TANGENTIAL BOUNDARY BEHAVIOR IN
THE UNIT DISK AND THE EXCEPTIONAL
SETS 0F FUNCTIONS AND THEIR
FRACTIONAL INTEGRALS

 

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
JOSEPH THOMAS MATTI
1970.

‘IHttu‘:

0-169

This is to certify that the

thesis entitled

TANGENTIAL BOUNDARY BEHAVIOR IN THE UNIT
DISK AND THE EXCEPTIONAL SETS OF FUNCTIONS AND
THEIR FRACTIONAL INTEGRALS

presented by

Joseph Thomas Matti

has been accepted towards fulfillment

of the requirements for

11E 1 9, degree infi/

/Vr//_7/5we7

Major professor/

Date ZZZ? /flll ///fl

 

     

aunhme f’ I 1
9II HMS & SUNS' ‘ I
. [BOOK MIMI" INC.

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ABSTRACT

TANGENTIAL BOUNDARY BEHAVIOR IN THE UNIT
DISK AND THE EXCEPTIONAL SETS OF FUNCTIONS AND
THEIR FRACTIONAL INTEGRALS

By

Joseph Thomas Matti

Let ¢(x) be a non-negative, non-decreasing function in

(0,1) which is integrable there. Define §(n) =‘I1 ¢(x)dx.

1_.1_
n

This definition is due to P. B. Kennedy [9] who uses it to gen-

eralize some reSults concerning the function ¢(x) = (1-x)a-1,

O < a s 1, and the corresponding discrete function §(n) = n-a.
A similar generalization is made for the case ¢(x) = (1-x)a-1,
a > 1. ‘We note that the factor n.“ is present in the defini-

tion of the integral of fractional order fa as given by Weyl:

Q

f(x) = z ane

-m

inx inx

~a
, fa(x) = Rd 2 n ane

where kc is a constant depending on 0 alone. We observe further
on
n
that ¢(x) = 2 y x where
n
n=0

 

a(a+l)...(a+n-l) a n-a

Yn = n! so that
on ¢<x> *=' 2 my“.
n=0

We consider the class of all general ¢ satisfying the
property (*), and we define the Q-fractional integral of f as

m
f§(X) ‘ 2 §(n)a nx. We proceed to show that certain results

e
n=1 n

for fractional integrals hold true for the generalized case. In

. . . i
addition, several results concerning function f(x) = 2 one nx

whose coefficients satisfy the condition 2 naIcnI2 < m are
c I2
n

@(n)

are given in terms of exceptional sets whose h-measure is zero,

 

shown to extend to those f for which 2 < m. Most results

the general ¢ being sometimes defined in terms of h by the
equation ¢(x) = hCI£;)' Certain other classes of functions are
discussed which are defined in terms of h alone. Finally we
discuss the behavior of elements of several function classes which
are defined in the unit disk D as the functions take on func-

tions near the boundary, the values being restricted to the set
ie
R[T,e] = {z e D: 1 - IzI 2 T(Ie - IzII)}

where T is an increasing function for which T(O) = O.

TANGENTIAL BOUNDARY BEHAVIOR IN THE UNIT
DISK AND THE EXCEPTIONAL SETS OF FUNCTIONS AND
THEIR FRACI' IONAL INTEGRAIS

BY

Joseph Thomas Matti

A THESIS
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Mathematics

1970

TO CAROLINE

ii

ACKNOWLEDGEMENTS

I wish to thank my wife and parents for their patience
and encouragement during the completion of this work.
I am deeply indebted to Professor John Kinney for his

guidance in the writing of this thesis.

iii

INTRODUCTION

This paper deals basically with three principal ideas:
measure functions, fractional integrals, and tangential limits.
Each of these topics will be discussed in a general sense, i.e.
the discussion will not be in terms of specific functions, but
rather in terms of classes of functions possessing certain pro-
perties. Each of these tOpics deals then with a generalization
of a concept which is itself a generalization of a more basic pro-
perty. The measure functions h(r) defined below extend the con-
cepts of Hausdorff dimension and capacities which include Lebesgue
measure as a Special case. Similarly our generalized fractional
integrals extend the ordinary fractional integrals which are de-
fined by Riemann and Weyl and which reduce to ordinary integration
and differentiation in the integer cases. Likewise our T-tangential
limits have as examples the usual tangential limits which in turn
are generalized Stolz angles or radial limits. We now discuss each
of these ideas in greater detail.

1. Throughout, let h(r) be a real-valued, non-decreasing,
continuous function defined for r 2 O and satisfying the conditions:
h(O) 8 0, h(r) >.0 if r >»0, he») >.1. For any plane set E and

*

any 0'< p < a let hp (E) = inf 2 h(rv), the infimum taken over
v=l

all countable systems of open circular disks, with radii 0 < rv < p,

* *
covering E. Then h (E) - lim hp (E) is the h-measure of E.
9-0

Similarly let E be an arbitrary bounded set, let {CV}:, be a
family of circles with radii {rv}:, covering E. Let
Mh(E) = inf é h(rv), the infimum taken over all such coverings.
If in addition h(r)/r2 is non-increasing we call h a measure
function, though in fact the resulting M are not generally
additive and hence are not measures in the sense of Carathéodory.
However they do determine the same exceptional sets as a class of
completely additive set functions introduced by Hausdorff; however
this latter class is not well-suitedifor application to the theory
of functions. These general h have been previously discussed
by Rung [11] in the first form, by Carleson [4] in the latter.
Similar set functions are defined by Aronszahn-Smith [1].

Next let “(x) be a distribution concentrated on
E<: [0,2n] in the sense that IE? du 3 IE du (= 1 say). Then

2
V = sup I‘" h( 1'

h XE[0,2n] o Ix-tI

is finite, then E" is said to be of positive h-capacity, otherwise

 

)du(t) is the h-potential of E. If Vh

E is said to be of zero h-capacity. Equivalently, let u be a

distribution spread on the unit circle C = {IzI = l}, concentrated
1
it
Ie -re

a u such that V is bounded uniformly in x as r a 1, then

on E<: C, and let V a I:" h( ix )du(t). If there exists

E is of positive h-capacity, it is of zero h-capacity otherwise.
In the case h(r) = r“, 0 < a < 1, these potentials have

been discussed by duPlessis [5] and SalemPZygmund [12]. Frostman

[6] and Carleson [4, p. 15] give an account of the relation between

these capacities and the earlier definitions.

2. Let f(x) be integrable in an interval (a,b). Let
F1(x) denote the integral of f(t) over (a,x), Fa(x) the integral
of Fa 1(t) over (a,x), a = 2,3,... . By induction it can be

shown that

l
F(o)

 

(*) Fae) = I:(x-t)°”1£(c)dc, x e [a,b]

and F(a) = 01-1)!

Extend F(o) to all values a > 0 by letting it be the Euler Gamma
function. Then define the integral of fractional order a of f

to be the function Fa defined by (*) with the noted change. We
thus have a definition of a fractional integral, this one defined

by Riemann and Liouville, which coincides in the case where o is

a positive integer with the ath integral of f. The fact that Fa
is not necessarily periodic even if f is makes this definition
unsatisfactory in the theory of periodic functions. This gives
reason to consider a second definition of fractional integral as
pr0posed by Weyl, a definition better suited to trigonometrical

series: Let f be periodic and integrable,

a) .
f ~ 2 c eZninx’ c = 0.
n o
-m
a, (321'! inx 1
Define fa(x) = z c ._______.- Io f(t) Ya(x-t)dt where Ya(x)

-m n (2nin)a

has the complex Fourier coefficients via) a (2nin)-a, Yo = 0. A
more detailed discussion of these concepts is given by Zygmund [14,
p. 222]. Also duPlessis [5] notes that for f 6 L9 that the
Riemann-Liouville version and the Weyl version of the fractional

integral differ by only a bounded function.

Derivatives of fractional order may be defined in a like
fashion; we use the Cauchy Integral Formula in the first case, while
in the second the method is quite the same as with fractional integra-
tion. In particular, we note that fa(x) = KG 2 nacneZflinx denotes
the fractional derivative of order a, a >10, where HQ 18 a COB-
stant depending on o alone.

Heywood [8] obtains certain results concerning the function
C(x) - (l-x)-¥, y < l. P.B. Kennedy [9] has abstracted basic pro-
perties of this function to obtain generalizations of Heywood's
results. Letting ¢(x) be a non-negative and non-decreasing func-
tion in (0,1), with ¢(x) E L(0,l), we have the case 0 s y < 1
of the Heywood function. If there exists an integer p 2 1 such
that ¢'(x),¢"(x),...,¢(p—1)(x) are all absolutely continuous for
O S‘x s l and vanish at x B l, and are such that ¢(p)(x) has
constant sign, and I¢(p)(x)I is non-decreasing in (0,1) wherever
¢(p)(x) exists, then we have properties satisfied by the Heywood
function in the case where y < 0. If ¢ satisfies the first

stated conditions we then define

§(n) = I‘ll ¢(x)dx .

1‘.
n

In the second case we define
-. -l 1
Mn) = n phi“ )(1 - 3)"

Here we note (see Salem-Zygmund [12]) that

[l - rei(x-t)]-a = l + arei(x-t) +u..+_o(a+1).;;(g+n-1) rneni(x-t)+}..

in which we note that the coefficients

o(o+1)---(o'HI-1l; 1 o-l . .
n! F(d) n by Stirling s formula.

