I nternat. J. Math. & Math. Sci.
Vol. 2 #3 (T979) 415-426
415
SMOOTHNESS PROPERTIES OF FUNCTIONS IN R (X)
AT CERTAIN BOUNDARY POINTS
EDWIN WOLF
Department of Mathematics
East Carolina University
Greenville, North Carolina 27834
(Received January 31, 1979)
ABSTRACT.
Ro(X)
X
Let
the algebra consisting of the (restrictions to
X.
with poles off
let
be a compact subset of the complex plane
Rp(X)
Let
m
.
We denote by
X of) rational functions
denote 2-dlmenslonal Lebesgue measure.
be the closure of
0(X)
R
in
For
p
>_ I,
Lp(X,dm).
In this paper, we consider the case p
2. Let x e X be both a
2
bounded point evaluation for R (X) and the vertex of a sector contained in
Int X.
For those
about
L be a lne which passes through
Let
y
If(y)
E
L
f(x)
X
x
that are sufficiently near
2
for all f
R (X).
and bisects the sector.
x
we prove statements
KEY WORDS AND PHRASES. Rational functions, compact set,
point evaluation, admissible
1980
LP-spaces
bounded
function.
Mathematics Subject Classification Cod: Primary 0A98, Secondary 46E99.
E. WOLF
416
.
INTRODUCTION.
i.
Let
be a compact subset of the complex plane
X
X.
poles off
Lp(X)
Let
Whenever
and
p
The closure of
(X)
in
Lp(X)
will be denoted by
both appear, we will assume that
q
p > I,
For
denote 2-dimenslonal Lebesgue measure.
m
Lp(X,dm).
of) rational functions with
X
the algebra consisting of the (restrictions to
R0(X)
We denote by
p
-i
+
q
-i
let
Rp(X).
i.
In "Bounded point evaluations and smoothness properties of functions in
RP(x) ’’,
[6, p. 76], we proved the following:
THEOREM i.i.
Let
be an admissible function and
integer.
Suppose that
function
g
> 0
Lq(X)
f
E
and that there is an
E
represented by a
Then for every
x
such that for
RP(x)
(z-x) j + R
(DJxf)
J--O
f
where
R
Rp(X)
for all
(iii)
X
X having full area density at
in
s
(i)
x
a nonnegative
(z-x) -s(l z-xl)-ig v Lq (X).
such that
there is a set
every
p > 2
s
app lim
R(y)
satisfies
y
E,
and
O.
It is natural to ask whether a similar result holds for the case
The problem in extending the proof of Theorem i.i to the case
p
2
p
2.
is that
-i
2
L loc"
Fernstrm and Polking have shown at least one way in which the
case
p > 2
differs from
z
set
X
such that
uation for
2(X).
R
R2(X)
2
p
+ L2(X)
Int X.
but no point in
They have constructed a compac!
X
In this paper we consider the case
is a bounded point evaluation for
point.
[2, pp. 5-9].
We will assume that
x
2(X)
R
is a bounded point eval-
p
2
when
x e X
and is a special kind of boundary
X is the vertex of a sector contained in
SMOOTHNESS PROPERTIES OF FUNCTIONS
417
To prove our theorem we will need the representing functions used in [6]
We will also use results
and a capacity defined in terms of a Bessel kernel.
of Fernstrm and Polking to construct a representing function for
x
with
support outside the sector mentioned above.
2.
REPRESENTING FUNCTIONS.
In this paper
A point
DEFINITION 2.1.
2(X)
L
R
2(X)
If(x)
will denote the identity function.
z
x e X
is a bounded point evaluation (BPE) for
C
if there is a constant
< C{
I Ifl2dm}
112
such that
for all
f e
R2 (X).
It follows from the Rlesz representation theorem that if
R2(X) then
R2(X).
f
BPE for
for all
DEFINITION 2.2
Such a
y
(X)
[ (z-y)-lg
I
f(x)
such that
We define the Cauchy transform of
such that
is a
fg dm
x.
is called a representing function for
g
(y)
for each
g
there is a function
X
x
g
to be
dm
z-Yl-iIgldm < ""
The
The following lemma was proved by Bishop for the sup norm case.
proof for our case is similar and is found in [6, p. 73].
