文档介绍：
Calculus with bounds
b
e
X
No wfor
R de
ne A
as consisting of the b
half
b
A
densities on X suc h that for some
dep ending on A
rb
b lb
is in the space
The extra subscript
is supp osed to indicate that
the space is de
ned b y b ounds and the inclusion of
giv es a little
ro om
in the estimates
Notice that
E
b
b
e
e
X
X
inf E
inf E
lb rb
b
b
In fact the second part of the pro of of Theorem
applies unc hanged to
sho wthat
b
e
X
A
de
nes a b ounded op erator
b
M
m
A
H
H
m
M if
and
b
b
It su
ces to ha v e
b ecause of the inclusion of
inthe
de
nition of the k ernels
No w the general calculus with b ounds is the sum
of three terms
m
b
X
b
os
b
m
b
b
e
X
X
H
X
b
os
b
lb
rb
b
The L b oundedness in
leads to p osition prop erties in v olv
ing the action of the
rst t w o summands on the third
Pr oposition
If
and
p osition of
op erators giv es
m
b
b
b
X
H
X
H
X
b b
lb lb
b
os
rb rb
Pr oof
Multiplying on the left b y
it su
ces to consider the case
b
can b e p osed in to
An y B
H
X
lb b
B
B
B
B
B
rb
where
is the pull
bac k of a b oundary de
ning function
C
X
from
rb
the righ t factor and
C
X
tak es the v alue
near the b oundary
and has supp ort in a collar neigh bourhood
The support of B is therefore
disjoin t from the righ t b oundary of X
so it is of the form of a C function
o