It is the observation that in the case ¢(x) = (l-x)-a we have

§(n) - na-l (ignoring constants which depend on 0 alone) which
leads us to consider the class of functions ¢ for which the follow-
ing holds: Let ¢(x) = 2 aixk and ak = A[§(k) + 0(j(k))] where
j(k) a 0 and A is a constant depending on ¢ alone. Consequently
em 3 A z §(n) x“. This leads us to make the following definition:
Let f(eix) = z Ck eikx, let ¢(x) possess the prOperties described
above; then

fg (X) = E “Makeikx = c I?" ¢(ei(x-t))f(t)dt

is called the ¢-fractiona1 integral of f. Here C is a constant
depending only on ¢, §(-n) is taken to be -§(n) and ¢(ei(x-t))
is the natural extension of o to complex values. The case
¢(x) = (l-x)a-1, 0'< a.< 1, gives the ordinary fractional integral,
while positive powers of (l-x) give the ordinary fractional
derivative. Fractional integrals and derivatives have been dis-
cussed by duPlessis [5], Kinney [10] and Rung [11].

In some cases, it is more convenient to speak of a Y/Q-
fractional integral defined as fY/§(eix) = z %%E%-cneinx where
I fits the description of the ¢ discussed above. Also, in the
event that the function I does not enter into the discussion,
we may choose to define fY(eix) a z ‘1’(k,)ckeikx without mention

of I, allowing even the possibility that for a given Y(n) no

suitable I exists. This would permit for example the definition

of f'(x) by 2 ncnelnx even though the natural choice

¢(x) 8 (lax)...2 fails to be L(0,l).

3. The idea of examining the behaviour of a function de-
fined in the unit disk D as it approaches the boundary is at least
as old as Abel. His concept of summability and the theorem of
Fatou are probably the best-known examples of radial limits. A
condition which is slightly more general than the radial limit at

a point e19 on the unit circle is the concept of a Stolz angle,

S(eie,q), at e19 and of opening a. In this case the function

is permitted to assume non-radial values as it nears the boundary,

9

but only values interior to S(e1 ,o)-

i.

i
S (e e,a)

Now let 7(r) be a real-valued function, increasing with r; and

with 1(0) 8 0. Define the set R[T,v] as follows:

R[T,v] I {z E D: 1 ' lzI 2 T(IZ ' eivI)l° If zj “ eiv’

zj E R[T,v], j 3 1,2,... implies that lim f(z ) exists, then we
w
say that T-lim f(z) = lim f(z ) or that the T-tangential limit
iv jaw '
zae,
exists at e1v . More briefly we say that the r-limit exists at
e1v if lim f(z) exists whenever z is confined to R[T,v].
iv
zde

In the case where f(r) =kx, R[r,v] is just the usual

Stolz angle. If ¢(r) = rY, y > 1, then R[T,v] is the ordinary

tangential limit, a parabolic approach whose order of tangency is
y - l. Tangential limits of this latter type have been previously

discussed by Kinney [10], Cargo [3] and in a sense by Rung [11].

et'
R [7. W]

In Chapter I we discuss some prOperties concerning h-measure
and the associated exceptional sets. Our first result corresponds
to the following theorem of Salem-Zygmund [12] concerning the points
of divergence of a Fourier series whose coefficients satisfy a
certain order of convergence:
Theorem. If 0 < a s l and if the series 2 nQ(a: +-b:) converges,

then the set of points of divergence of the Fourier series

8

E2 + (ancos nx + bn sin nx) is a set of (l-a)-capacity zero if

HMS

a # l and of logarithmic capacity zero if a = 1. In the course of
this proof, necessary and sufficient conditions are established for

a set o! to have positive (l-a)-capacity. We extend these results

from capacities to the general h-measures. By choosing

¢(x) - h(i%;9 and f(n) B I1 ¢(x)dx we have the result stated
1

1..—
n

above, with the requirement that 2(a: +~b:)-l/§(n) < m, by taking
h(r) - r1-a, ¢(r) 8 (l-r)a-1, §(n) = n.a. This idea of requiring

a level of convergence beyond mere square summability in order to
obtain broader results is also discussed by Kinney [10] and Carleson
[A].

At this point we offer some examples which show the sig-
nifigance of the presence of the factor l/§(n). Consider the
following example:

Let ¢(t) = 2 an sin bnt’ ¢'(t) = z anbn cos bnt° Say, for example,
an ' 2-“, bn = 3“, then 2 ani< a so that ¢(t) converges for
every t, while ¢'(t) converges almost nowhere. Geometrically

we are considering the behavior near the boundary of D .of the
expression g(t) = E anrneint. Then unless the coefficients con-
verge rather rapidly, the function g(t) may behave badly as r ~ 1
on a rather large set of t, giving rise to such undesirable pro-
perties as the image of the function taken along a radius failing
to have a tangent line. Basically then the rate at which a series
converges determines its smoothness near the boundary; our require-
ment that the series converge with the factor 1/§(n) present is
then equivalent to the function not behaving too badly near the
boundary. For the example ¢(t) cited above we observe that the
order of convergence determines a path defined in terms of a suit-
able Lipschitz condition, within which the oscillation of ¢ is
contained. For example, in the case an 8 2-“, bn - 3n, the func-
tion ¢. oscillates within a path p determined by a function
which satisfies a Lip %§§-% condition, while it will pass outside
any path p' interior to p.

 

We note here that the boundary results obtained may be quite good
metrically yet quite bad tapologically, i.e. the points of dis-
continuity may form a too large set. To illustrate we consider
the following example of a lacunary series which is due to Mg‘Weiss
[13]:

Suppose zIanI = no, 2 a: < an (or even 2 a: nu < on),

b
bn > q bn_1 where q 2 2. Let ¢(z) =- 2 anz “. Then ¢(z) con-
verges a.e. on the unit circle C. However, if y denotes an
arbitrary complex number, and 0,8 are any two real numbers, then
for e19 E C, a S 6 s B, ¢(z) = y for non-denumerably many 9
in [a,B], i.e. within any [o,B], every number y is attained by
¢ on a dense subset of [a,B]. Beyer [2] obtains a similar re-
sult for Hausdorff dimension and Rademacher functions.

Following the results which corresponds to the stated
Salem-Zygmund results, we move to a discussion of equi-summability
and equi-convergence of series and integrals of certain functions
satisfying Lipschitz conditions. Also we discuss the class To

defined by Carleson [4]: let Im(z)I s 1, then

10

d 2 ' 2
w 6 TO! c: k I: :2); I: rdr IofiIm'(rele)I d9 < co co

2 a _ n
“'ZIanI n < m for m — E anz .

In each case, we obtain an h analogue of the capacity results.

As an example of a function in his class To’ Carleson
offers the class Bo which is that subclass of the class of
Blaschke products which satisfies the convergence condition~
2(1 - IanI)1-a < a. He proceeds toigeneralize a result of Frost-
man [7] concerning radial limits, giving a necessary and sufficient
condition, in terms ofliorel series, under which the (l-o)-capacity
of the set where the radial limits fail to exist is zero. We
modify this result slightly (Carleson's theorem is already expressed
in terms of general h-measure) to make it adaptable to a general
tangential limit argument which we use later.

We conclude the section with a result on our generalized
fractional integral. duPlessis has shown that if f E Lq, then
the fractional integral fa/q' O < o‘< l, is finite except on a
set of zero B-capacity, where B >'l-a if q > 2 and B = l-a
if 1 s q s 2. Letting ¢(r) - h(i%;) where h is a measure
function, we show this result holds true for a generalized frac-
tional integral having an exceptional set given in terms of h-
measure.

In Chapter II we are mainly concerned with tangential limits
of certain classes of functions defined in the unit circle. We
first discuss the class Bh of all Blaschke products which satisfy
the condition that z h(l - IavI) < a. For the previously mentioned

example Ba of Frostman (he uses h(r) = rd, 2(1 - IanI)a < co),

 

11

there is shown to hold a local property, namely that for a given
1 ..
IanI

Ieie-a

e, the convergence of the series 2 implies the existence

n
of a radial limit there. The theorem is extended to ordinary tan-

gential limits by Cargo [3] who shows that for y 2 1, the convergence
1 - a
InI

Of 2 -—T__-——-
Ie e-anIY

implies that at 9 there is a tangential approach

whose order of tangency is y - 1. 'We make the natural extension of

1 - IanI

 

this local result to the series 2 and T-tangential

19
7(Ie -ahI)

limits. Frostman shows that his local condition holds (and hence
radial limits exist) except possibly on a set of a-capacity zero

0 < at< l; Cargo's result is expressed in terms of ay-capacity
where it is assumed av < 1. Similarly our result is seen to hold
off a set whose ham-measure is zero (where hOT is assumed to be
a measure function). Finally this same result is also shown to be
true in a different way using the previously stated result of
Carleson concerning Borel series.

The class Sa = {f(z) = 2 cnzn: 2 naIcnI2 < m} has been
discussed by Kinney [10] and is noted to be similar in nature to
the Fourier series discussed by Saleerygmund [12] and to the class
To discussed by Carleson [4]. Kinney establishes the existence of
tangential limits for all functions in this class, as well as for
their fractional integrals and fractional derivatives of certain
orders. Since, as noted by Carleson, Ba<2 Ta, this includes the
result of Cargo stated above. We here show that these results can

2
c
be extended to the class SQ - {f(z) - 2 cnzn: Z lan

9 (n)
§(n) is defined in terms of m satisfying the earlier prOperties

 

< a] where

12

(p. 4 ). Then for certain T, T-tangential limits exist except

possibly on a set of h-measure zero for each f 6 S along with its

Q
generalized integral fylg'

Our final results are related to the function class A8
which is discussed by Rung [11]. A function f belongs to the
class A8, 8 > -1, if the following integral is finite:
IDIIf'(z)I2(1 - IzI)sdxdy < w. This condition is shown to be

equivalent, where f(z) - 2 anzn, to the convergence of the series

2
I8 I
2 821 . We define the class A(Y,¢) according to the convergence

 

of the integral. IDIIfY(z)I2 ¢(IzI)dxdy < m, where Y, m have the
properties given earlier. The convergence of the integral is shown
Ito be equivalent to the convergence of the series 2 Y2(n)§(n)IanI2.
These equivalences and some related properties are then used to

prove some results on tangential limits.

