11 Wavelet Methods for Pointwise Regularity
Let x0 £
!Rn;
define
\a\N
and
P(X-XQ)
= y^Pj( x -Xp) .
j0
(This series converges because of (1.3)). Given x, define jo by
(1.5) 2~jo \x-x0\ 2.2~ j o ;
then
3J0 JJO
Using the mean-value theorem, we bound the first term by
T\x-x0\N+1 sup H^Aj-COHoo
C2~(N+l)jo J2 2^N+Vj 0(2-i) (using (1.4))
3Jo
C0(2-jo) (using (1.3))
ce(\x-x0\).
The second term is bounded by
W | | A
j
( / ) | |
0 0
+ J2 I s - s o n ^ A ^ / ) ! ! ,
jjo ^ \*\N
] T
\e(2-j)+
J2
\x-x0\a2^je(2~^)
jjo \ \*\N
C2~Njo ^2Nj6{2-j)
33o
0(\x-xo\) (using (1.3)) .
Thus, under assumption (1.3), the uniform modulus of continuity of /
can be characterized using the Littlewood-Paley decomposition. The
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