Painleve null sets

2.7.

PAINLEVE NULL SETS*

Suppose that E is a compact plane is called a Paenleve null set (or P.N. can be analytically continued onto E. 9

J

set and that N is an open neighborhood of E. A set set) if every function regular and bounded in N \ E In this case we also say that E has zero analytic

capacity. The p r o b l e m of the structure of P.N. sets has a long history 9 Painlev6 proved that if E has linear (i.e., l-dimensional Hausdorff) measure zero, then E is a P.N. set, though it seems that this result was first published by Zoretti [27], Painlev~'s theorem has been rediscovered by various people including Besicovitch [9] who proved that if f is continuous on E, as well as regular outside E, and if E has finite linear measure, then f can be analytically continued onto E. Denjoy [16] conjectured that if E lies on a rectifiable curve, then E is a P.N. set if and only if E has linear measure zero. He proved this result for linear sets. Ahlfors and Beurling [3] proved Denjoy's conjecture for sets on analytic curves and Ivanov [28] for sets on sufficiently smooth curves. Davie [19] has shown that it is sufficient to prove Denjoy's conjecture for C l curves. On the other hand Havin and Havinson [23] and Havin [22] showed that Denjoy's conjecture follows if the Cauchy integral operator is bounded on L 2 for C l curves. This latter result has now been proved by Calder~n [20] so that Denjoy's conjecture is true. I am grateful to D. E. Marshall [29] for informing me~ about the above results. Besicovitch [24] proved that every compact set E of finite linear measure is the union of two subsets El, E2. The subset El lies on the union of a finite or countable number of rectifiable Jordan arcs. It follows from the above result that E~ is not a P.N. set unless El has linear measure zero. The set E2 on the other hand meets every rectifiable curve in a set of measure zero, has projection zero is almost directions and has a linear density at almost none of its points. The sets El and E2 were called, respectively; regular and irregular by Besicovitch [24]. Since irregular sets behave in some respects like sets of measure zero, I have tentatively conjectured [30, p. 231] that they might be P.N. sets. Vitushkin [10] and Garnett [11] have given examples of irregular sets which are indeed P.N. sets, but the complete conjecture is still open. A more comprehensive conjecture is due to Vitushkin [6, p. 147]. He conjectures E is a P.N. set if and only if E has zero projection is almost all directions. It is not difficult to see that a compact set E is a P.N. bounded complex measure distributed on E, the function

that

set if and only if for every

(1)

Fc ) = I

J

E

is unbounded outside E.~ measure ~ on E such that

Thus

g is

certainly

not

a P.N.

set

if

there

exists

a positive

unit

;~-gl E 9 is bounded outside E, i.e., if E has positive linear capacity [2, p. 73]. This is certainly the case if E has positive measure with respect to some Hausdorff function h, such that

O

([31])o Thus in particular E is not a P.N. set if E has Hausdorff dimension greater than one. While a full geometrical characterization of P.N. sets is likely to be difficult there *W. K. HA~IAN. Imperial College, London SW7, England. ~See the editors' note at the end of the section.

2235

still seems plenty of scope for further work on this intriguing class of sets. Editors' Note. As far as we know the representability of ~l~ functions bounded and analytic off E and vanishing at infinity by "Cauchy potentials" (I) is guaranteed when E has finite Painleve's length, whereas examples show that this is no longer true for an arbitrary E ([32, 33]). We think the question of existence of potentials (I) bounded in ~ \ E (provided E is not a P.N. set) is one more interesting problem (see also Sec. 5 of [17]).

2236