10 CALDERON' S FORMULA AND A DECOMPOSITION OF L
2 (Rn)
Suppose x i 3Q an d (y, t) e T(Q)\ the n / l(Q) \x-y\. Con -
sequently, ^— ^ 1 an d i t follow s fro m (1) i n (1.1 ) tha t tp t(x - y) =
0. Thi s gives u s property (1.6). I t follow s tha t fo r an y fixed x th e su m
S/(Q)=2""
sQao(x}
^
as at m o s t a
fi
n*te
numbe r of nonzero terms. Thi s shows
that fj(x) i s a well-defined function . Fro m the definitions o f f.
9
s
Qi
a
Q
and T(Q) w e see that
j
"=-J/(G)=2_l/
= E £ f
x
f ft^-y){? t*f){y)dy^
i/=-y
The last expression is precisely ^ ^ for e = 2~
7_1
an d S = 2
J
\ hence, (1.5)
follows from Theorem (1.2).
Using propert y (5 ) i n Lemma (1.1 ) an d Plancherel's theorem , w e obtai n
(1.9):
£4
=
f°°
/j^)lV(0|2
^y = ll/ll2.
Property (1.7) follows from (4 ) in (1.1) and Fubini's theorem.
Finally, we need to show property (1.8) i s satisfied. Sinc e t e [^ , /((?) ]
when (y,t)e T(Q) , w e have:
\D\(x)\ 7 - / / \[D
y
xpt(x-y)](pt*f)(y)\dy^
SQJJT(Q)
t
S
i/L^ ~ '"''' T f k \llrJ"- *
/)WI
^T
'T(Q) l ) I 1JT(Q)
\tr2M\(Drp)t(x-y)\2dy^\
'T(Q) * J
f8upi(D'f)(x)iVMy,"",",/V(e)i,/272RU€
=2|yi+"+i/2[/(0]-w-"-,/2[/(G)r/2
f ^
V/ 2
IIDV H
\—
= Cyl{Q)-M-»'\
Theorem (1.4) has an easily proved converse. A version of this converse is:
If atoms are defined as functions a
Q
havin g properties (1.6), (1.7), (1.8) and
Previous Page Next Page