
Fritz	O
John	O
and	O
Louis	O
Nirenberg	O
who	O
introduced	O
and	O
studied	O
it	O
for	O
the	O
first	O
time	O
.	O
</s>
<s>
According	O
to	O
,	O
the	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
of	O
bounded	B-Algorithm
mean	I-Algorithm
oscillation	I-Algorithm
was	O
introduced	O
by	O
in	O
connection	O
with	O
his	O
studies	O
of	O
mappings	B-Algorithm
from	O
a	O
bounded	B-Algorithm
set	I-Algorithm
belonging	O
to	O
Rn	O
into	O
Rn	O
and	O
the	O
corresponding	O
problems	O
arising	O
from	O
elasticity	O
theory	O
,	O
precisely	O
from	O
the	O
concept	O
of	O
elastic	O
strain	O
:	O
the	O
basic	O
notation	O
was	O
introduced	O
in	O
a	O
closely	O
following	O
paper	O
by	O
,	O
where	O
several	O
properties	O
of	O
this	O
function	B-Algorithm
spaces	I-Algorithm
were	O
proved	O
.	O
</s>
<s>
The	O
mean	B-Algorithm
oscillation	I-Algorithm
of	O
a	O
locally	O
integrable	O
function	O
u	O
over	O
a	O
hypercube	B-Operating_System
Q	O
in	O
Rn	O
is	O
defined	O
as	O
the	O
value	O
of	O
the	O
following	O
integral	O
:	O
</s>
<s>
uQ	O
is	O
the	O
average	O
value	O
of	O
u	O
on	O
the	O
cube	B-Application
Q	O
,	O
i.e.	O
</s>
<s>
A	O
BMO	O
function	O
is	O
a	O
locally	O
integrable	O
function	O
u	O
whose	O
mean	B-Algorithm
oscillation	I-Algorithm
supremum	O
,	O
taken	O
over	O
the	O
set	O
of	O
all	O
cubes	B-Application
Q	O
contained	O
in	O
Rn	O
,	O
is	O
finite	O
.	O
</s>
<s>
The	O
supremum	O
of	O
the	O
mean	B-Algorithm
oscillation	I-Algorithm
is	O
called	O
the	O
BMO	O
norm	O
of	O
u	O
.	O
and	O
is	O
denoted	O
by	O
||u||BMO	O
(	O
and	O
in	O
some	O
instances	O
it	O
is	O
also	O
denoted	O
||u||∗	O
)	O
.	O
</s>
<s>
The	O
use	O
of	O
cubes	B-Application
Q	O
in	O
Rn	O
as	O
the	O
integration	B-Algorithm
domains	O
on	O
which	O
the	O
is	O
calculated	O
,	O
is	O
not	O
mandatory	O
:	O
uses	O
balls	O
instead	O
and	O
,	O
as	O
remarked	O
by	O
,	O
in	O
doing	O
so	O
a	O
perfectly	O
equivalent	O
definition	O
of	O
functions	O
of	O
bounded	B-Algorithm
mean	I-Algorithm
oscillation	I-Algorithm
arises	O
.	O
</s>
<s>
BMO	O
functions	O
are	O
locally	O
Lp	O
if	O
0	O
<	O
p	O
<	O
∞	O
,	O
but	O
need	O
not	O
be	O
locally	O
bounded	B-Algorithm
.	O
</s>
<s>
Constant	O
functions	O
have	O
zero	O
mean	B-Algorithm
oscillation	I-Algorithm
,	O
therefore	O
functions	O
differing	O
for	O
a	O
constant	O
c>0	O
can	O
share	O
the	O
same	O
BMO	O
norm	O
value	O
even	O
if	O
their	O
difference	O
is	O
not	O
zero	O
almost	O
everywhere	O
.	O
</s>
<s>
As	O
the	O
name	O
suggests	O
,	O
the	O
mean	O
or	O
average	O
of	O
a	O
function	O
in	O
BMO	O
does	O
not	O
oscillate	O
very	O
much	O
when	O
computing	O
it	O
over	O
cubes	B-Application
close	O
to	O
