SMOOTHNESS PROPERTIES OF FUNCTIONS IN R ( X ) AT CERTAIN BOUNDARY POINTS

Let X be a compact subset of the complex plane . We denote by Ro(X) the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dlmenslonal Lebesgue measure. For p >_ I, let Rp(X) be the closure of R 0(X) in Lp(X,dm). In this paper, we consider the case p 2. Let x e X be both a bounded point evaluation for R 2 (X) and the vertex of a sector contained in Int X. Let L be a lne which passes through x and bisects the sector. For those y E L X that are sufficiently near x we prove statements


E. WOLF i. INTRODUCTION.
Let X be a compact subset of the complex plane .We denote by R0(X) the algebra consisting of the (restrictions to X of) rational functions with poles off X. Let m denote 2-dimenslonal Lebesgue measure.For p > I, let Lp(X) Lp(X,dm).The closure of (X) in Lp(X) will be denoted by R p(X).
-i -i Whenever p and q both appear, we will assume that p 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 s a nonnegative integer.Suppose that p > 2 and that there is an x X represented by a function g Lq(X) such that (z-x) -s(l z-xl)-ig Lq v (X).Then for every > 0 there is a set E in X having full area density at x such that for every f E RP(x) where R R p(X) satisfies (iii) app lim
for all y E, and It is natural to ask whether a similar result holds for the case p 2.
The problem in extending the proof of Theorem i.i to the case p 2 is that -i L 2 z loc" Fernstrm and Polking have shown at least one way in which the case p > 2 differs from p 2 [2, pp.5-9].They have constructed a compac!set X such that R2(X) + L2(X) but no point in X is a bounded point eval- uation for R2(X).In this paper we consider the case p 2 when x e X is a bounded point evaluation for R2(X) and is a special kind of boundary point.We will assume that x X is the vertex of a sector contained in Int X.
To prove our theorem we will need the representing functions used in [6] and a capacity defined in terms of a Bessel kernel.We will also use results of Fernstrm and Polking to construct a representing function for x with support outside the sector mentioned above.
In this paper z will denote the identity function.
DEFINITION 2.1.A point x e X is a bounded point evaluation (BPE) for R2(X) L2(X) if there is a constant C such that I R2 If(x) < C{ Ifl2dm} 112 for all f e (X).
It follows from the Rlesz representation theorem that if x X is a BPE for R2(X) then there is a function g (X) such that f(x) fg dm for all f R2(X).Such a g is called a representing function for x.DEFINITION 2.2 We define the Cauchy transform of g to be (y) [ (z-y)-lg dm for each y such that I z-Yl-iIgldm < "" The following lemma was proved by Bishop for the sup norm case.The proof for our case is similar and is found in [6, p. 73].that-g L2(X) and that [ fg dm-0 for all LEMMA 2.1.Suppose L 2 f R2(X) Suppose that (y) is defined and + 0 and that (z-y)-ig (X).
Then (y)-l(z-y)-lg is a representing function for y.
Let c(y) (z-x)(z-y)-ig dm i + (y-x)(y).From the above lemma there follows Let g (X) be a representing function for x X.
Then c (y)-i (z-x) (z-y)-Ig is a representing function for y whenever c (y) is defined and + O, and (z-y)-ig L 2 (x).
E. WOLF   3. CAPACITY DEFINED USING A BESSEL KERNEL.
Denote the Bessel kernel of order i by G I where G I is defined in terms of its Fourier transform by f e where 2 denotes the space of functions UI, DEFINITION.i the norm is defined by 2 is the Sobolev space of functions in L 2 whose distri- DEFINITION.L I bution derivatives of order i are functions in L2.

equals the space of functions
The Calderdn-Zygmund theory shows that i LI2 and that the norms are equivalent [4].
We recall the definition of the capacity F 2.
DEFINITION.Let E be an arbitrary set.Then F2(E) inf Igrad 12dm where the infimum is taken over all e L I such that m >_ i on E. Hedberg has used this capacity to characterize BPE's for R2(X) [3].DEFINITION.Let E be an arbitrary set.Then CI, 2(E) inf .f.2dm where the infimum is taken over all f () such that f f(z) >_ 0 and Ul(Z) >_ i for all z e E.

and implies that the capacities
The equivalence of the norms on 'i LI F2 and C1 2 are equivalent.

A FUNDAMENTAL SOLUTION FOR
We will use 8 (81,82 to denote a double index that may be (0,0), (0,i), or (i,0).We set 181 81 + 82 Letting z x + iy, we denote the first order partial derivatives by 81 82 D 81 82 @x @y The differential operator 2 x + 2 i.i) z zw as a hi-regular fundamental solution.Hence ---H(z,w) and w t __t H(z,w) where is the formal ad]oint of and is the z Dirac measure supported at z.We note that for 8 (0,0), (0,i), (i,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 z 0 e X is a BPE for R2(x) L2(x) if and only if 2 () such that f(z) i( i there is a function f e Ll,loc z_z0) for all z e \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 8 (0,0) as well as (0,i) and (i,0).Before stating it we introduce more notation.