CHAPTER I
Some Properties Concerning h-Measure and

the Associated Exceptional Sets

1. 0n the convergence of certain Fourier Series

Salem and Zygmund [12] established the following concern-
ing the points of divergence of the Fourier series of a particular
class:

Theorem: If 0 < a s 1 and if the series 2 na(a: +-b:) con-
verges, then the set of points of divergence of the Fourier series

a
.42

+z‘In os +b 3'
2 1 an c nx n in nx
is of (l-a)-capacity zero if a # l and of logarithmic capacity

zero if a B 1.
Similar classes have been discussed by Kinney [10] and Carleson
[4].
a. Necessary and sufficient conditions for convergence.
Letting h(x) denote a measure function having the pro-
perty that h(I%;) is non-negative, increasing and integrable in

(0,1), we define ¢(x) I h(i%;) and

Mn) - I1 h(xmx.
1-}.
n
We assume also that h(ab) '5 h(a)h(b) and ¢(x) '--'- 2 §(n)x“. Then

we can show that the following extension holds:

13

l4

TheoremII

If 2 3%;; a: + bi) converges, then the Fourier series

ao/2 +~2: an cos nx +bn sin nx converges except possibly on a
set whose h-measure is zero.

Choosing h(r) = r1-a we have the Saleerygmund result
noted above in the non-lograithmic case. We first must determine
an equivalent condition in terms of Fourier series to a set having
positive h-measure.
lEEEfl.l- In order that a set E have positive h-measure it is
necessary and sufficient that there exist a positive distribution
u concentrated on E such that if the Fourier-Stieltjes series

of du(x) is
-1--+2a cosnx-I-b sin
d“ " 2n n n nx

then the series 2(a.n cos nx +bn sin nx)§(n) is the Fourier series
of a bounded function.

Proof.

Suppose E has positive h-measure. Then there exists a

positive distribution u with du ~ %;'+-2 an cos nx +bn sin nx

concentrated on E such that

 

 

2n 1 ) 2n ( 1 )
V = h (t) = h dp.(t)
IO (‘eit-reix‘ d“ J‘O ‘l-relOK-t)‘

is bounded uniformly in x as r d 1. By our assumption

®(re19) 3 z 9(n)r“ei“°. Then let

 

. 2n 1 a 211 i(x-t)
u Io “(Hem-or”) To ¢<re )th)

where we are using the natural extensions of our functions to

15

complex values.

in(x-t)

U = I 2 i(n)rne du(t) =

I(Z §(n)rn[cos n(x-t) +’i sin n(x-t)])(%F-+ 2 ancos nt +-bnsin nt)dt
which has real part
2 @(n)rn(an cos nx +bn sin nx)

and this is uniformly bounded in x as r a 1 due to the finite-
ness of V.

Conversely the sufficiency is established by noting that
the above argument can be used to show the real part of U is
bounded as r- l' for some positive distribution u concentrated

on E. For every r < l and every choice of argument 8,

l - reis iBI-ei6

- I1 - re with ~n/2 < e < n/Z.

Then taking real parts
Re[¢(1 - telB)] = Re[¢(|l - reiBI-eie]

where the right hand expression is bounded by the integrability of

¢. It follows that the boundedness of the real part of U implies
the boundedness of V. This completes the proof of Lemma 1.

At this point we establish the following relation between the func-
tions ¢ and Q: i‘§(fi) E ¢(l-x) as x'~ O. For extending Q

from the discrete case we find

x'l‘Qé) " % Ii.“ ¢(x)dx = ¢(§) where l - x < g < l,

l6

and as x -o 0, g -0 l-x and ¢(§) -° ’Ml-x).
Before proving the main theorem we need the following lemma:
_L_ggn_a_2_. Let H(x) - z: 9(k)cos let, snot) a 2‘; 9(k)cos kx. Then
IHn(x)| < A-H(x) +-B where A, B are positive constants depending
on Q alone.
Proof.

Without loss of generality we assume that 0 < x S n.
Let x be arbitrary but fixed.

If n s £3 choose Cx so that Cx§(1/x) 2 @(l).

n n

Then 2 @(k)cos RX 5 2 Mk) s n e(1)s -1- - c -e(1/x) = Ogi- 0(1/x))
kal k.1 X X
1 [1/x] n
If n >';3 let S1 +S2 = 2 +' 2 §(k)cos kx and let
1 [l/x]+l

tn = [l/x] + 1
S1 is covered by the previous case. Apply Abel‘s formula to

82 with uk - 9(k), vk - cos kx and note that 2 cos k: - 0(l/x)

(See Salem-Zygunnd [12, p. 27] or Zygnund [14, p. 2])

n-l
$2 = k: Tack) - sac-unik + §(n)Tn
.‘m

where Tk ' 2: cos kx = 0(l/x)
s2 = [Nun - e(n)10(llx) + §(n).0(l/x) = 0(1/x §(1/x))
since m,n > l/x and hence 9(m) s §(l/x) and §(n) s §(l/x).

We observe that as x a O, and for prOper choice of C , that

1
C1 + H(x) 3 A2 $(l-x).

- Hf) .. ¢(l-x) - h(l/x)

X

 

l 1
ix) = ¢(e x) - 2 §(n)einx
l-e

H(x) ‘ Re 2 §(n)einx

when

h(i/x) g h(i)h(£) ~ ¢(l-x)

17

Then since %'§(1/x) g ¢(l-x)
IHn(X)I < C2 i’§(1/x) 5 C2 ¢(1-x) 5 CZ[C1 +-H(x)]

so that IHn(x)I < A-H(x) +-B.

This proves the lemma and we now proceed to prove the main theorem.

8 G
Let S denote a Fourier series 32-+-2 (a cos nx +'b sin nx)
2 2 1 n n
a a +-b
such that 2 -EETESE~< 9. Then we wish to show that the set E of
1

points of divergence of S has h-measure zero. For suppose E
has positive h-measure. Then there exists a positive distribution

' G
u, concentrated on E, such that if du ~ %;'+'2 (ancos nx +-anin nx),
1

then the series 2(ancos nx + anin nx)§(n) is the Fourier series
of a bounded function. Choose m(n), a positive monotonic function

increasing infinitely with n and such that z %%%% (a: +-b:)
converges. Ital: An '3 an(w(n))k, Bn . bn(w(n))%. If x 6 E then
n

the series Sn(x) - 2(Akcos kx +~B sin kx) is unbounded, otherwise
1

k

S would converge for an element of E. Let n(x) s n denote a

positive integer valued measurable function of x. By a well-known

argument (see Zygmund [14, p. 253]) the integral I:" S (x)du

n(x)

can be made to increase in absolute value with n for a suitable

 

2n
. d '

choice of n(x) for each n We now show that Io Sn(x)(x) x is
bounded, this contradiction proving the theorem.

1 2 2

Since 2 §(n) (A.n +-Bn) < a, then
2Q(A cos nx +-B sin nx) - 1 is the Fourier series of a function
1 n n 3
2 9 (n)

F E L .

_ 1 2n 3
Akcos kx +-Bksin kx = E'Io Q (k)F(t)cos kt cos kx dt +

1. 2n % . .
+.n Io Q (k)F(t)Sin kt Sin kx dt

18

-1-:"I‘ e (k)F(t)cos k(t-x)dt

Let snot) = 2: i350.) cos lot

n(x) . 1 2n
Sn(x)(x) . g (Akcos let +Bksin kx) - ;I‘o F(t)G W )(t-x)dt
1 = we sn(x)(x)du<x) .. ‘I2"I:" r(t)c ”I )(t x)dp,(x)dt

By Schwarz's inequality:

2 2 2
1112< jfl"1-:<t>dtj":“f cm)(t-xmnm(c-y>de<x)de<y>

Then Ii" (t-xm <t-y>dc —-- 11 a (x-y)

Gn(x) n(y) n(x.y)

where n(x,y) B min[n(x),n(y)], Hn(x) = 2: §(k)cos kx

2 2
Let A = Io" F (t)dt. This gives

2 2 2
111 <A OW

Hn(x,y) (x-y) I du (X)du (y)

2 2
< Ayo"jo"{|nn(x)<x-y>\ + IunIy)<x-y>\}du(x)de(y>
2 2 co
= ZAIoflIonIHn(y)(x-y)Idu(x)dp(y). Then since H(x) = 2 §(k)cos kx,
2n 2n
Io I Hn(y)(x -y)Idu(X) 5 AIIo H(x-y)du(x) +-B

2 Q
The Fourier series of fi'Ion H(x-y)du(x) is 2 §(n)[ancos ny + Basin ny]
l
which is by assumption the Fourier series of a bounded function.

