1.1. DEFINITIONS AND NOTATIONS. 5
1.1. Definitions and notations.
We conclude this section with few definitions and notations. However, the
reader should be aware that many objects are defined as the paper progresses;
see Section 3 for all gauge elements and their equations, Section 4 for the Fourier
analysis and related objects/spaces and Sections 5-6 for the functions spaces used
in the analysis of the nonlinear problem.
While at fixed time our maps into the sphere are functions defined on R2,
the equivariance condition allows us to reduce our analysis to functions of a single
variable |x| = r [0, ∞). One such instance is exhibited in (1.2) where to each equi-
variant map u we naturally associate its radial component ¯. u Some other functions
will turn out to be radial by definition, see, for instance, all the gauge elements in
Section 3. We agree to identify such radial functions with the corresponding one
dimensional functions of r. Some of these functions are complex valued, and this
convention allows us to use the bar notation with the standard meaning, i.e. the
complex conjugate.
Even though we work mainly with functions of a single spatial variable r, they
originate in two dimensions. Therefore, it is natural to make the convention that
for the one dimensional functions all the Lebesgue integral and spaces are with
respect to the rdr measure, unless otherwise specified.
For the Sobolev spaces we have introduced
˙
H
1
e
and He
1
in (1.6) as the natural
substitute for
˙
H
1
and
H1.
In a similar fashion we define
˙
H
2
e
and He
2
by the norms
f
2
˙
H 2
e
= ∂r
2f 2
L2
+
r−1∂rf 2
L2
+
r−2f 2
L2
, f
2
˙
H 2
e
= f
2
˙
H 2
e
+ f
2
L2
as the as the natural substitute for
˙
H 2 and H2.
For a real number a we define a+ = max{0, a} and a− = min{0, a}.
We will use a dyadic partition of
R2
(or [0, ∞) after the dimensional reduction)
into sets {Am}m∈Z given by
(1.14) Am =
{2m−1
r
2m+1}.
We will also use the notation Ak = ∪mkAm as well as Ak,A≥k which are
similarly defined.
Two operators which are often used on radial functions are [∂r]−1 and [r∂r]−1
defined as
[∂r]−1f(r)
=

r
f(s)ds,
[r∂r]−1f(r)
=

r
1
s
f(s)ds
A direct argument shows that
(1.15)
[r∂r]−1f
Lp p
f
Lp
, 1 p
We also have a weighted version
(1.16)
w[r∂r]−1f
Lp p
wf
Lp
, 1 p
assuming that g(r) = w(r)r
2
p
is an increasing function satisfying
g(r) (1 )g(2r)
for some 0. The proof is straightforward.
Previous Page Next Page