THE MAIN RESULT.
It is no restriction to assume that the boundary point x e X is the origin (x 0).Also, we may assume that X { zl < 2}.In taking 0 to be the vertex of a sector in Int X we mean that there are numbers a, , 0 -< a < < 2, and a number a, 0 < a < 2, such that if (r,8) are polar coordinates, and S {(r,8) la < 8 < 8, 0 -< r < a}, then Int S Int X.
8-a 0 < r < a}.Since y Int X Let L be the mld-llne L {(r,8)18 --is a BPE for R2(x), we may use f(y) to represent the value of that linear R 2 functional at a given f e (X).We want to study f(y) f(0) for R 2 f e (X) as y approaches 0 along L. L 2 First we will construct a function g (X) which represents 0 for R2(X) and which has support disjoint from a sector surrounding L. This second sector S' is a subset of S defined by S' {(r,8)la + < 8 < ----, 0 -< r < a}.LEMMA 5.1.Suppose that 0 is a BPE for R2(X) that is the vertex of L 2 a sector S in X.Then, there is a function g e (X) such that: (1) g represents 0 for R2(x), (il) m((supp g) f S') 0, (iii) For all n >-0, where F is a constant independent of n.
PROOF.Choose X C 0(R I) such that (t) I i For each integer k set For those values of z in Int S define Xk(Z) so that the following three conditions are satisfied: (2) (z) 0 for z X ' S', and (3) There are constants F I and F 2 such that for all k k (z) k (z) 2k I-x < Fl2k and I-y < F 2 The constants F I and F 2 are independent of k.
Given > 0 choose the functions k of Lemma 4.2.On the complement of X we have k%k %k since supp %k C .Thus, I k%k I on 0 A(0,1/4)\X.Choose X e C O with X(z) I near X.Set h(z) E(z)H(0,z) i where H(0,z) ---.z For each double index 8 (0,0), (0,i), and (i,0) there is a constant F8 such that IDSh(z) < FsIzl -I-181 These inequalities follow from those of Section 4 and the fact that X and its derivatives are bounded.Set fE h 0 [ k khk where h k h.
Since supp %k C , the above inequalities imply that E. WOLF (*) IDS(z) < F82k(l+ISl)" Henceforth, we will limit the number of symbol.sdenoting constants by letting F denote any constant The inequalities (*) combined with Lemma 4.2   imply that e 2 -< F IDS(z)DYk(Z 12dm(z) Finally, by the subadditivity of the capacity CI,2, we have The net {fe } is bounded in L I. We can choose a subsequence {fe J 2 Let f(z) lim f (z) + (I-x)H(0,z) for that converges weakly in L I.
J-J 2 z e \X.Then f Ll,loc, and f(z) H(0,z) for z e \X.Note that since f (z) 0 for all z e X f% S' f(z) 0 for a e z e X S' zeXfNS'.
We have The integral f JDShk,kl2dm will be nonzero only for those k such that n i.e., k --n-i, n, n + i.Thus, by (*) and Lemma 42, If the sum of the infinite series is less than i, the theorem is nearly proved.Suppose the sum is greater than or equal to I. Then Since the capacity series converges by Theorem 3.1, we may choose 6 2 such that for IYl < 62 F$ClYl) 22nc2-n)-2Cl 2CAn \X) < " n=l Then If(y) f(O) < e,(lY[)llfll 2 for [Yl < min(l,2) and y e L.
This concludes the proof.

REMARKS. (i)
If 0 e X is a BPE for R2(X), there always exists an admissible function as in the hypotheses of Theorem 5.1 (see [5, p. 74]).
(ii) The theorem may be extended by techniques of Wang [5] to include bounded point derivations of order s so that a statement similar to Theorem l.l(ii) holds for y e L % (0,).
(iii) For certain sets X a point 0 e X which is a BPE for R2(x) may not be the vertex of any sector having interior in Int X. Suppose, however, that 0 is a cusp for a curve whose interior is in Int X.Let L be a llne segment which bisects the cusp at 0 and let C denote the interior of the cusp near 0. Then if y L C and z e X\C, ly-zlT(lyl) >-lyl where r is a monotone decreasing function such that lim+v(r) =.Depending on how rapidly r approaches at 0 (or how rapidly the cusp "narrows"), we can show that functions in R2(X) satisfy an inequality similar to that of Theorem 5.1.
The next theorem is proved in[6, p.82].we define the Bessel capacity which Fernstrm and Polking use to describe BPE's for R2(X). Now