This demonstrates the boundedness of I, proving the theorem.

b. Analogue of the class To

A further equivalence concerning the convergence of the
series 2IanI2na is given by Carleson ([4])who shows that
T (a) 2 Is Izna where Iw(z)I s l and w(z) B 2 a 2n IzI < 1
0’ 1 n n s

and we say that w 6 Ta if the integral

19
Ta(w)= k(w)‘f:( ——)—J‘o r drj’ifl‘ (reieflzde is finite,

where k(u)) denotes a certain constant 31; s k(w) s 1. (By f g
is meant that there exist constants m and M such that

mf s g s Mf, so that f,g are bounded together). We show here

2 2n
that under the assumption that the limit of marl—$911.: {71: h(l-r)dr

exists as n .. an, we have the following:

_.-._.—_ {u}: k(w)I1MT:rd

TheoremZ. Let wET , 0 1r

2
hY «14‘2" m,n—e °)| d6<m}
where 03(2) is defined as above. Then Th Y0») z|an‘2 ol/Q (n).

Proof.
Thwfln =Kf1—Q—rldr rZJ‘ 1»er”: "M, e(ie)\de

lhl- r 22 2 1
= 211K 3‘0 —%_—?— dr f0 Xi‘an‘ Y (n)r n

2
= 11K 1‘]. 1151-2?! 2‘8 ‘2 ‘1’ Sn! r211 d
O ‘1' n n

g 2 n Y n 1 2n b 1-1‘
TTK Z‘an‘. 3—6—11) is—nLiLlJ'or Lil-r dr

which with our assumption establishes the result. The case
Y(n) = n, @(n) = n-a is the class discussed by Carleson.
A somewhat different argument is used later (see II, § 3 )

on a class which is similar to the class
Toy,” a {0): J‘: ¢(t)dt f: r dr L2)" ‘Waeieflzde < on}.

Here we choose ¢(x) - h(i—l'? to be increasing, non-negative and
L(0,l). Using a theorem due to Kennedy [9] we can show that

w 6 T(‘1’,¢) is equivalent to the convergence of the series

2
z‘an‘z Lam)- §(n).

The later result is true for a more general

(b, and the details being the same, we defer the proof until that point.

20

Carleson [4] shows that the convergence of the series
2 2 a .
2(an +-bn)n is equivalent to a certain continuity property of the

function represented by thetmigonometricseries. We obtain a

1
§(n)

I: Esgl sinzr dr exists finitely and h(ar) z h(a)h(r). Carleson
r

 

2
similar result for 2(a: + bn) under the assumptions that

treats the case h(t) = tha(log %3-8, 0 < a < l, B > 0, of the
following theorem.

Theorem 3. Let g(x) be a real or complex-valued function defined
by g(x) .. '23 +33: (an cos nx + bn sin nx). Then

f: If," ‘g(x+t) " g(x-t)[2 h t dx dt
(‘2

Sh(g)

2 2
2(‘an‘ +-\bn\ )An

where A.n is of the order 1/§(n).
Proof.

According to Parseval's relation

2
Sh(g) = 4n 2(lan12 + lbn|2)f: glf§E£-h(t)dt and

n/2 ginzx h(x)‘

. 2
fk Sin nt 0 2 dx. The theorem follows by

0 T h(t)dt 3 Effigy

‘1 5.1....
h(n) §(n)

 

taking limits and noting

c. Some conditions for equiconvergence and equisummability
of certain integrals and related series.
A theorem similar to this last result and concerning the
equi-summability of the series conjugate to the Fourier series and
the corresponding integral may be obtained in the case where f(x)

has period 2n and is of a certain Lipschitz class (see Salem-

21

Zygmund [12] for the case h(r) = rl-a).
a 00
Theorem 4. Let 52-+-z (an cos nx +bn sin nx) be the Fourier

1
series of a continuous function f(x) of period 2n and belong-

t

 

ing to lip n, where “(t) 8 h???“ Then the difference
f (xi-t) - fix-4t: °° . r“
Kn‘ri-r t2 h(t)dt - E (an Sln nx - bn cos nx) §(n)

tends uniformly to O in x, as r‘# 1-. Similarly the difference

is bounded uniformly in x if f € Lip n .

We shall assume the following properties concerning h:

 

 

l 1
that I th(t) dt 3 __h(t)

and that fay-«35551.
t

, that f: sin ; h(t) dt exists finitely
' t

Lemma 1. Suppose g(x) 6 Lip fi has period 2n,

0 on
_Q_ .
g(x) ~ 2 +-z an cos nx +-Bn 81n nx.

l
, a m
0 . n .
Let g(r,x) = 3"4'2 (an cos nx +-Bn 81n nx)r be the corresPonding

harmonic function.
Then g(r,x) - g(x) = 0(n(l-r)) as r ~ 1', uniformly in x.

Proof.

 

Let Pr(t) a (l-r) 2 denote the Poisson kernel.
2(1-2r cos t +'r )

Then Pr(t) < -]-'- and also Pr(t) < 1-r

1 , so that
-r 4r sin t/2

n|g<r.x> - g(x)! = miles“) + g(x-c) - 23(x)]1>r<t)dt\

1-
s TEL) r 0(T](t))dt + (l-r) fir «gig-Ht

o<n<1-r>) + o<<1-r>1‘§'-"-|‘{_r

0(“(1-r))

 

22

Lemma 2. Let g(x) and g(r,x) be as in Lemma 1. Then

aggr,x2 . 0(fl51-r2)
ax l-r '

Proof.

We note first that I 23%27'dt " implies

_L_
h(t)

 

 

l N l l 1 dt
dt = —— for dt = —— + K —2——
.r tzhm t h(t) I t2h(t) t h(t) J‘ t h(t)

Then 55§§&51»= -%.I:[g(x+t) - g(x-t)]P£(t)dt where

-(1;r)2r sin t

 

 

 

 

t
P'(t) “ so that P'(t)‘ < and also
r (l-2r cos t +r2)2 ‘ r (1'?)3
, 2(l-r)t
P (t) < . Hence
‘ r ‘ (4r sinzt/Z)2
‘aggrfiz‘ S 1

l-r 11 t
ax (1-: 3 lo OWN)?“ + (1-r> $1-: 0(ch )dt

= 061%; - n(l-r» + (H) ° “319W-.-
t

which is, as in Lemma 1, on the order of I%;'fi(1-r).

Now to prove the theorem,

&
let g(t) - f(x+t) - f(x-t) ~ - 2 2 sin nt[ansin nx - bncos nx]
m l
and g(r,t) = - z 2 sin ntLansin nx - bncos nxjrn.
1

For given x and r (r < l) the series

as
agritz h(t) = -2 2 sm at
t

1 t2 h(t)[an sin nx - bn cos nx]rn

is uniformly convergent in t for t >13 > 0. Hence we can inte-

grate termwise over (e,T). We note that

u: n. “-95 «t s n at: 4-1 at = owe-15% -- 0.. we»

23

and

. h t h T
‘I: sin nt - -f§l’dt‘ S‘Té‘l'.

Thus we may conclude that

sin nt h(tldt

2

hgtz m n
I: g(r,t) t2 dt -2 g (ansin nx - bn cos nx)r f: t

 

 

sin nt n sin x K-n ~ 1
where I: —-:§~*'h(t)dt I -——'f: ---'h(x)dx = = K

h(n) x2 h(n) §(n)
so that K(h) I: Eiffgl'h(t)dt = g (ansin nx - bucos nx)r“. 3%;;

Therefore to complete the proof of the theorem it suffices to show

that

n = f: S-L:§flh(t)dc - J11. 3194:3215)- dt is 0(1)

if g E lip fl, and 0(1) if g 6 Lip n, as r d 1-, uniformly in

x. The proofs being identical we let 3 6 Lip n.

8 a ggr,t2-ggt2 l-r g(r,t)hgtz
D D1 + D2 fir t2 h(t)dt + f0 t2 dt.

The first integral is bounded since

‘Dll s 0(n(l-r)-Ef%i£l) = 0(1) using our above assumptions

on b. To establish the boundedness of D we note that since

2
g(r,0) - O, we have g(r,t) a t a8§5&£z‘t=9’ 0 < e < t which by
Lemma 2 is on the order of trߤ%igl so that

1- -
\D2\ --- o 1%; - n(l-r) fo r E?)- dt)= o '1'}; n(l-r) 2&2). ~(1-r)) = 0(1)

This concludes the proof of the theorem. A nearly identical argument

establishes the following:

24

Theorem. Let f be as in the previous theorem; then the difference

 

fijx+t)+f(x-t)-2f(x)_ ” n 1
K2(n)f:_r t2 n(t)dt - fi (ancos nx +-bnsin nx)r .Q(n)

is o(l) if f 6 lip n and is 0(1) if f 6 Lip n.

2. Some results on Blaschke products: the class Bh

As is well known, any bounded holomorphic function in the
unit disk D can be represented as a product of a non-vanishing
function and a function containing all its zeros, this second
function being a Blaschke product B(z,av) where

sea-z

a
m V V
B a B .. ' 'T <1
(2 av) 2 g l'zav lav , ‘2‘

 

and {av}:, 0 < ‘av| < 1, is a sequence of complex numbers such
that g (l - ‘av|) < o. By this series condition we know that
‘av"~ l as v dim, i.e. that the zeros av cluster near the
boundary as v becomes large. By considering the subclass of

the Blaschke products consisting of those whose associated sequence
has the property that 2(1 - |an|)1-a < m, 0 < a < l. Frostman
[7] in effect Speeds up the rate of convergence, forcing the zeros
to cluster sooner at the boundary. In doing so he obtains some
results which were not possible in the general case. This sub-
class, Ba’ is also discussed by Carleson [4] and he shows it to
be a subset of the previously mentioned Ta' We are interested
here in a local condition on the boundary in which Frostman shows

that 8(2) 6 Ba tends to a radial limit at e of modulus 1

so 1-‘8 |
provided 2 +< a. He then proceeds to show that this is
V'l |e -av|

 

25

true except possibly on a set whose (l-a)-capacity is zero. Our
discussion will be concerned with the class Bh which consists of
all Blaschke products having the property that g h(l - |av|) < a,
where h is a measure function. Now let T(x) be a function
having the properties that T is increasing, w(o) = 0, f(x) s x
if x s 1. The following generalization of Frostman's result,

in the case 1(x) - x, is due to Carleson:

Theorem 5. Let {2v}:=l be a sequence of complex numbers in the

unit circle and {Av} a sequence of real numbers, 0 < Av‘< 1.