LEMMA 2.1.
Suppose that-
R2(X) Suppose that
Then (y)-l(z-y)-lg is
(y)
f
Let
[ fg
and that
is defined and
+
0
a representing function for
(z-x)(z-y)-ig
c(y)
L2(X)
g
dm
i
dm- 0
and that
for all
L 2 (X).
(z-y)-ig
y.
+ (y-x)(y). From
the above lemma
there follows
COROLLARY 2 1
Then
c
Let
(y)-i (z-x) (z-y)-Ig
is defined and
+ O,
and
g
2
L (X)
be a representing function for
is a representing function for
(z-y)-ig
2
L (x).
x
y whenever
X.
c (y)
418
E. WOLF
CAPACITY DEFINED USING A BESSEL KERNEL.
3.
Denote the Bessel kernel of order i by
G
where
I
G
I
is defined in
terms of its Fourier transform by
l(Z)
2
f e L (C)
For
(i+
Iz12) -1/2
we define the potential
U
2
i
DEFINITION.
L
2
IUI
is the Sobolev space of functions in
I
f e L
2
where
2"
L
bution derivatives of order i are functions in
The Calderdn-Zygmund theory shows that
LI2
f
UI,
denotes the space of functions
the norm is defined by
DEFINITION.
I Gl(z-y)f(y)dm(y)"
If(z)
2
i
L
2
whose distri-
2.
equals the space of functions
and that the norms are equivalent [4].
We recall the definition of the capacity
Let
DEFINITION.
12dm
Igrad
inf
m
>_
R
2(X)
i
[3].
(I z l)-iv
Then
where the infimum is taken over all
F2(E)
e L
I
such that
The next theorem is proved in [6, p. 82].
THEOREM 3.1.
Let
2
2
e L (x).
REMARK.
0
L (X).
v
a function
be an arbitrary set.
Hedberg has used this capacity to characterize BPE’s for
E.
on
E
F 2.
X be a BPE for
Suppose that
Then
2(X)
R
is an admissible function such that
22n(2 -n) -mr (An\X)
2
n=l
The theorem is, in fact, true if
decreasing function defined on
tkat is represented by
<
.
is any positive non-
(0,).
Now we define the Bessel capacity which Fernstrm and Polking use to
describe
BPE’s for R2(X).
Let
DEFINITION.
inf
.f.2dm
f(z) >_ 0
and
E
be an arbitrary set.
where the infimum is taken over all
f
Ul(Z)
>_
i
for all
z e E.
Then
f
CI, 2(E)
()
such that
SMOOTHNESS PROPERTIES OF FUNCTIONS
2
’i and
The equivalence of the norms on
F2
and
4.
A FUNDAMENTAL SOLUTION FOR
C1
2
2
LI
419
implies that the capacities
are equivalent.
We will use
(0,i), or (i,0).
(81,82
8
181
We set
(0,0),
to denote a double index that may be
81 + 82
z
Letting
x
+
we denote
iy,
the first order partial derivatives by
81 82
81 @y 82
@x
D
The differential operator
i.i)
w
z
x+
2
z
2
Hence
as a hi-regular fundamental solution.
__t H(z,w)
t
where
---H(z,w)
and
is the formal ad]oint of
Dirac measure supported at
We note that for
z.
1
-1-1 1
IDs.<0,z) <_Tlzl
8
w
and
is the
z
(0,0), (0,i), (i,0)
z+0.
The next lemma links BPE’s to the function H(w,z).
A proof which includes
this as a sp.eclal case is in [2, p. 3].
LEMMA 4.1.
A point
there is a function
z e
f e
z
0
2
is a BPE for
e X
Ll,lo c ()
R
such that
2(x)
L
2(x)
f(z)
i(
z_z0)
i
if and only if
for all
\X.
The next lemma we need is proved by Fernstrm and Polking in [2, pp. 13-15].
It is interesting that this lemma holds for
and (i,0).
