Estimates of analytic capacity

5.7.

ESTIMATES OF ANALYTIC

CAPACITY* o

Let u denote the Lebesgue measure on r over the interval ~ , I ) ~

We always

generic notation

subsets of

tion f on ~

for compact

~m~

. In what follows we let z range over

suppose that p > I and % > 0.

~

and

E will be used as a

~ . For any locally integrable

(complex-valued)

func-

we denote by

its mean value over the disc ~(D,%)={~:ge~,l~-%l~%}. ~P'~ will stand for the class of all functions f on ~ that are locally integrable in power p and satisfy

(see [43]) for references on related function spaces). Investigation of removable singularities for holomorphic functions in these classes gives rise naturally to the corresponding capacities Tp,% defined as follows (compare [44]). If Ap,%(E, I) denotes the class of all ~P'~

that are holomorphic

in the complement of E (including ~) and satisfy the conditions

~(oo) :0,

n)b.~..~ i ,

th en

where ~'~oo): ~

~(~).

It is an important feature of these capacities that they admit simple metrical estimates which reduce to those of Melnikov (cf. Chap. V in [35]) for a special choice of the parameters when they yield Dol~enko's result on removable singularities in HSlder classes. Writing d i a m M for the diameter of M, M

CC,

and defining for ~, ~ ~> O, and ~, e > O,

where the infimum is taken over all sequences of sets Mn, Mn = ~ Ec~Mm,

we may state the following

THEOREM I. Let ~ p + ~ p + ~ stants c and k such that

inequalities ,~=m~(~,~)

, with d i a m M n < E such that

(cf. [45]). and define

~=~+~

Then there are con-

for all E. Problem

I.

What are the best values of the constants

c, k occurring

in (I)?

Theorem I is of special interest in the case ~ = 2, because it characterizes removable singularities of holomorphic functions of bounded mean oscillation (cf. [46]) as those sets E whose linear measure

*JOSEF KRAL.

Ceskoslovenska

Akademie Ved Matematicky

Ustav, Praha

I, ~itnaz ulice 25, CSSR.

2241

vanishes. This is in agreement with the example of Vitushkin (cf.[10]). The capacity of E corresponding to the broader class of functions of bounded mean oscillation may be positive even though E has zero analytic capacity corresponding to bounded functions which is defined by

CE) =

eACE,4 ,

where now A(E, I) is the class of all functions holomorphic off E and vanishing at ~ whose absolute value never exceeds I. Nevertheless, by the so-called Denjoy conjecture (which follows from combination of results in [20, 19, 23, 22]) the equivalence

=0

=0

is true for E situated on a rectifiable curve. The upper estimate of X(-) by means of I(.) is generally valid (cf. [34]) while the lower estimate of y(E) by means of a multiple of I(E) is possible only for E situated on sets Q of a special shape. Problem 2. Let ~ c ~ be a compact set. existence of a constant c such that

Find geometric conditions on Q guaranteeing the

(E) clCE), EcG.

(2)

The following theorems may serve as sample results. THEOREM 2 (cf. [28]). ferentiable, I~'] = I and

Let

Q=~(), where

|

#

I

~:-~

is simple and continuously dif-

I

IU?(t)-~(~)l @t: s 0 1 , then (2) holds with C=t2~)-~(~V(Q)§ -4 . If Q is a straight-line segment then V(Q) = 0 and Pommerenke's equality ~ ( D = ~ ( E ) (cf. [47]). This leads to the following. Problem 3.

Is it possible to improve c in Theorem 3 to 88

holds

~ ?

Remark. It was asserted in [45] that Theorem 3 holds with this value of the constant. Dr. J. Matyska kindly pointed out that there was a numerical error in the original draft of the corresponding proof in [48]. LITERATURE CITED i.

2. 3. 4. 5. 6. 7. 8.

2242

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9.

10. 11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

36. 37. 38. 39. 40.

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t

2243

41. 42. 43. 44. 45. 46. 47. 48.

2244

E. P. Dolzhenko, "On the 'removability' of the singularities of analytic functions," Usp. Mat. Nauk, 18, No. 4, 135-142 (1963). C. A. Rogers, Hausdorff Measures, Cambridge Univ. Press, Cambridge (1970). J. Peetre, "On the theory of ~ p~-spaces,'' J. Funct. Anal., 4, 71-87 (1969). R. Harvey and J. Polking, "A notion of capacity which characterizes removable singularities," Trans. Am. Math. Soc., 169, 183,195 (1972). J. Kr~l, "Analytic capacity," in: Proc. Conf. "Elliptische Differentialgleichungen," Rostock 1977 (to appear). F. John and L. Nirenberg, "On functions of bounded mean oscillations," Comm. Pure Appl. Math., 14, 415-426 (1961). Ch. Pommerenke, "Uber die analytische Kapazitgt," Arch. Math., 1!, 270-277 (1960). J. Fuka and J. Kr~l, "Analytic capacity and linear measure," Czech. Math. J. (to appear).