If h o r is a measure function and

m G A
23 7-1(Av)‘f3_1 L121-2(-t;2<1r < co then 2 Ja-j—D- < on except
v=1 T (AV) r V'1 T z 2”

possibly on a set E with hoT-measure 0.

Proof.
We note first that z h(Av) < w and consequently, since

7(AV) 5 Av’ z hoT(Av) < a as well. This follows since

a -1 3 hoigrz a -1 1
a >.2 T (A ) dr 2 2 T (A )h(A )(--9|
N V [(10%) r2 N V " r
{10. > .. MA >

G
= 2 h(Av)[1 - -—§-¥!-] z z'-§-¥- for N sufficiently large.
N N

3
-l
T (Av)

- 3 ” -
Also since 2 T 1(Av)f _1 @ dr 2 z 1- 1(A\,)J‘3 h—oT-zfil dr
N T N l r

(Av)
for N sufficiently large, we have that z T-1(Av) converges.

For any integer p, denote by 0P the open set where
a A

g T(|z-sz) > 1

Let u E rhT where Ph is defined as the class of all non-negative

completely additive functions of a set such that

26

u(r,a) s h(r) for all a,

where g(r,a) is the value taken by u for the circle

C 8 {2: ‘z - a‘ s r}. For a discussion of these measures and

their existence see Carleson [4, p. 11-12] and references cited

there. Also we may choose u so that it vanishes outside 0 .
About each point zv, v 2 p, we put a circle of radius

(T-1)2(Av)’ so that within these circles we have

1(‘2 - zv‘) < T-1(Av)° Let Gp denote the exterior of these

circles, i.e. where f(lz - zv‘) 2 T‘1(Av). 0n the circles, u

distributes a mass which does not exceed lg h¢(AV) B 3;. Then

for p sufficiently large,

A
Map) = Icdez) s ij2 ,( ”v ) du<z>

A 3 dub-32v)
$2 —"'"""'
V -1 r
T (Av)
SZAhm‘ZB +A I3 mdr.
v r -1(A ) v -l(A ) 2
T V ‘1' v r
The first term above is less than K'Z A , the lower limit being
h(A )
the negative of the positive expression 2 A.“, _lv Both of
1' (A)
v

these converge by virtue of the fact Av S T-1(AV) and z h(Av) < m.
Likewise, since AV s T-1(Av) and by our hypothesis, the second
integral also converges.

Hence n(Gp) s 33, and so by a result of Carleson [4, pp.
10—12] the hOT measure of 0p is less than 32-36-¢.:p where
e = s' +'eg- Finally we note that except for the points 2

P P
v - 1,...,p-l, Op contains E so that

V,

27

11 0' =0.
Mh(E) S l”35,114,513

The converse to this theorem is true under the hypothesis that

I 25%2-3 0(Eé519 as we now show. 'We shall make use of these
t

results in Chapter 11.

Theorem 6. Given a bounded set E, a necessary and sufficient

m A
condition for the existence of a Borel series 2 ;ii;:;—T3-
v=l v
divergent on E, with 2 T-1(Av) I3_1 he-T---2(-E2--dr <,c, is that
T (Av) r
Mh0T(E) g 0'
Proof.

That the condition is necessary is precisely the result
of the previous theorem. It remains to show sufficiency: Let
MhT(E) - 0. Cover E by a family of circles with radii

-l -l -n
T r such that hT T r = h r s 2 . Denote b
(up) 2 ( (my) E<:“) y

z the centers of these circles: ‘z - z | < T-1(r ). The
an up “H

 

 

a: m r 9 A
Borel'series z 2 Ann 8 z «y diverges on E
n=1 p']. T<1z-znp‘) v.1 qu'zml)
while 2 T-1(A.) j3_1’ EIéEl dr converges if z h(AV) does
v 1' (Av)
as we shall show below.
a: C CD -n
3. 2 h(Av) ' 2 Z h(r ) S 2 2 = l.
v=1 n=1 ”-1 n“

- 3 .
It remains to show 2 T 1(A ) j 1 hT r dr converges if
V -
T (Av) r

z h(A ) converges.
v

-1 3 hrgrz -4 thr2 3 -l
21' (A) - dr'°(£ ‘. )‘T (A)
V IT 1(A\)) r2 r 'r 1(A\)) V

= 00: (Rep) + 0(2 mp).

28

3. Finiteness of the generalized fractional integral.

N. duPlessis [5] has shown the following concerning the
finiteness of the fractional integral fa/q of order a/q:
Theorem. If f E Lfl[0,2n] then:

(a) For 0 < ai< 1, 2 < q < o, fa/q is finite everywhere except
in a set which is of zero B-capacity for every 5 > l-a.
(b) For OT< ae< l, l s q s 2, fa/q is finite everywhere except

possibly in a set of zero (l-a)-capacity.

We wish here to extend this result to a generalized frac-
tional integral with an exceptional set of the general h-measure;
as with duPlessis, our two functions,¢ and h, will not be
independent of each other.

Let h(x) be a measure function having the property that
¢(x) 3 h(I%;) is a non-negative, non-decreasing function, integrable
on (0,1). Further assume that h(ab) s h(a)h(b). Let

@(n) -.f: 1 ¢(x)dx and note that §(n) g'fi'h(n) (see §l). Extend

n
6 from a function of a discrete variable to a continuous variable

by letting @(x) a % h(x). Finally we assume that

 

I2" 1 Qs/q

1 2n 1 . l k 1 £ 1
o W (Es-7PM" ”d I. M {1.11: W” (FT-Tr)“
both exist finitely.

a inx
Let f(x) =12 c e , c = 0.
Define the generalized fractional integral as the function

fs (qu) ' 2‘. e1/q(n)cneim‘

-ca
Next choose a < 1 such that a = r/q where for q > 2, c > 0,

94119-1)

r = 1+s(q-1) , so that 2 < r < q and r' - e = q'. (Here p'

29

denotes the conjugate of p in the sense that 1/p + l/p' = 1)
Finally let ha(x) B x §a(x). We now prove the following:
Theorem 7. Let f 6 LS[O,2n].

f§(q,x) is finite except in a set of h-measure zero if
1 s q s 2, in a set of ha-measure zero if q > 2.

Letting h(x) = xlqa, O < a < 1, so that ¢(x) = (1_x)a-1

and §(n) - n-a, we have (l-a)-capacity and the ordinary fractional
integral. The proof depends on the following lemma which is of

independent interest. First we define

Vh I: sup If," h(fiq)dp(t).

 

x€[0,211]
LEMMA:
(1) For every 6 >»0, 1 < q < 2,
1
2n 1 1/Q' 1 (q-e)'
MQ'6[IO W 9 (m)dp(t)] S A(§,e)Vh
where A is a constant depending on 9,3 only.
(2) For 2 S q s.a:
2n 1 Wl/q
n m cw]
where A is a constant depending on i only.
Proof.
2 2 -1
<1) I." his “1qu“” ‘ I." 6 lq<1ifTr>dvx<t>
where Vx(t) = I: h(fiqfldpxs)
2 1&1 2 - 1
(Jo11 W ’q q‘TTx c N m)“ = ( o" 9 l’qq—th )dv "" exmdv “q “We

3O

- 12$)
5( :11 Q q dvx (12)) - (Izndvx(t)) ((1:6) by Hb'lder's inequality,
_<L:L,
2 -
(*) S I011 1 Q3/q(.‘_x_.]-'_t_l_)du(t) . V“! E)

The latter inequality holds since
..ét3§

1 -1+ 1
Q )BQ e/q. l . .e/q

(Ix-tl x-t ‘x-tl

 

 

The inequality (*) and our assumption above gives (1).
Proof (2): The Special case q = 2 is proved first and used to

prove the remaining cases.
2 2n 1 k 1
M2[ o FEET ‘ (mwwfl
=1? 1:" [13," rz—s *si—r ' r—r *x—r]

Using the substitution x-t = (s-t) - u we find, on the

innermost integral, that

2151181 3
Ion Ix-t‘ é (Ix-t‘)‘x-s| Q (|x-s|)dx

 

‘ f3" #7 ° FIT ‘k‘fifiF-IT’ ' T5? '11—11 ' ‘g‘ll-Jhs-tTHS't‘d“
‘T‘T ' “TE—tr”? M will ‘21??? ' “(TEE—1P“

 

I2" 211 211

l k l .
Io Io m <1 (FE-f) ‘x—IJQ J"‘(‘f—“dedpl(t)dMS)
s muff? ITIJ @(Tfigr>de(c)du<s)

5 8(2) . Vh(u(2n) - u(o)) giving (2) for q = 2.