8
(0,0)
as well as
Before stating it we introduce more notation.
DEFINITION.
For a compact set
DEFINITION.
LEMMA 4.2.
We denote
Let
(0)
{z12
Let
X
let
{zlDist(z,X)
X
DEFINITION.
X,
-k-2
< }.
{z12 -k-I <_ zl <_ 2-k+l}
<_
zl <_ 2-k+I}-
be compact and suppose that
by
.
(0,i)
E. WOLF
420
22kci,2 (Ak\X)
[
> 0
Then for each
k > 0
and for each
<
""
there is a function
e C
k
such that
k(Z)
(i)
I
Izl
for
Xe,
\
near
z
IDSk(Z) 12am(z)
!
(il)
for
<-
F2-2k(I-]I)CI,2(\X)
(0,0), (0,i), and (i,0).
8
and
The constant
F
is independent
k.
of
THE MAIN RESULT.
5.
It is no restriction to assume that the boundary point
(x
origin
0).
Also, we may assume that
to be the vertex of a sector in
,
a,
0 -< a <
< 2,
L be the mld-llne
is a BPE for
R
2(x),
y
as
la
f(y)
2
f e R (X).
along
First we will construct a function
2(X)
R
S’
LEMMA 5.1.
S
a sector
(iii)
g
is a subset of
{(r,8)la +
S’
(il)
0 < r < a}.
then
Since
in
< 8 <
g
L 2 (X)
0
m((supp g) f S’)
For all n >-0,
for
0,
2(x),
R
Int X.
Int S
Int X
y
f(y)
f(0)
for
which represents
L.
0
for
This
defined by
----,
0 -< r < a}.
Then, there is a function
represents
are
L.
is a BPE for
0
Suppose that
X.
S
(r,8)
to represent the value of that linear
and which has support disjoint from a sector surrounding
second sector
(1)
8-a
We want to study
0
approaches
0
In taking
such that if
-< r < a},
< 8 < 8, 0
{(r,8)18
L
we may use
functional at a given
2
f e R (X)
{(r,8)
S
< 2}.
is the
we mean that there are numbers
a, 0 < a < 2,
and a number
polar coordinates, and
Let
Int X
zl
{
X
x e X
R2(X)
that is the vertex of
2
g e L (X)
such that:
SMOOTHNESS PROPERTIES OF FUNCTIONS
n+l
Ig]2dm <
1
F
F
where
PROOF.
X
Choose
k
0(RI)
C
,2
(\X)
n.
such that
Ii
-< I or
if
t
if
<tbl
>- 2
t
i
set
(2kl1)/ Y. x(2J1.l)
Xk(.)
2kCl
is a constant independent of
(t)
For each integer
2
k--n-i
A % X
n
421
For those values of
z
for
Int S
in
\Int
z
S.
Xk(Z)
define
so that the following
three conditions are satisfied:
(2)
k(z)
(z)
(3)
There are constants
(i)
k
C
0
for
(z)
I-kx
F
The constants
X
we have
A(0,1/4)\X.
where
and
I
> 0
Given
of
<
F
F
’
S’, and
F
and
I
I-ky
and
such that for all
< F
2
are independent of
2
choose the functions
k%k %k
since supp
X e CO
Choose
2
(z)
Fl2k
with
k
%k
X(z)
2k
.
k.
of Lemma 4.2.
C
k
I k%k
Thus,
I near X.
On the complement
0
Set
I on
E(z)H(0,z)
h(z)
i
---.
z
H(0,z)
For each double index
F8
X
z
(0,0), (0,i), and (i,0)
8
there is a constant
such that
IDSh(z)
<
FsIzl -I-181
These inequalities follow from those of Section 4 and the fact that
,
its derivatives are bounded.
Since
supp
%k
C
Set
fE
h
[k
khk
0
the above inequalities imply that
where
h
X
k
and
h.
