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
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