Next for q > 2:

51-3
2n 2n 1 q . l /q q
lx-tl ‘1
9.1;. 2/

2n 2n 1 l q 2n 1 % q q

Soro{[o WNWNMQ [o memo] } dx
using Holder's inequality and the fact that 9&2-+-%-= 1

q-2 2 2n l 5 1
‘Vs Mzgo mi (Ezr’dt‘tfl
s‘Vq-Z . constant ° V = const.‘Vq-1

h h h

This completes the proof of the lemma.

Proof of the theorem:
Q

11o: °° 1 'k
let f E Lq, f N Z Cke , f ((193) " 2 Q lq(k)¢eke1 x
«a Q -ao

n 1/q
Let 8 ago (k)ce
n k
-n
To show: Sn is bounded outside a set of h-measure 0 if
1 s q s 2, outside a set of ha-measure 0 if q > 2.
Suppose Sn is unbounded in a set E of positive H-measure where
H is either h or ha accordingly. Then there exists a dis-
Zn 1
d
tribution n(x) concentrated on E such that Io n(T;:ET) n(t)
is bounded for all x. As noted earlier (§l), it can be shown
[14, p. 253] that there exists a function n(x) s n taking integer

values such that f2" Sn (x)du(x) exists and is unbounded as

(X)

n 4 a. ‘We show this to be impossible.

n(x)
jg" Sn(x)(X)du(X) = If," z W“

<k>cke“°‘do<x>
-n(x)

32

n(x)
= $212“ f(t) 2: “q
o o -n(x)

ik(x-t)

(k)e dtdp, (x)

We note here that, as has been shown, (§1)

23 “Ml/q ilk(x~t)

w W “Ix—T ' , q‘Tx—T”

Theref°re ‘fin So(x>(x)d“(x" ‘ C ' I:"|f<t>‘°(@:fi T§%?T'§1/q

(Tia—mp. (x9 dt
. . 2n 1 sI/q
s C Mq(f) Mq'[Io T—ré (m)dp(x):]

For 1 s q s 2: Use (2) of the lemma, interchanging the roles

of q, q'. The resulting contradiction proves the theorem for

this case.

For q >.2: define a,r, as given earlier so that

.nlu-I
I
«Inc

2<r<q,and q'Ir'-s.

211 l Ql/q 2n 1 a l l/r
d t be 8 M. ‘--- d
MqILfo T‘x-C‘Q (l KIT) HI< )] come 19-69;) “(5‘12 (lx-t‘)) 11.03]
which is finite by (1) of the lemma with r' for q, r for q'.
This completes the proof of the theorems 'We note that the limiting
case of ha-measure, i.e. as e ~»O in the definition of r, gives
precisely h-measure. This corresponds to the capacity case where

(l-a+e)-capacity, i.e. every 3 > l-a, has (l-a)-capacity as the

limit ing case .

CHAPTER II

On Generalized Tangential limits for Certain Function Classes

1. T-tangential limits of functions in Bh

a lanl a -z
LBC B(z,{an}) ' “H1 a igazg', 0‘< ‘an‘ < 1, n = 1,2,...
= n

 

Q
and z (l-‘anl) < w, so that B(z,{an}) is a Blaschke product.
n-l

It is known that B(z,{an}) has radial limits at almost every
point of the unit circle C and further that this implies an
angular limit at each such point (See Cargo [3] for a brief dis-
cussion and references).

Let R[T,e] = {2: l - ‘2‘ 2 T(‘arg z - e\)} where
|arg z - 9| is the length of the shorter segment of C which
joins ei9 and z/‘z‘, and where T satisfies the following pro-
perties:

T(0) = O,

T is increasing

T'(x) exists finitely at 0

T(ax) s anCx) where a is constant and 9a is a

constant depending on a and T alone

“(x-met»; o as

 

 

T'(X) x d 0
l l
'rC"T"——— 2
\ele-ak| T(‘eie-ak‘)

33

34

Next let h denote a measure function such that
h(ab) s h(a)h(b). Let Bh denote the class of all Blaschke pro-
ducts having the property that 2 h(l - ‘ah|) < a.

In the case T(r) - ry

, y 2 l, R[T,e] is the path which
meets the circle with order of tangency y-l. It has been shown
(see Cargo [3] for references cited there) that corresponding to
each such path, there exists a Blaschke product having no tangential
limit whatever on C. Cargo however has shown that for l s y-< Ila,
tangential limits of order y-l exists for Blaschke products in
the class Bh’ with h(r) B fa, except for sets on the unit circle
whose ay-capacity is zero. This includes, in the case y a l,
the results of Frostman [7] noted in Chapter I concerning radial
limits and a-capacity.

Following the lead of Cargo of composing the measure func-
tion.with the tangential function to form a measure function, we
show here that this resulttcan be extended to show that every

Blaschke product belonging to Bh has a T-tangential limit at

every point of C except possibly for a set of hoT-measure zero.

a. Local condition

Theorem 1. Let {an} be a Blaschke sequence such that

a» l-lanl

 

 

(1) 2 .
n31 T(|e19-ah\)

Then B(z,{an}) has a T-tangential limit of modulus l at e19.

Proof.

Without loss of generality we may assume 9 = 0. For if

1 - |a l
n
< co = lim B 2 a exists then the same roof
2 T(‘1-an|) 2‘1 ( ,{ {1}) p

35

 

 

shows that
19
l-‘ane ‘ -ie . .
2 < o = lim.B(z,{a e }) exists i.e.
T(‘1-a e-ie‘) z-ol n
n
1"‘an‘ is
' 4—-< m =1 lim.B(ze {a }) exists
T(‘eie-a \) z~l ’ n
n
but 2 a l== zeie-a e19 and by change of notation (z for zeie)

we have lim. B(z,{an}) exists.

24c
By virtue of (l) and our definition of T, we note since ‘an‘ a 1
that at most finitely many of the an lie on the radius to 1;

hence we may assume without loss of generality that no an lie on

this radius.
on 1- ‘an‘
We show first that 2 <1m. For suppose not,
“’1 TCTarg an‘)
1 - |an|
then we will show that z - ww
T(‘l an|)

Let K 3 D(l,lA/2), let A be the right angle to l bisected by

 

the radius to 1.
Since 2(1 - \anl) < co and
T(‘arg ahl) has a positive

minimm value outside K we have

\ 1 ' ‘3 l
2 n = w .
1 a 6K T(|arg an“
n
I If infinitely many an 6 K lie in

A then 1 - ‘an‘ > C-T(‘1 ‘ an‘)

 

where C >,0 and hence

36

If only finitely many an 6 K lie in A, consider only those

an E K-A. Let pn be the perpendicular distance from 1 to the

radius thru an.

 

P
Then pn < ‘arg an‘ , 11—3—1- > cos 11/4
:1

\l-anl < pn2t < ‘arg an|-2%
T(‘1-8n|) s C T(|arg anl), C >10

'T'_|—1 2 l 1 that th d' i
T( l'an ) C T([arg anl) so e correspon ing ser es

mst diverge .

1 - HI I
This establishes g T([argna 1) < co.
n

 

Next, using a well-known device, choose a sequence {mu}, 0 < wn s l,
wn - 0 such that
l - a
. 1 n1

(2) n21 wn'r(‘ar8 an[) < O

 

37

Let Sn = {2: \z - anl < wnT({arg an{)}.

We show that for any fixed positive integer k that

B(z,{an}) converges uniformly on D - U S

 

1=kj°
1a 1 . ~z
Let b(z,{an}) . a: lfznz and C(z,{an}) = b(z,{an}) - 1
(l+‘z‘)(l-‘an‘)

 

Then ‘C(z,{an})\ s ‘1 'Efizl

l-‘anl {ah-2‘

1-\a | 2(1-‘an|) .
an-z wnT({arg an]) 1f 2 E D ' Usj

 

< 2

This inequality, the convergence (2) and the fact that

{C(z,{an})‘ < 2 implies that n[l + C(z,{an})] converges uni-

formly on D - U S .
jak .1
It remains to show that R[T,O] meets at most finitely many of

the disks Sn; it suffices to show that for j sufficiently large,

20 E Sj n D implies 1 - {20‘ < T(‘arg 20‘).

Since 20 6 Sj’ ‘zo - aj“< w T(\arg 20‘) so that

J

(3) 1 ~ {20‘ < l - ‘aj\ +-wj T(‘arg zo|)

Also for j sufficiently large

uhfilars ajl)
{arg zo - arg aj‘ < arcsin( - ) < nij(‘ar8 ajl)

lajl

 

w T(|arg 81‘)

 

as we see from the following diagram, where e = arcsin i la T
J

38

‘lanlzL-dm7q

\‘L\ l/ The latter inequality holds

   

\I because nij(|arg aj‘) is

half the circumference of SJ.

Hence we can say that
(4) T(‘arg zo|) 2 T(|arg aj‘ - ‘arg zo - arg ajl)
> T(|arg aj| - nij(\ar8 811))-
If we can show that
T(‘arg a1} - nij(|arg aj\)) 2 1 - |aj| +-ij(|arg ajl)

for j sufficiently large, then it follows from (3) and (4) that
T({arg zol) > 1 - {20‘ as was to be shown.
It suffices to show that

T(‘arg 81‘ - nij(‘arg aj|)) l - {aj{ . .
T(‘arg 831) 2 T(‘arg an)+'wj for j suff1c1ently

 

 

large.

That the right hand side goes to 0 is immediate since wj d 0
1 ' lai‘
a d < m.
n 2 T(‘arg op
That the left hand side does not go to 0 is a consequence of the

 

facts that ‘arg aj‘ is bounded, w «~ 0, {arg aj‘ a O as j a m and

J

39

 

 

lim ”x ' ”(’91 - lim “(X " ”(’01 (1 - KT'(x))

x-O fix) 3.0 t' (x)
where we cite our assumptions on T and the fact that K = nmj a 0.
Q
Thus T-tangential limits exist on R[T,O] c: U Sn’ for j suf-
n=j

ficiently large.

