On 𝐿¹-norms of meromorphic functions with fixed poles

Baranov, A.

doi:10.1090/S0002-9939-06-08349-3

On $L^1$-norms of meromorphic functions with fixed poles
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Proc. Amer. Math. Soc. 134 (2006), 3003-3013 Request permission

Abstract:

We study boundedness of the differentiation and embedding operators in the shift-coinvariant subspaces ${K_B^1}$ generated by Blaschke products with sparse zeros, that is, in the spaces of meromorphic functions with fixed poles in the lower half-plane endowed with $L^1$-norm. We answer negatively the question of K.M. Dyakonov about the necessity of the condition $B’\in L^\infty (\mathbb {R})$ for the boundedness of the differentiation on ${K_B^1}$. Our main tool is a construction of an unconditional basis of rational fractions in ${K_B^1}$.

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