Restriction spaces of AA^\infty

  • Dietmar Vogt

    Bergische Universität Wuppertal, Germany


In the present paper it is shown that for certain totally disconnected Carleson sets EE the restriction space A(E)={fE:fA}A_\infty(E)=\{f|_E : f\in A^\infty\} has a basis. Its isomorphism type is determined. The result disproves a claim of S. R. Patel in [12]. To prove our result we analyze restriction spaces C(E)={fE:fC(R)}C_\infty(E)=\{f|_E : f\in C^\infty(\mathbb{R})\} and then, using a result of Alexander, Taylor and Williams, we show that A(E)=C(E)A_\infty(E)=C_\infty(E). Among our examples there are the classical Cantor set and sets of type E={xn:nN}{0}E=\{x_n : n\in\mathbb{N}\}\cup\{0\}, where (xn)nN(x_n)_{n\in\mathbb{N}} is a null sequence in R\mathbb{R} with certain properties.

Cite this article

Dietmar Vogt, Restriction spaces of AA^\infty. Rev. Mat. Iberoam. 30 (2014), no. 1, pp. 65–78

DOI 10.4171/RMI/769