A note on 𝐿^{𝑝}-bounded point evaluations for polynomials

Abstract

We construct a compact nowhere dense subset K K of the closed unit disk D ¯ \bar {\mathbb D} in the complex plane C \mathbb C such that R ( K ) = C ( K ) R(K) = C(K) and bounded point evaluations for P t ( d A | K ) , 1 ≤ t > ∞ , P^t(dA | _K), ~ 1 \le t > \infty , is the open unit disk D . \mathbb D. In fact, there exists C = C ( t ) > 0 C=C(t) > 0 such that \[ ∫ D | p | t d A ≤ C ∫ K | p | t d A , \ \int _{\mathbb D} |p|^t dA \le C \int _K |p|^t dA, \] for 1 ≤ t > ∞ 1 \le t > \infty and all polynomials p . p.