E. WOLF
422
IDS(z)
(*)
F82k(l+ISl)"
<
Henceforth, we will limit the number of symbol.s denoting constants by
F
letting
The inequalities (*) combined with Lemma 4.2
denote any constant
imply that
IDS(z)DYk(Z 12dm(z)
I+ I<l
22k(i+181)
F I
D k(Z) 12dm(z)
k--0 18+ I<I
z l_2_k+l
22kc 1 ,2
F
-< F
L2
I
e
_<
-<
k;O
CI, 2,
Finally, by the subadditivity of the capacity
II f e I12L 2
I 22kc I 2 (\X)
< F
k--0
1
{fe }
The net
is bounded in
2
\X.
since
f
Then
f
(z)
0
2
Ll,loc,
L
2
and
We can choose a subsequence
I.
f(z)
Let
L I.
that converges weakly in
z e
we have
f(z)
z e X f% S’
for all
lim f
(z) + (I-x)H(0,z)
H(0,z)
Jfor
f(z)
0
J-
z e
for
\X.
a e
{fe J
for
Note that
S’
z e X
zeXfNS’.
Set
0
t
g--- f.
(see [2, p. 3]).
g
Then
If
z
L2 (X),
and
X, g(z)
0.
is a representing function for
g
Clearly,
m((supp g)
We have
n
18+ 1-<i
f
k
such that
0.
n
< F
The integral
S’)
A nX
X
A
’
JDShk,kl2dm
k=O
A
n
I
X
DShkDk 12dm-
will be nonzero only for those
n
i.e., k --n- i, n, n
+
i.
Thus, by
(*)
and Lemma 42,
.
423
SMOOTHNESS PROPERTIES OF FUNCTIONS
Igl2dm
< F
18+ I<I
AOX
n
n+l
[
< F
k--n-i
IDShkD%kl2dm
k--n-i
AX
n
22kci 2(\X).
(i), (ii), and (iii).
This completes the proof of
We will use the next lemma to obtain representing functions for points
0
near
c(y)
lemma, and let
LEMMA 5 2
0 e X
e > 0
Ic(y)
be represented by a function
there exists a
I +
be as in the previous
g
is an admissible function and that
Then for any
PROOF
0,X,S, and
Let
be as defined in Section 2.
Let
Suppose that
then
L.
on the line segment
Y(Y)
> i
such that if
2
v e L (X).
2
L (x).
v(z)(Izl)-i
IYl < and
y e L,
E.
Since the capacities
r2
CI, 2
and
are equivalent, Theorem 3.1
imp lies that
[
n=l
To show that
IYll
c(y)
(r)
that
kllZ-y
is a constant
r-(r) -I
Izl
>
k
2
*(IYl)(lYl)I
-<
S’
By definition of
for any
such that
L
y
and
k21z-y
>-
{0}.
z e X\S’
IYl
k
there is a constant
for any
such
Similarly, there
L
y
I
and
z
XkS’
are both increasing,
and
#
<
is defined, we first note that
I g’(z-Y)-idml
where
Since
22n (2-n)-2Ci,2 (An \X)
(Izl)(lz-yl) -I
< k
and
I
,(lyl),(Iz-yl) -I
< k
-I
e L
2.
Hence
IYl
We claim that
g. (z-y)
g’-i
that
Igl2.-2dm
<
<
.
2
L (X)
n=l
(IY
and therefore
(2-n) -2
g
jgl2dm.
AOX
n
I
(X).
First observe
{0}.
E. WOLF
424
CI, 2
By Lemma 5.1 and the subadditivity of
gl2-2dm <
(2 -n)
n=l
_222nci ,2 (An\X)
The capacity series converges.
IYl
for
<
and
L.
y
Let
2
R (X).
v
function
v(z)(Izl) -I
Y(Y)
> i- e.
0,
such that
Ic(Y)
R
for
I +
2(X)
which is represented by
is an admissible function and that
Suppose that
L 2 (X).
lira
X, 0, and L are just as they have been.
X be a BPE
0
(r)
Since
is defined.
> 0
a
It follows that
In the following theorem,
THEOREM 5.1.