each	O
other	O
in	O
position	O
and	O
scale	O
.	O
</s>
<s>
This	O
property	O
is	O
,	O
in	O
fact	O
,	O
equivalent	O
to	O
f	O
being	O
in	O
BMO	O
,	O
that	O
is	O
,	O
if	O
f	O
is	O
a	O
locally	O
integrable	O
function	O
such	O
that	O
|fR−fQ|	O
≤	O
C	O
for	O
all	O
dyadic	O
cubes	B-Application
Q	O
and	O
R	O
adjacent	O
in	O
the	O
sense	O
described	O
above	O
and	O
f	O
is	O
in	O
dyadic	O
BMO	O
(	O
where	O
the	O
supremum	O
is	O
only	O
taken	O
over	O
dyadic	O
cubes	B-Application
Q	O
)	O
,	O
then	O
f	O
is	O
in	O
BMO	O
.	O
</s>
<s>
The	O
John	O
–	O
Nirenberg	O
Inequality	O
is	O
an	O
estimate	O
that	O
governs	O
how	O
far	O
a	O
function	O
of	O
bounded	B-Algorithm
mean	I-Algorithm
oscillation	I-Algorithm
may	O
deviate	O
from	O
its	O
average	O
by	O
a	O
certain	O
amount	O
.	O
</s>
<s>
For	O
each	O
,	O
there	O
are	O
constants	O
(	O
independent	O
of	O
f	O
)	O
,	O
such	O
that	O
for	O
any	O
cube	B-Application
in	O
,	O
</s>
<s>
Conversely	O
,	O
if	O
this	O
inequality	O
holds	O
over	O
all	O
cubes	B-Application
with	O
some	O
constant	O
C	O
in	O
place	O
of	O
||f||BMO	O
,	O
then	O
f	O
is	O
in	O
BMO	O
with	O
norm	O
at	O
most	O
a	O
constant	O
times	O
C	O
.	O
</s>
<s>
such	O
that	O
its	O
over	O
every	O
arc	O
I	O
of	O
the	O
unit	O
circle	O
is	O
bounded	B-Algorithm
.	O
</s>
<s>
The	O
space	O
VMO	O
of	O
functions	O
of	O
vanishing	B-Algorithm
mean	I-Algorithm
oscillation	I-Algorithm
is	O
the	O
closure	O
in	O
BMO	O
of	O
the	O
continuous	O
functions	O
that	O
vanish	O
at	O
infinity	O
.	O
</s>
<s>
It	O
can	O
also	O
be	O
defined	O
as	O
the	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
whose	O
"	O
mean	B-Algorithm
oscillations	I-Algorithm
"	O
on	O
cubes	B-Application
Q	O
are	O
not	O
only	O
bounded	B-Algorithm
,	O
but	O
also	O
tend	O
to	O
zero	O
uniformly	O
as	O
the	O
radius	O
of	O
the	O
cube	B-Application
Q	O
tends	O
to	O
0	O
or	O
∞	O
.	O
</s>
<s>
where	O
fi	O
∈	O
L∞	O
,	O
α	O
is	O
a	O
constant	O
and	O
H	O
is	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Similarly	O
f	O
is	O
VMO	O
if	O
and	O
only	O
if	O
it	O
can	O
be	O
represented	O
in	O
the	O
above	O
form	O
with	O
fi	O
bounded	B-Algorithm
uniformly	O
continuous	O
functions	O
on	O
R	O
.	O
</s>
<s>
Let	O
Δ	O
denote	O
the	O
set	O
of	O
dyadic	O
cubes	B-Application
in	O
Rn	O
.	O
</s>
<s>
The	O
space	O
dyadic	O
BMO	O
,	O
written	O
BMOd	O
is	O
the	O
space	B-Algorithm
of	I-Algorithm
functions	I-Algorithm
satisfying	O
the	O
same	O
inequality	O
as	O
for	O
BMO	O
functions	O
,	O
only	O
that	O
the	O
supremum	O
is	O
over	O
all	O
dyadic	O
cubes	B-Application
.	O
</s>
<s>
All	O
bounded	B-Algorithm
(	O
measurable	O
)	O
functions	O
.	O
</s>
