MathSciDoc: An Archive for Mathematician ∫

Let$H$^{∞}be the algebra of bounded analytic functions in the unit disk$D$. Let$I=I(f$_{1},...,$f$_{N}) be the ideal generated by$f$_{1},...,$f$_{N}∈$H$^{∞}and$J=J(f$_{1},...,$f$_{N}) the ideal of the functions$f∈H$^{∞}for which there exists a constant$C=C(f)$such that |$f(z)|≤C(|f$_{1}$(z)|+$...;$+|f$_{N}$(z)$|),$z$∈$D$. It is clear that $$I \subseteq J$$ , but an example due to J. Bourgain shows that$J$is not, in general, in the norm closure of$I$. Our first result asserts that$J$is included in the norm closure of$I$if$I$contains a Carleson-Newman Blaschke product, or equivalently, if there exists$s$>0 such that $$\mathop {\inf }\limits_{z \in D} \sum\limits_{k = 0}^s {(1 - |z|)^k } \sum\limits_{j = 1}^N {|f_j^{(k)} (z)| > 0.} $$

No keywords uploaded!