(y)
Thus,
e > 0
we can choose for any given
we get
e > 0
Then for any
> 0
there is a
such that if
A(0,),
y e L
If(y)
for all
Choose
i
e(lyl)llfl 12
<
2
R (X).
f
PROOF.
f(0)
2
g e L (X)
Let
be a representing function for
by Lemma 5.2 so that for
L
y
IYl
and
<
0
as in Lemma 5.1.
i’ Ic(Y)
1/2.
>
Then by Corollary 2.1,
i
c(Y)-I
c(y) -1
f(0)
f(y)
II
Thus, for y e L
If(Y)
and
f(0)
Y(Y)Yl < I
-<
21yl
f(0)]z(z-y)-igdm
[f
-i
[f- f(0)][l + y(z-y) ]gdm
[f
f (0)
(z-y)-igdm.
r
J If- f(0)l Iz-yl-Zlgldm.
There exists a monotone, increasing function
(Izl)-l(l zl)-Iv(z) L2(x)
and
so that the function
(r)
for
(r).(r).
y e L
and
llm (r)
r+0+
0
(see [6, p. 74]). Moreover, we may choose
-I is also
monotone increasing. Let
r(r)-l(r)
Then recalling that
z e X\S’
such that
kllZ-y
{0}, we have
>
Izl
and
k21z-y
>
lYl
.
SMOOTHNESS PROPERTIES OF FUNCTIONS
If(y)
f(0)
_<
F(lyl)llf il 2 {
Igl2dm} I/2.
(2-n) -2
n=l
AX
If the sum of the infinite series is less than
i,
the theorem is nearly
I.
Suppose the sum is greater than or equal to
proved.
If(Y)
f(0)
<
<-
425
Then
FI (lyl)]Ifl 12 n--1
I 22n(2-n)-2Cl, 2 (%\X)
F$(IYl)*([Y[)IIf[ [2 I 22n(2-n)-2C 1 2(An \X)n=l
Since the capacity series converges by Theorem 3.1, we may choose
IYl
for
<
F$ClYl)
62
n=l
If(y)
Then
f(O)
<
6
2
such that
22nc2-n)-2Cl 2CAn \X) < "
e,(lY[)llfll 2
[Yl
for
<
min(l, 2)
and
y
e L.
This concludes the proof.
REMARKS.
(i)
If
R2(X),
is a BPE for
so that a statement similar
s
to Theorem l.l(ii) holds for
y e L % (0,).
(iii)
X
2(x)
For certain sets
a point
which is a BPE for
0 e X
Int X.
may not be the vertex of any sector having interior in
however, that
0
is a cusp for a curve whose interior is in
be a llne segment which bisects the cusp at
0.
of the cusp near
where
(see [5, p. 74]).
The theorem may be extended by techniques of Wang [5] to
include bounded point derivations of order
R
there always exists an
as in the hypotheses of Theorem 5.1
admissible function
(ii)
0 e X
r
Then if
y
L
C
0
and
approaches
at
we can show that functions in
R2(X)
of Theorem 5.1.
r
0
Let
and let
C
z e X\C,
ly-zlT(lyl) >- lyl
is a monotone decreasing function such that
on how rapidly
Int X.
Suppose,
L
denote the interior
lim+v(r)
=.
Depending
(or how rapidly the cusp "narrows"),
satisfy an inequality similar to that
426
E. WOLF
REFERENCES
I.
Caldern,
2.
Fernstrom, C. and Polking, J., Bounded
3.
Hedberg, L. I., Bounded point evaluations and capacity.
A. P., Lebesgue spaces of differentiable functions and distributions.
Proc. Sympos. Pure Math. 4, 33-49, Providence, R. I., Amer. Math. Soc. 1961.
point evaluations and approximation in
by solutions of elliptic partial differential equations. J. Functional
Analysis, 28, 1-20(1978).
Lp
J. Functional Analysis,
i0, 269-280(1972).
4.
Stein, E. M., Singular Integrals and Differentiability Properties of Functions,
Princeton University Press (1970).
5.
Wang, J., An approximate Taylor’s theorem for
(1973).
6.
Wolf, E., Bounded point evaluations and smoothness properties of functions in
RP(x), Trans. Amer. Math. Soc. 238, 71-88(1978).
R(X), Math. Scand. 33, 343-358