Since RET,O] meets at most finitely many disks Sn’
Q

B(z,{an}) 3 1'1 b(z,an) converges uniformly on RET,O]. For any

n=l N
fixed positive integer N, n b(z,an) is a rational function with
n=l
only finitely many poles, all of which lie outside of D U C.
N N
Therefore n b(z,ah) d H b(l,an) as 2 a 1, 2 within R[T,0].
n81 n-l

Since the convergence is uniform

N N
B(z.{an}) = 11-1: agree“) - :1: oE1ba'a“) = B(l.{an})

as 2 a l on R[T,O]. Since |b(l,an)| = l for all n, we con-

clude that the limit is of modulus l.
b. Global condition

Theorem 2. Let {an} be a Blaschke sequence with 2 h(l - {an|) < a.

Let T be defined so that h o T is a measure function.

«a I-m

 

19. z
n-l T(|eie-an‘)

am}.

Let E, = {e

Then ET has zero h o T-measure.
Proof.

For each positive integer n, let On be an open arc on
a

n -l
C with center 'T-r and length T (l - |a ‘).
Q an n
wt Cu: U0 ,F =(:.G o
krn k ‘1 n
n
Then U Fn and n Gn are disjoint sets whose union is C.
n81 n=l

40

Let f = F n E, G = EIW GWG ).
n n n

a:
We observe that for each N, U 0 is a cover for flGn.

k=N 1‘
°° -1
Then lim 2: h o T(T (1 - \anh = no}: h(l - |an\) = 0
Hence h o T(G) '3 0.
Next let n be fixed, let e19 E Fn'

For k 2 n,

I819 - ak|>fi'1'1(1 461k!)
1 ‘ \ak\

T(|e19-ak{)

 

< T(n)

Suppose that h o T(fn) > 0, then there exists a positive dis-

tribution p concentrated on fn such that for all z

ff h o T(’T;-—’)du(9) < M < e

n e '2

For k 2 n, using our assumptions on T and h,

 

 

 

 

 

 

Ifn :(‘eLZ'fikb dMe) =ffnh(1-lakl>' ltd-:23) 'T(‘eile_akh
1 1 hOT du
hm eta-81.1)
sJ'f h(l-lak‘)oh0T(-—i-%-—-)‘ 1 - ‘1?“ - 1 '118T dMe)
n |e -ak| T(‘e -ak\) h( k

T(|eie-ak‘)

< x jfnh(1-\ak\)°hor('—Eé-:;kT)do(e) < M-K~h(l - {ak|).

Then

 

 

co 1 - {ad on 1 - \ak|
du(6) ' Z ‘ du(9)
Ifn kPn T(‘eie-ak‘) k-n‘ffn T(|eie-ak‘)

<M~1<zh(1 - {akp <oo
which contradicts the assumption that

= m on f CZE . Hence h0T(f ) = O.
n T 11

That E,r has zero hOT-measure follows from the above and the sub-

additivity of the measure function.

c. An alternate proof using an analogue of a theorem of

Carleson.

We assume here that I 13-93223- 3 061-591). Then, as noted
t

in the proof of theorem 6, §2, chapter I, we have that

2 T-1(Av) I3_1 hI-ZQ-der converges if z h(Av) converges.
T (Av) r
In theorem 5, §2 of chapter I, we now let Av = 1 - ‘3v|. There-
1-\a\
v

fore we have that the local condition 2 < m holds

 

T(\eie-avl)
except possibly on a set E with hoT-measure 0. This result,
together with theorem 1 of the present chapter, shows that the
T-tangential limits exist on the unit circle on the complement of
E. These results may then be used to replace theorem 2 of this
chapter.

In particular we note that the measure function h(r) = rd,

0 < a < 1, satisfies all the above properties. Thus the Carleson

result serves as a proof of the Cargo result mentioned above.

42

2. T-Tangential Limits of Functions in SQ and of Their

Generalized Fractional Integrals.

Kinney [10] has established the following concerning func-
tions f in the class
a n a 2
Sa={f(z)= Z cnz :2n‘cn‘ <00]:
n=0
(1) Given 0 < v < a, f 6 Sa has tangential limit with order of
1-
tangency T-l for every T < TZg' except possibly for a set
with (l-y)-capacity 0.
(2) If 0 < y < a-Zr, then the fractional derivative of order r
of f has tangential limit for every T'< l33L—- except
l-a+2r
possibly for a set with (l-y)-capacity 0.

(3) If 01< y < a +-2q < 1, then the fractional integral of order

q of f has tangential limit for every T < Ijgjfa'

except
possibly for a set of (l-y)-capacity 0.
Let h(x) be a measure function, let ¢ and V be functions
such that ¢(x) and l;é%:§l- are non-negative, non-decreasing
and integrable in (0,1). Also assume that ¢(ax) s Ca ¢(x) where

c is a constant depending on a and W alone. Define
a

%
Mo) = {“1 ¢(x)dx and Y(n) = (1 Lfif‘ldx and let

 

n 2 n
°° n m
S = {f(z) = 2 c Z : 2 < a}. Define 8(9) to be the
Q n 2 . .
n=0 Q (n) c eine
function whose Fourier Series is defined by 8(9) ~ 2 -Eszay

Extend ¢ and T to complex values by writing

is
¢(2) '5 2 @002“ and Lf-Efl '5 z; Y(n)zn

43

Let f = Z XLEL cneme be the Y/Q-fractional integral of f.

Finally let R[T,v] = {2: l - ‘2‘ 2 7(‘2 - eiV‘)} where T is

an increasing function such that T(0) = 0. We now show:

l-x 1+6 -1 l 1
Theorem 3. If 3(l-x) = ¢(1-x) and T(x) = B o(fi)(x) and
f E SQ, then there exists a T - lim fY/Q(z) except possibly
iv

i zae
for e V E E where E has h-measure 0.

Before proving the theorem we note that by taking

a/Z-l a/2

h(x) = xl-Y, y < l, ¢(x) = (l-x) and hence §(n) = n-

then we obtain Kinney's results given above by choosing in turn

 

(1) m) = x“. Y0!) = n'°"2

<2) «g(x) = 23’2”. You) = rid/2“

<3) nx) = xw“. Mn) = n‘°"2'q

1’M
For (1) we choose 5(l-x) = (l-x)1-a+€, 7(x) = xl-a+e and similarly
for the remaining cases.
Proof: Note that
C eine 'c- e‘ine ‘C ‘2

n - 2 .. H .9.— . _n___ = n ..
j_n‘S(e)‘ d9 -I_n 2 §(n) z §(n) d9 const. 2 §2(n) <

2
so that S E L .

We write

 

% -e
= y_(l-ze44)
fY/§(2) k‘f ~19 S(e)de

l-ze

1
In I, §3 we have shown that if 6(x) = h(i:;) and A(n) =‘f1 5(x)dx,

1A

110: i (x-k) n

q = 1, then gA(x) - z A(k)ake = kj5(e )g(t)dt
(where g(x) = 2 akxk) is finite if g E L except on a set E

of h-measure O.

44

l l

h( it‘

_ 2 i(x-t) g =
Let 8 ‘ ‘3 " 5(3 ) h(‘1_ei(x-t)‘) ‘eix

). Then

\S:(9)‘ = I?“ h(Y—Ecl—IET)‘S(9)‘2de is finite off E, where
e -e

iv fl 1 2
E = e : h(—-—-—-—-—) 8(9) d9 = m and h-measure (E) = 0.
{ I-n \eiv-eiO‘ ‘ ‘ }

_ 1+3 _
Let 9(1-x) I fil—fir- , 'r(x) = B 10(hl) 91"). Then B(l-x) is de-

creasing, so 9 is an increasing function and
BO!(1-x)) 2 kdB(l-x) from the above assumptions on w.

Now let 2 E R[7,v], then for suitable k > 0,

 

' ' -1 1 l
‘ele-z 2 min ‘ele-z 2 k°B o-C-r--7-)
‘ z€R[T,v] ‘ h ‘e1v_ele‘
'9 l l
B( e1 -z ) 2 k °-(—-.-——)
‘ | B h ‘elv-eie‘
-i9
1(Ll-ze 1) 1
so that _. . s k°h(—————"")
‘1-ze 19‘I+e ‘eiv-eie‘

Choosing a > O,

‘av+a ¢%(l-ze-ie)

2
_. -S(9)d9| s
V'a l-ze 19

 

'16 .
UK: figfifl)‘ -\S(6)\2de| -|§:‘_": 1e19-z;€'lde‘

H

E;—
_“‘-s 8 g 19
(1_ze-19)%+e/2’

 

- 2
by Schwarz inequality using f = (l-ze )

s RU?" h(I-fil—i—é-[Msmnzdemjfi \eie-zfi'ldel

1
Choose e v G E, z E R[7,v]. Then the first factor is finite and

the second factor goes uniformly to 0 with a. Since

4S

5 19
n 19(l-ze 4) = v-a v+a + ,
ffl< 1_ze_ie S(9)d9 Ln +Iv-a [3+8 and the first and

 

i . . .
the last are analytic at e v, then the T-llmlt ex1sts.

