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