Approximation by rational modules in Lp and BMO

where each ri(z)~9?(X). Let a= ;(8/2x+ i2/+) be the usual CauchyRiemann operator in the complex plane and let d* = 8 c ii. We note that a function satisfies a*f = 0 in an open set U if and only if f= h + Fk with h and k analytic. A major problem in approximation theory is describing the closure of the rational module 92(X) ~?i in various norms (see [3]). It has been shown in [S] that W(X) gi is dense in C(X), the space of all continuous functions on X, for any compact X when the interior k of X is empty. However, the presence of an interior really complicates the situation. An outstanding open problem is whether S?(X) P, is dense in the space { f~ C(X): J*f= 0 in k} for all compact sets X. For 1 6 p 6 00, let L,P(X) be the closed subspace of Lp(X) which consists of functions analytic in 2. If I/ is any space of functions on X, we denote by [ VJP the closure of V with respect to the LP(X) norm, and we write p = {f: f~ V} where p is the usual Cauchy transform of ,fi In (6) the author proved that [9(.X’) g,], = [W(X) + L:(X)” 1, for all 1 d p < K. using the theory of singular integrals and Schwartz lemma. In this note, we use a somewhat constructuve method to prove the same result for the Lp norms and then extend it to the BMO norm case. To each compact subset X of @ (E lR*) one associates the restrictive spaces BMO(X) = BMO(C)lX z BMO(C))lI(X) and VMO(X) = VMO(@)~,EVMO(C)/J(X), where Z(X)= {~EBMO(C): f’=O a.e. on Xi and J(X)= {~EVMO(@): f=O a.e. on X}. Let [9(X) P,],,, and [g(x) + L,” cm A 1 BMCI be the closure in BMO(X) of 9(X) 9, and a(X) + L,:(X) h, respectively. Holden’s extension theorem [ 1 ] implies that