3. T-Tangential Limits of Functions in A(Y,¢)

Let h(r) be a measure function, D the unit disk. The
following theorem, which follows from a lemma of Ahlfors, has
been established by Rung [11]:

Theorem A. Let U(z) be a real-valued, non-negative, measurable

function defined in D such that

I I U(z)dxdy < m.
D

 

Then lim

1
U(z)dxdy = 0.
:40 h(r) 'ffg

Me .1“)
This theorem is used by Rung to obtain results concerning the
class A8, 3 > -1: we say that f holomorphic in D belongs

to the class As if

folf'(z)‘2(1"Z|)sdxdy < a .

He goes on to determine the "statistical" orders of certain func-
tions of the Picard type; the order of a function is said to be
"statistical" if this order is obtained by restricting the choice
of 2, as for example in Rung's case, to a Stolz domain.

Let ¢ be a function which satisfies either of the follow-
ing properties:

(i) ¢(x) is non-negative, non-decreasing and integrable

in (0,1)

or (ii) there is an integer p 2 1 such that

46

¢(x),¢'(x),...,¢(p-1)(x) are absolutely continuous
in [0,1] and vanish at 1. Also we suppose that
¢(p)(x) has constant sign and ‘¢(p)(x)\ is non-
decreasing in the subset of (0,1) where ¢(p)(x)
exists.

In the first case define @(n) - I1 ¢(x)dx; in the second
case define §(n) = n-p‘¢(p-1)(1 - l/n)‘.

The function ¢(x) = (l-x)s serves as an example of (i)
if -l.< s s 0 and as an example of (ii) if s > 0.

We shall discuss here the class A(Y,¢) where Y is a
general function and ¢ satisfies either of the properties given
above. Then f holomorphic in D belongs to A(Y,¢) if
IDI‘fY(z)‘2¢(‘z|)dxdy < a.

Here we shall write f(z) = 2 anzn, z - reie

fY(z) = Z ‘1’(n)anzn

By letting Y(n) = n, ¢(x) = (l-x)8 we have the class As.
Also the Stolz domain is the example 7(r) = r of the tangential
domain R[T,9] - {2: 1 - ‘2‘ 2 ¢(‘eie-z‘)} in terms of which our

"statistical" results are given.
a. An equivalent series condition

We state here a result of P.B. Kennedy [9] (Thms. l & 2).
Theorem B. Let f(x) = 2 cnxn; O S x < 1; cu 2 0 and let m be
as above. Then f(x)¢(x) G L(0,l) if and only if 2 @(n)cn is

convergent.

47

The following theorem gives an equivalence between ACY,¢)
and a class of functions with a certain order on their Taylor
coefficients.

n i9 .
Theorem 4. Let f(z) - z anz , z 8 re . Then f E A(Y,¢) if
and only if z Y2(n)|an\2§(n) < a»

Proof.

InflffifileVDdxdr/ =

n2" 1 n9 n
Jo

o z ‘1’(n)anrne1 ° 2 Y(n)§hr e'ine¢(r)rdrd9 =

j: 2 Y2(n)\anl2r2“+1 ~ ¢(r)dr

These equalities together with theorem B, above where we are
using cn - [Y(n)‘ah|]2, gives the desired result.
To obtain the proof of the following theorem,*we make the
following assumptions on the functions ¢, y, f and h:
(1) If a > 0, h(ar) s qah(r) where Ca is a constant depending
on a alone.
(2) “Yank is subharmonic in D for k > o.
(3) If \z - §| < (1 - \gm, 0 < c < 1 then ¢(‘z‘) 2 C¢,c¢(‘§‘)°

Theorem 5. Let f(z) be holomorphic in D; k,> 0 and

J‘1)“f~y(z)‘k¢<‘2|)dxdy < co.

wanmzna - \th

Then lim
h(‘z-eie‘)

 

= 0 except possibly for a

Z-Oeie

set of e16 with zero h-measure.

Proof. Let g E D

48

Let r = (1 - |§|)t, O < t < 1 so that D(§,r) CD.

By (2) above,
nr2\fY(§)‘kS JOJP ‘fy(z)‘kd"dy
D(§,r)

Lat Z 6 D(§,r). By (3):

<4) "(1 - |§l>2tzlfy(§>lk ° C¢,t¢(‘§‘) S H lfy‘ZHkNZDdxdy
D(§,r)

Here we apply theorem.A of Rung [ll] cited above with
U(z) = ‘fY(z)‘k¢(|z|), R B 2‘; - ele‘ and using the assumption

(1) so that h(2|§ - eie‘) s C h(‘g - eie‘). From this we find

H; \f,<z>lk¢qudxdy

 

D (e .R) =
lit:e h (R) 0
§~e
except on a set of e16 with h-measure 0. Together with the

inequality (4), this gives the desired result.
Next we let y(t) be a function such that ¢.y is a
function satisfying either properties (i) or (ii) as described

above. We define x(n) to be the function which is defined so

2
that 2322;.121 is the discrete function correSponding to the
x (n)
continuous function ¢y. For example, if ¢~y satisfies prOperty

(i) then

2
w =J'1 ¢y(t)dt ,
x (n) 1

5IH

Let fx(z) =»z x(n)anzn. We now use x to extend the previous

theorem.

49

Theorem 6. Let f(z) = z anzn, f E A(Y,¢). Then

lfx<z>\{¢v<lzl>}”a - M)
= o
19 [h(‘z - eie‘)

2"8

except possibly on a set of 19

 

e of h-measure 0.
Proof.
let a we .
3(2) 2 Y(n)
2 2 2 2
We note that Y - x3.. |a |n - (— f) a Y2 - e - \a | so that
Y n n 2
by theorem 4, g 6 A(Y,¢y) since here ¢v is to Q? 88 ¢
is to Q in theorem 4. Therefore we apply theorem ‘X
5 with k = 2 to obtain:
2 2
lsy<2>| ¢¥<\2|>(1 - |2|>
= 0
16
Zdeie h(|e - 2|)
except possibly on a set of e16 of h-measure O.
The theorem follows upon noting that 8Y(2) B fx(z) and taking

of square roots.

b. The generalized tangential limit

Let G E ACY,¢). We use the results of either of the two

preceding theorems to determine an order for G

Let T be an increasing function with T(O) = 0 and such

that hoT is a measure function.

Let 2 e R[T,6] so that

l - ‘z‘ 2 7(‘819 - z‘)

and h(1 - \z|) 2 how-(lai6 - 2|).

50

1 2 1
h0T(‘eie-z‘) h(1- z )
lcy(z)|2¢<|2|><l - lzhz

16 a O as 2 a e19, z 6 R[T,9]
ho¢(|e -z|)

 

Then

i
except possibly for a set of e 9 whose hoT-measure is zero,

which implies that for the same exceptional set

lcY<z>|2¢<lzha - Izhz

h(1 - ‘21)

 

~ 0 as 2 4 ale, 2 6 R[T,9] .

As an example, let h(r) = rd, 0 < a s l, and let
¢(r) = r6, 1 s 9 s lla. Then the resulting order holds for z
confined to a tangential approach of order of tangency 9-1

for all ei9

outside a set whose aB-capacity is zero. In
addition, the statistical result is more simply stated if we
choose ¢(r) - (l-r)8, s > -1.

In determining a tangential approach, as in [10], for
instance we note that we are covering the circles bad points,
i.e. points where limits fail to exist, by a system of circles
with certain radii. The smaller the radii, the larger the per-

missible tangential approach, i.e. the approach missing these

circles.

 

51

By describing our measure function hor in the previous example
using the technique of Cargo (see §l) of composing the measure
function h with the approach function T, we are able to realize
a wider approach with a still small exceptional set. Naturally
this leads to a loss in the result which describes the order of
the function.

Applying theorem 5 to the above example we find

\GY(z)‘(1 - \z\)°’2*1‘“/2 a o, z a .19, z e RLT,93

except possibly on a set of e19 of aB-capacity 0. Where
a > 0 is small we are permitted to choose values of 9 2 l which
are large, maintaining the relation as s 1. This permits then
a tangential approach of larger order of tangency, while at the
same time, once 9 is fixed, gives rise to an extremely small
exceptional set since we can choose a (and hence as) to be
arbitrarily small. However the smaller a then diminishes the
order of the overall result. Therefore the wider approach and
smaller exceptional set are obtained at the expense of the
statistical type order.

If, in theorem 6, we choose Y(n) = n, ¢(t) = (l-t)s,

'3'1, and let y(t) = (1-t)25'2 where

s > -1 and so @(n) = n
s +-2e > 1, we then find x(n) a nB so that fx(z) = fB(z), the
fractional derivative of order 9 of f. We have then the follow-
ing result obtained by Rung [11, p. 329]: Except on a set of
h-measure O,

s+29

11m tifB(z)1(1 - 121) 2 g 0 .

 

19
2-06 [h(‘eie-z‘)

52

This result is applied to the Picard function Q(z) which is
holomorphic in D and omits there the values i;2n n i and which

belongs to the class A2+6 (see Rung [11, pp. 329-330] for details).

Choosing B = O we find

11,, iomla - lzb1+6 __,
z-oej’e fho'rfleie-zl)

0

 

except for e19 which belong to a set of hoT-measure 0.
Again letting h(r) = fa, f(r) 3 r6, 0 < a s 1 and 6 2 l, with

z 6 R[T,9] we find

mama-14f“
(1 _ ‘2‘)072

d 0 as z d e19

except at most on a set of aé-capacity O. The result of Rung
that limie ‘Q(z)|(1 - ‘2‘)5+6 = 0, z confined to a Stolz angle,

zae
for almost every 9, is obtained by choosing a = 6 = l.

BIB LIOGRAH'IY

10.

11.

12.

13.

14.

BIBLIOGRAHIY

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