LECTURE NOTES ON DYADIC HARMONIC ANALYSIS

EXERCISE 1.11. Let

f

E L

1[1I'], {KN}

a summability kernel, show that

lim II!* KNII1

=

0.

N-oo

9

We can consider families { Kr}

depending on a continuous parameter

r

instead

of the discrete N. For instance the

Poisson kernel defined below, depends on the

parameter 0 :::;

r

1, and we replace the

limit "limN_,oo" by "limr_,1" wherever

necessary.

Remember that for

f

E

£

1

[TI']

the

N-th partial sum is given by

SNJ(x)

=

L

}(n)e2rrinx =

1

1

f(t)DN(X-

t)

dt

=

f

*

DN(x).

JnJ:SN

0

Here DN is the Dirichlet kernel, and

f01DN

=

1 but the sequence

{IIDNIIdNEN is

not

uniformly bounded , this is the cause of many difficulties when trying to show

a.e.

convergence of partial Fourier sums.

Mathematicians used summability methods to overcome this difficulty. The

Cesaro

method considered averages of partial sums:

1

N

11

aN f(x)

=

-N

1

L

Skf(x)

=

f(t)FN(x-

t)

dt

=

f

*

FN(x),

+

k=O

O

where FN is the Fejer kernel which is

positive and

J01

FN

=

1, therefore IIFN

11

1

=

1

for all

N.

The Poisson method considers an analytic extension to the unit disc,

F(z)

=

'L,nO }(n)zn, where z

= re2rrix.

Notice that the real and

imaginary parts

of F(z)

=

u(z)

+

iv(z) are given by:

u(re2rrix)

=

~

nf;oo }(n)rlnle2rrinx

=

~

11

f(t)Pr(x-

t)

dt

=

~f

*

Pr(x),

v(re2 rrix)

=

~i

f=

sgn(n)}(n)rlnle2 rrinx

=

~

1J(t)Qr(x-

1

t)

dt

=

~f

* Qr(x),

n=-oo

0

where Pr is the Poisson kernel on the disc, which is also positive and

J01

Pr

=

1;

and Qr is the conjugate Poisson kernel on the disc.

EXERCISE 1.12. Find closed

formulas for DN, FN, Pr and Qr. Check that FN

and Pr are summability kernels but not DN. More precisely show

that IIDNII

1

~

lnN. Compare limr_,

1

Qr(t) with (1.2) and observe that as r---+ 1,

F(z) approaches

~f(x)

+

i~Hf(x) (the limit should be taken in the sense of distributions).

More details can be found in

[Duo],

and

[Kat].

1.2.3. Convergence a.e. The following theorem

connects maximal operators

and a.e. convergence.

THEOREM 1.13. Given a family of linear operators, {Tt}tEA in LP(X,J-L), in-

dexed by a closed set of real numbers, A. LetT* f(x)

=

suptEA ITtf(x)l be the

maximal operator associated to {Tt}tEA. If T* is of weak type (p,

p), p

2

1, and to

is in the closure of

A,

then the

following sets are closed in LP:

{!

E £P(X, J-L): lim Ttf(x)

=

f(x) a.e.}

=

Ata·

t----+t

0